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Beginning in the early 1930's and extending through the early
1960's, a number of researchers have proposed various schemes and hypotheses
to explain population dynamics and commonly observed population interactions
in the field. Leading authorities have been Smith (1935), Nicholson (1933),
Nicholson & Bailey (1935), Solomon (1949), MIlne (1957a,b, 1958),
Thompson (1929a,b, 1930a,b, 1939, 1956), Andrewartha & Birch (1954), Lack
(1954), and Holling (1959), Watt (1959), Chitty (1960), Pimentel (1961), to
mention some of the more vociferous authorities. Toward the end of this
period considerable conflict of opinion developed with the introduction of
ideas by Turnbull (1967), Turnbull & Chant (1961), van den Bosch (1968),
Force (1972), Huffaker (1958), Huffaker et al. (1963, 1971). Presently the
debate continues with publications by Ehler (1976), Ehler & Hall (1982),
Hokkanen (1985), Hokkanen & Pimentel (1984), Goeden (1976), Myers &
Sabbath (1980). The matter of population interaction is too complex for simple
discussion; the subject must be treated in a mathematical manner, separating
the various kinds of population systems (eg., single host single natural
enemy, single host multiple natural enemies, multiple hosts multiple natural
enemies, patchy distributions versus relatively uniform ones, etc., etc.) In
general the modern theories (since the 1960's), which encompass many
different types of interactions between species, accord well with the
observed outcomes of experimental populations in both laboratory and natural
settings. This is especially true for simpler systems (e.g., laboratory
studies, studies of single species), partly because there is more data
available on such systems. In each of the many systems that have been studied
during recent times, from the simple single-species systems to the
multispecies systems, the theories indicate a complex range of dynamics that
can arise from simple regulatory mechanisms. This range generally includes
stable equilibria, stable cyclic behavior and chaotic dynamics. The presence
of these behaviors is standard to systems, which include time-lags or
time-delays, such as the developmental time between oviposition and adult
emergence in insects or the time between infection and subsequent
infectiousness of diseased individuals. The actual dynamics of any particular system
depends on the strength of the interactions among the member species. Thus
increased intensity of competition leads from stable equilibrium to cyclic
behavior in single-species and competing-species systems. Increasing the
effects of delayed density dependence in host-parasitoid systems (by
decreasing the contagion in attacks among hosts) leads from stable
equilibrium (when contagion is significant) to unstable cyclic behavior when
search is more independently random. Inherent overcompensation in some
host-pathogen models leads directly to cyclic behavior without any
intervening sphere of stable equilibria (Bellows & Hassell 1999). For many of the systems theory has been
developed for both homogeneous and patchy environments. In general, patchy
environments permit greater degrees of stability for most interactions, even
permitting global persistence of interactions that are intrinsically unstable
in homogeneous environments. A thorough updated review on the subject of
population regulation was presented by Bellows & Hassell (1999), which
shows the complexity of considerations necessary and offers a clearer
explanation for some of the possible population interactions. These authors
emphasized that natural regulation of populations necessarily involves
interactions among species. By understanding the potential and likely
outcomes of these interactions and the relationships between particular
mechanisms and their consequences, we can better interpret the outcomes of
biological control experiences and better direct future efforts toward
achieving goals of population suppression and regulation. Issues of natural population regulation lie
at the core of biological control. Characteristic of "successful"
biological control are the reduction of pest populations and their
maintenance about some low, non-pest level. Such outcomes are frequently
recorded as being achieved (e.g., DeBach 1964), but documented evidence is
less common (Beddington et al. 1978). The reduction in density of the winter
moth, Operophtera brumata Cockerell in Nova
Scotia following introduction of natural enemies is one such example, while in
the laboratory similar outcomes have been reported. The objective of
biological control programs is to enhance such natural control of
populations, and an understanding of the principals involved in biological
control necessitate an appreciation of mechanisms of population regulation. Biological control has as a principle aim
the reduction of pest species.
In this context the objectives are two-fold, first to reduce or suppress the density of the species and secondly to regulate the pest species around this
new lower level. Thus there are two concepts, suppression and regulation,
which encompass the objectives of biological control. While mechanisms of
population suppression are in many cases as simple as increasing the level of
mortality acting on a population, issues of regulation, or what will be the
dynamical behavior of the population once the new mortality factor has been
added, are more complex and can be affected by density-dependent responses of
both the pest and natural enemy population, natural enemy search behavior,
patchiness of the environment, additional natural enemies in the system, and
other interactions, both behavioral and stochastic, among the populations.
(Please see Legner et al. 1970, 1992, 1973, 1983, 1983, 1975, 1980). These questions of population suppression
and regulation have been the subject of a considerable amount of research,
both theoretical and experimental. It is then well to consider features of
interacting population which can contribute to either suppression or
regulation (or both). The discussion begins with single species systems and
interspecific competition, proceeds to interactions between a host or prey
and a natural enemy, and concludes with considerations of systems with more
than two species (of either prey or natural enemy). (Bellows & Hassell
1999). The topics are developed generally within an analytical framework of
difference equations but, where these are significantly distinct, also
consider the implications of continuous-time systems. The implications of
heterogeneous environments are also addressed, where resources such as food
plants or prey are distributed in patches (rather than homogeneously) over
space. In general theories and mechanisms are considered which are supported
by experimental evidence as having some effect on the dynamical behavior of
populations. Although there is an abundance of information on the effects of
herbivory on the performance of plants, there is little data on the effects
of insect herbivory on plant population dynamics (Crawley 1989). For this
reason most of the discussion on hosts and natural enemies is centered on
interactions of populations of insect predators and parasitoids and their
prey, interactions for which there exists a large body of literature on experimental
investigations (Bellows & Hassell 1999, Hassell 1978). Single age-class
systems Single-species population dynamics has
relished a long history of both theoretical and empirical development,
centering largely around mechanisms of population growth and regulation. The
structure in which the concepts are developed is one of population growth in
discrete time, where the population consists largely of individuals of only a
single generation at any one time. Such populations are characteristic of
many temperate insects and additionally of many tropical insects which occupy
regions with pronounced wet and dry seasons. The algebraic framework is
straightforward: Nt+1 = Fg(Nt)Nt. (1) Here N is the host population denoted by
generations t and t=1, and Fg(Nt) is the per capita net rate of increase of the population
dependent on the per capita fertility F
and the relation between density and survival g (which is density dependent for g<1). The fundamental concept represented in
equation (1) regarding population regulation is that some resource, crucial
to population reproduction, occurs at a finite and limiting level (when g=1,
there is no resource limitation and the population grows without limit).
Individuals in the population compete for the limiting resource and, once the
population density has saturated or fully utilized it, the consequences of
this intraspecific competition bring about density dependent mortality and
growth rates reduced from the maximum population potential. Such competition
can be by adults for oviposition sites (e.g., Utida 194_, Bellows 1982a), by
larvae for food (e.g., Park 193_, Bellows 1981), or by adults for food (e.g.,
Park 193_, Nicholson 1954). The dynamics of populations subject single
species competition in discrete generations can span the range of behaviors
from geometric (or unconstrained) growth (when competition does not occur),
monotonically damped growth to a stable equilibrium, damped oscillations
approaching a stable equilibrium, through cyclic behavior. The type of
behavior experienced by any particular population is partly dependent on the
mechanisms and outcome of the competitive process. Species with contest
competition have more stable dynamical behavior, while species with scramble
competition may show more cyclic or oscillatory behavior (May 1975, Hassell
1975). Most insect populations appear to experience monotonic damping to a
stable equilibrium (Hassell et al. 1976, Bellows 1981). The exact form of the function used to
describe g is not particularly
critical to these general conclusions and many forms have been proposed
(Bellows 1981), although different forms may have specific attributes more
applicable to certain cases. Perhaps the most flexible is that proposed by
Maynard Smith & Slatkin (1973), where g(N) takes the form g(N) = [1+(N/a)b]-1. (2) where the relationship between proportionate survival and
density is defined by the two parameters a, the density at which density-dependent survival is 0.5, and b, which determines the severity of
the competition. As b
approaches 0, competition becomes less severe until it no longer occurs 9b=0), when b=1 density dependence results in contest competition with the
number of survivors reaching a plateau as density increases, and for b>1 scramble competition occurs,
with the number of survivors declining as the density exceeds N-a. Multiple age-class
systems.--Most
populations are separable into distinct age or stage classes, and this is
particularly important in competitive systems. In most insects the
preimaginal stages must compete for resources for growth and survival, while
adults must additionally compete for resources for egg maturation and
oviposition sites. In such cases, competition within populations divides
naturally into sequential stages. Equation (1) may be extended to the case of
two age classes (May et al. 1978) and, where competition occurs primarily
within stages (e.g., larvae compete with larvae and adults with adults), At+1 = g1(Lt)Lt (3a) Lt+1 = Fga(At)At (3b) where A and L denote the adult and larval
populations. In such multiple age-class systems, the dynamical behavior of
the population is dominated by the outcome of competition in the stage in
which it is most compensatory. Hence in a population where adults exhibit
contest competition for oviposition sites while larvae exhibit scramble
competition for food, the population will show monotonic damping to a stable
equilibrium, characteristic of a population with contest competition. This
result is extendable to n age
classes, so that any population in which competition in at least one stage is
stabilizing or compensatory (i.e., contest), the dynamics of the population
will be characterized by this stabilizing effect (Bellows & Hassell
1999). A review of insect populations showing
density dependence in natural and laboratory settings indicates that most
such populations exhibit monotonic damping towards a stable equilibrium
(Hassell et al. 1976, Bellows 1981). This does not preclude the possibility
of scramble competition in insect populations (e.g., Nicholson 1954, Goeden
1984), but does imply that compensatory competition exists in at least one
stage in most studied populations. More complex approaches to constructing
models of single-species insect populations can be taken which involve many
age-classes and great detail in description of biological processes. Many of
these have been designed to consider only the problem of describing
development of the population from one stage to another and do not bear
directly on mechanisms of natural population regulation. Others consider
internal processes which may limit population growth (e.g., Lewis 19__,
Leslie & Gower 1958, Bellows 1982a,b) and consequently do touch on
population regulation. In one comparative study, Bellows (1982a,b) found
little difference in dynamical behavior between simple one and two age-class
models and more complex systems models with several age classes. hence at
least for single-species population models, the distinction between two and
more age classes in the analytical framework may be of little consequence.
This may not be the case for systems with more than one species (Bellows
& Hassell 1999). The preceding unfolding is particularly
applicable to homogeneous environments and uniformly distributed resources.
For many insect populations, however, resources are not distributed either
continuously or uniformly over the environment but rather occur in
disjunctive units or patches. For such cases equations (1) through (3)
generally will not apply, for the distinction between homogeneous and patchy
environments has significant consequences for population dynamics.
Populations competing for resources in patchy environments may be expected to
show the same range of qualitative behaviors-- stable points approached
either monotonically or by damped oscillations, periodic cyclic behavior and
disarray, but the formulations representing them shed new light on the
importance of dispersal, dispersion and competition within patches. Considering an environment divided into j discrete patches (e.g., leaves on
trees) which are utilized by an insect species, adults (N) disperse among the patches and
distribute their compliment of progeny within a patch. Progeny deposited in a
patch remain in the patch and compete for resources only within the patch and
only with other individuals within the patch. The population dynamics is now
dependent partly on the distribution of adults reproducing in patches OE and partly on the density
dependent relationship that characterizes preimaginal competition. Population
reproduction over the entire environment (i.e., all patches) can be
characterized by the relationship by deJong (1979): Nt+1 = jFZOE(nt)ntg[Fnt] (4) (Z = summation sign) where n is the
number of adults in a particular patch and OE(n) is the proportion of patches colonized by n adults. DeJong (1979) considered four distinct
dispersion distributions of individual adults locating patches. In the case
of uniform dispersion, equation (4) is equivalent to equation (1) for
homogeneous environments. For three random cases, positive binomial,
independent (Poisson), and negative binomial, the outcome depends somewhat on
the form taken for the function g.
For most reasonable forms of g,
the general outcomes of dividing the environment into a number of discrete
patches are a lower equilibrium population level and enhanced numerical
stability in comparison to equation (1) with the same parameters for F and the function g. Two additional features arise:
(1) there is an optimal fecundity for maximum population density and (2) for
a fixed amount of resource, population stability increases as patch size
decreases and the number of patches increases (the more finely divided the
resource the more stable the interaction) to an optimal minimum patch size.
The addition of more patches of resource (increasing the total amount of
resource available but holding patch size constant) does not affect stability
per se but increases the equilibrium population level (Bellows & Hassell
1999). In the same way that competition for
resources among individuals of the same species can lead to r1estrictions on
population growth, competition among individuals of different species can
similarly cause density dependent constraints on growth. Although Strong et
al. (1978) suggested that competition is not commonly a dominant force in
shaping many herbivorous insect communities, it certainly is an important
potential factor in insect communities, especially those which feed on
ephemeral resources (e.g., Drosophila
spp.) and additionally in insect parasitoid communities (e.g., Luck &
Podoler 1985). The processes and outcomes of interspecific competition in
insects have been studied widely in the laboratory (e.g., Crombie 1945, Fujii
1968, Bellows & Hassell 1984) as well as in the field (Atkinson &
Shorrocks 1977). Homogeneous
Environments Single age-class systems.--Many of the same mechanisms implicated in intraspecific
competition for resources (e.g., competition for food, oviposition sites,
etc.) also occur between species (e.g., Crombie 1945, Leslie 194_, Park 1948,
Fujii 1968, 1970). The dynamics of these interspecific systems can be
considered in a framework very similar to that for single species
populations. Equation (1) can be extended to the case for
two (or more) species by considering the function g to depend on the density of both competing species (Hassell
& Comins 1976), so that the reproduction of species X depends not only on the density of
species X but also on the
density of species Y (and
similarly for species Y): Xt+1 = Fgx(Xt+alpha
Yt)Xt (5a) Yt+1 = Fgy(Yt+Beta
Xt)Yt (5b) Here the parameters alpha and Beta
reflect the severity of interspecific competition with respect to
intraspecific competition. Population interactions characterized by equation
(5) may have one of four possibilities: the two species may coexist, species
X may always exclude species Y, species Y may always exclude species X, or
either species may exclude the other depending on their relative abundance.
Coexistence is only possible when the product of the interspecific
competition parameters alpha
Beta<1 (when alpha
Beta>1 one of the species is driven to extinction). For coexisting
populations, the dynamical character of the populations is determined by the
severity of the intraspecific competition and may take the form of stable
equilibria approached monotonically, stable cyclic behavior, or chaos
(Hassell & Comins 1976). It is conventional to summarize the
character of the interspecific interaction by plotting isoclines which define
zero population growth in the space delimited by the densities of the two
populations. In these simple, single age-class models with linear
interspecific competition, these isoclines are linear. When they have an
intersection, the system has an equilibrium (stable for alpha Beta<1); when they do not
intersect the species with the isocline farthest from the origin will
eventually exclude the other (e.g., Crombie 1945). The biological
interpretation applicable to this analysis is that each species must inhibit
its own growth (through intraspecific competition) more than it inhibits the
growth of its competitor (through interspecific competition) for a persistent
coexistence to occur. Multiple age-class systems.--Many insect populations compete in both preimaginal and
adult stages, perhaps by competing as adults for oviposition sites and
subsequently as larvae for food (e.g., Fujii 1968) and in some cases the
superior adult competitor may be inferior in larval competition (e.g., Fujii
1970). The analytical properties of such multiple age-class systems may be
considered by treating separately the dynamics of the adult and preimaginal
stages (Hassell & Comins 1976): Xt+1 = xtgxl(xt+alpha
1yt) (6a) Yt+1 = ytgxl(yt+Beta
1yt) (6b) xt+1 = XtFxgx
alpha(Xt+alpha alpha Yt) (6c) yt+1 = YtFygy
alpha(Yt+beta alpha Xt) (6d) where x and y are the preimaginal or larval
stages and X and Y are the adults. Here larval
survival of each species is dependent on the larval density of both species, and
adult reproduction of each species is dependent on the adult densities of
both species. Larval competition is characterized by the larval competition
parameters alphal
and Betal, while
adult competition is characterized by alphaa
and Betaa. The simple addition of competition in more
than one age has important effects on the dynamical behavior of the
competitive system. The isoclines of zero population growth are now no longer
linear, but curvilinear, and multiple points of equilibrium population densities
are now possible. It is even possible to have more than one pair of stable
equilibrium densities (Hassell & Comins 1976). Such curvilinear isoclines
are in accord with those found for competing populations of Drosophila spp. (Ayala et al. 1973). More complex systems can be visualized with
additional age classes and with competition between age classes (e.g.,
Bellows & Hassell 1984). The general conclusions from studies of these
more complex systems are similar to those for the two age-class systems, vis.
that more enigmatic systems have non-linear isoclines and consequently may
have more complicated dynamical properties. More subtle interactions may also
affect the competitive outcome, such as differences in developmental time
between two competitors. In the case of Callosobruchus
chinensis and Callosobruchus maculatus, the intrinsically
superior competitor (C. maculatus) can be outcompeted
by C. chinensis because the latter develops faster and thereby
gains earlier access to resources in succeeding generations. This earlier
access confers sufficient competitive advantage on C. chinensis
that it eventually excludes C.
maculatus from mixed species
systems (Bellows & Hassell 1984). Patchy
Environments Many insect populations are dependent on resources
which occur in patches (e.g., fruit, fungi, dung, flowers, dead wood).
Dividing the resources for which populations compete into discrete patches
can have significant effects on the consequences of interspecific
competition. Two general views of competition in a patchy
environment have been proposed. In the first coexistence is promoted by a
balance between competitive ability and colonizing ability (Skellem 1951,
Cohen 1970, Levins & Culver 1971, Horn & MacArthur 1972, Slatkin
1974, Armstrong 1976). An alternative view proposed by Levin (1974) is that
competition in a patchy environment may result in a persistent coexistence if
both species inhibit their own growth less than their competitors, so that in
any patch the numerically dominant species would exclude the competitor; each
species would have a refuge in those patches where it is numerically
dominant. A idea has been proposed by Shorrocks et al.
(1979) and Atkinson & Shorrocks (1981), where each patch is temporary in
nature but is regularly renewed. Such resources may be typical for many
invertebrates (Shorrocks et al. 1979). In this case the competitively
inferior species is not constantly driven out of patches because the patches
are ephemeral in nature. Because of this, coexistence can occur when
competition between the species can be more severe than in the homogeneous
case because its frequency of occurrence is reduced by the fraction of
patches which contain only one species. This view emphasizes the importance of
aggregated spatial dispersion among patches in the populations of the
competing species. Atkinson & Shorrocks (1981) investigated the
consequences of this by using the negative binomial distribution of
individuals among patches in a two-species competitive model. The conclusions
of this work were primarily that coexistence of competitors on a divided
resource is possible under many more scenarios than in the homogeneous case.
Specifically, coexistence is promoted by dividing a resource into more and
smaller breeding sites, by aggregation of the superior competitor, and
especially by allowing the degree of aggregation to vary with density. Equation (1) may be extended for single
species populations in a homogeneous environment to include the additional
effect of mortality caused by a natural enemy. The particular details of the
algebra espoused would depend to some extent on what biological situation it
is desired to express. Following previous work (Nicholson & Bailey 1935,
Hassell & May 1973, Beddington et al. 1978, May et al. 1981), the insect
protolean parasites or parasitoids are considered. Such systems have
attracted much attention for both theoretical and experimental studies
(Hassell 1978). Pursuing the discrete framework of the preceding sections,
the dynamics of these interactions may be summarized by: Nt+1 = Fg(fNt)Ntf(Nt,Pt) (7a) Pt+1 = cNt{1-f(Nt,Pt)} (7b) Here N
and P are the host and
parasitoid populations; Fg(fNt)
is the per capita net rate of increase of the host population, intraspecific
competition is defined as before by the function g with density dependence for g<1; the function f
defines the proportion of hosts which are not attacked and embodies the
functional and numerical responses of the parasitoid, and c is the average number of adult
female parasitoids which emerge from each attacked host. In such analytical
frameworks, different dynamics can result depending on the sequence of
mortalities and reproduction in the hosts life cycle (Wang & Gutierrez
1980, May et al. 1981, Hassell & May 1986). Equation (7) reflects the
case for parasitism acting first followed by density dependent competition as
defined by g (May et al. 1981
give a discussion of alternatives). This model then represents an
age-structured host population in which density dependence (if any) occurs in
a distinct post-parasitism stage in the life cycle. This design has a long heritage, and has
been utilized with many versions of the functions f and g. Bellows
& Hassell (1999) stated that early workers incorporated no density
dependence in the host (g=1)
and functions for f which
implied independent random search by
individual parasitoids (e.g., Thompson 1924, Nicholson 1933, Nicholson
& Bailey 1935). A simple reference to Nicholson (1933) and Nicholson
& Bailey (1935) will reveal how emphatic these authors were to
distinguish nonrandom searching by
individuals from random
searching by populations (see section on "Searching". Thus
it is difficult to understand the current statements by Bellows &
Hassell, although they have been made before by Varley et al. (1973) and
Milne (1957a,b). In the cases referred to by Bellows & Hassell (1999),
the model design becomes somewhat simpler: Nt+1 = FNtexp(-aPt) (8a) Pt+1 = Nt{1-exp(-aPt)} (8b) where f(N,P) is
represented by the zero term of the Poisson distribution in keeping with the
assumptions of independent search by parasitoid adults. The parameter a is the area of discovery an adult
parasitoid, characterizing the species searching ability. This model
incorporates a somewhat mechanized search behavior for the parasitoid, with
search for hosts being continuous and successful subduing of hosts
instantaneous upon discovery, with no such limits on search as physiological
resources or egg depletion. These works laid useful groundwork but, as
they reflected an interest in biological control and population regulation,
proved inadequate because such simple systems did not include regulatory
population dynamics; quite simply, there is no direct density dependence in
equation (8) and thus no stabilizing feature in the model. In contrast, these
simple systems suggest a destabilizing effect of parasitism on the host
population (delayed density dependence), with such matched populations
exhibiting oscillations of ever-increasing magnitude until extinction
occurred. Experiments conducted in the laboratory (under artificial
conditions) were applied to examine the suitability of such models and
affirmed that such simple systems were characterized by unstable oscillations
(Burnett 1954, DeBach & Smith 1941). Subsequent work considered the dynamics of
more complex forms of equation (8) which have attempted to capture additional
behavioral features of predation and parasitism. Holling (1959a,b, 1965,
1966) introduced the idea of characterizing the act of parasitism or
predation by component behaviors, such as the separate behaviors of attack
and subsequent handling of prey. This view permitted different types of
functional responses to be characterized by different component behaviors
(Holling 1966). Parasitism and predation in insects are largely typified by
Type II functional responses, viewed by Holling as characterized by two
parameters, the per capita search efficiency a and the time taken to handle a prey Th. These were incorporated into the structure of
equation (8) by Rogers (1972), who added the limitations of handling time to
independently searching parasitoids. Equation (8) becomes Nt+1 = FNtf(exp(-aPt/(1+aThNt))) (9a) Pt+1 = Nt{1-f(exp(-aPt/(1+aThNt))} (9b) The result of this addition to the earlier
design was increased biological realism, but decreased population or system
stability. The addition of handling time increased the destabilizing effect of
parasitism without contributing any stabilizing density dependence (Hassell
& May 1973). Truly the principles involved in type II functions responses
(as in equation (9b) are inversely density dependent and thereby
destabilizing, contributing to the instability caused by the delayed density
dependence. In more realistic situations, the outcome of
search by a parasitoid population may not be typified by independent random
search. Many processes (spatial, temporal and genetic) will combine to render
some prey individuals more susceptible to predation than others. This unequal
susceptibility between individuals will result in non-independence of
attacks. One approach to capturing this non-independence is to employ the
negative binomial distribution to characterize the distribution of attacks,
so that the function f becomes
f(N,P) = [
aP ]-k [1
+_______] [ k(1+aThN)] (10) and
the simplest case with no host density dependence becomes Nt+1 = FNt[{1+aPt/(k(1+aThNt))}-k] (11a) Pt+1 = cNt{1-{1+aPt/(k(1+aThNt))}-k} (11b) Here once again a is the per capita search efficiency of parasitoid adults and Th is their handling
time. The differential susceptibility of prey host individuals to attack is
characterized by contagion in the distribution of attacks among individuals,
representing the outcome that more susceptible individuals are more likely to
be attacked. This contagion is depicted by the parameter k of the negative binomial.
Contagion increases as k->0,
where as in the opposite limit of k->
attacks become distributed independently and the Poisson distribution is
recovered (equation 8). As May and Hassell (1988) have discussed, the outcome
of a parasitoid's searching behavior cannot usually be fully characterized so
simply as equation (10) (Hassell & May 1974, Chesson & Murdoch 1986,
Perry & Taylor 1986, Kareiva & Odell 1987). Nonetheless, the use of
equation (10) with a constant k
permits the dynamical effects of non-random or aggregated parasitoid
searching behavior to be examined without introducing a large list of
behavioral parameters. More complex cases, such as the value of k varying with host density, can be
considered (Hassell 1980), but have little effect on the dynamical aspects of
the host-parasitoid interaction. The simple change from independently random
search foreseen by early workers (equation (8)) to the more general case of
equation (11) can have profound effects on the dynamics of such systems.
Although equation (11) still contains the destabilizing affect of delayed
density dependence inherent in such difference-equation systems, the system
can not be stable when k takes
values between 0 and 1, implying some degree of contagion in the distribution
of attacks. This contagion is a direct density dependence in the parasitoid
population which can stabilize the otherwise intrinsically unstable system.
For values of k>1 the
contagion is insufficiently strong to stabilize the system. Hassell (1980) presents an application of
this analytical framework to the case of winter moth, Operophtera brumata
Cockerell, in Nova Scotia parasitized by the tachinid Cyzenis albicans
(Embree 1966). Drawing on quantitative studies from the field, values for the
parameters a and k were obtained and, in this case, Th approximated by 0. The
resulting model outcomes characterized well the known outcomes in the natural
system, vis. the host population declined and remained at a lower level
following the introduction of the parasitoid. The analytical framework
appears sufficiently general that it may have wider application to other
"successful' cases of biological control, and perhaps even to
"unsuccessful" cages where contagion or differential susceptibility
to attacks was insufficiently pronounced to contribute to stability. Future
examination of the roles of natural enemies may benefit from determining the
distribution of attacks in the host population. The preceding discussion has focused on
situations where there has been no implicit host density dependence, with the
function g=1. This may be an appropriate
framework for many situations, particularly where biological control agents
are established and populations are substantially below their environmentally
determined maximum carrying capacity. In other cases, however, the relative
roles of regulatory features of both host and natural enemy populations must
be addressed. Such situations are probably more characteristic of cases where
a host populations is without natural enemies prior to their introduction and
has reached an environmental maximum density. In these cases the function g will no longer be negligible, and
consideration of natural control must include the relative contribution of
both intraspecific competition and the action of the natural enemies. The design presented in equation (7) can be
used to explore the joint effects of density dependence in the host together
with the action of parasitism. This has been accomplished by Maynard Smith
& Slatkin (1973) for a two-age-class extension of this design with
independent random parasitism (the Nicholson-Bailey model) and by Beddington
et al. (1975) who employed a discrete version of the logistic model together
with random parasitism. To more fully examine the relative contributions of
intraspecific regulatory processes and parasitism a model must be used in
which parasitism can also act as a regulating or stabilizing factor. May et
al. (1981) approached this by using equation (10) for the function f (the proportion surviving
parasitism) with the addition of a discrete form of the logistic for the host
density dependence function g,
where g=exp(-cN). One important feature of these discrete
systems incorporating both host and parasitoid density dependence is that the
outcomes of the interactions will depend on whether the parasitism acts
before or after the density dependence in the host population. May et al.
(1981) envisaged two general cases, the first where host density dependence
acts first and the second where parasitism acts first (their models 2 and 3).
They employed equation (10) with no handling time (Th=0) for function f, and the two resulting systems are: Host density dependence acts before parasitism: Nt+1 = Fg(Nt)Ntf(Pt), (12a) Pt+1 = Ntg(Nt){1-f(Pt)}; (12b) parasitism
acts first: Nt+1 = Fg(fNt)Ntf(Pt), (13a) Pt+1 = Nt{1-f(Pt)}. (13b) Equation (12) is a specific case of equation
(7) with the specified functions for f
and g. Beddington et al. (1975) and May et al.
(1981) have explored the outcomes of such interactions by considering the
stability of the equilibrium populations in the host-parasitoid system. This
stability can be defined in relation to two biological features of the
system: the host's intrinsic rate of increase (log F) and the level of the
host equilibrium in the presence of the parasitoid (N*) relative to the carrying capacity of the environment (K) (the host equilibrium due only to
host density dependence in the absence of parasitism). This ratio between the
parasitoid-induced equilibrium N*
and K is termed q, q=N*/K. The relationship between F and q varies depending on the degree of contagion in the
distribution of attacks (the parameter k of equation 11), and further depends on whether parasitism
occurs before or after density dependence in the life cycle of the host. In
both cases the degree of host suppression possible increases with increased
contagion of attacks. The new parasitoid-caused equilibrium density may be
stable or unstable, and for unstable equilibrium the populations may exhibit
geometric increase or oscillatory or chaotic behavior. For density dependence
acting after parasitism and for k<1
any population reduction is stable. Additionally, special combinations of
parameter values in this latter case can lead to hypothetically higher
equilibria in the presence of the parasitoid. This only applies to over
compensatory density dependence, where it is possible to envisage parasitism
reducing the number of competitors to a density more optimal for survival
than would occur in its absence, leading to a greater density of survivors
from competition than when parasitism is not present (May et al. 1981). Also
see Bellows & Hassell (1999) for graphed figures. More generally, much of
the parameter space for both cases implies a stable reduced population
whenever k<1. This
reduction would be less for equivalent parasitism acting before density
dependence in the life cycle of the host rather than after. Patchy Environments In the same way that single-species and
competing species population may occur in heterogeneous or patchy
environments, populations which are hosts to insect parasitoids may occur in
discrete patches (Hassell & May 1973, 1974, Hassell & Taylor 198_).
The consequences of such heterogeneous host distributions on the dynamics of
the host-parasitoid system can depend significantly on the numerical
responses of the parasitoid population to prey distributed in patches.
Several mechanisms exist which tend to lead to aggregations of natural
enemies in patches of higher prey densities. Denser patches may be more
easily discovered by natural enemies (Sebalis & Laane 1986), search
behavior may change upon discovery of a host in such a fashion as to lead to
increased encounters with nearby hosts (Murdie & Hassell 1973, Hassell
& May 1974), and the time a predator spends in a patch may depend on the
encounter rate with prey (Waage 1980) or on the prey density (Sebalis &
Laane 1986). The result of each of these mechanisms is an aggregation of
natural enemies in patches of higher prey densities. Consider analytically the consequences of
such aggregations, a simple model of host and parasitoid distributions over
space. If an environment is divided into j patches of areas in the environment, the fraction of hosts in
each area can be specified by alphai
and the fraction of parasitoids in each area by Betai, with the condition that the entire population
is represented in the environment, so that Zalphai = 1, ZBetai = 1. [Z = summation sign] Equation (7) can be modified to express
this distribution over space, Nt+1 = FNt Zg(falphaiNt)alphaif(alphaif(alphaiNt,BetaiPt), (14a) Pt+1 = cNt Z alphai-f(alphaiNtBetaiP12t)} (14b) Adopting some of the simplifications
employed in equation (8) (i.e., independent random search by solitary
parasitoids, so f(P)=exp(-exp(-aP)
and c=1, and no host density
dependence, so g=1, gives the
explicit model: Nt+1 = FNt Z alphaiexp(-alpha
BetaiPt), (15a) Nt+1 = Nt Z alphai{1-exp(-aBetaiPt). (15b) The key parameters affecting the dynamical
behavior of this system are host fecundity F and the distribution of hosts and parasitoids over patches
(Hassell & May 1973, 1974). In equation (15) there is a general model for
exploring the effects of any specific host and parasitoid distributions. In
particular the case may be considered where the natural enemy distribution (Betai) is dependent in
some way on the host distribution (alphai),
Betai = c alphai. (16) In equation (16) the relationship between
the host and parasitoid distributions is determined by the parasitoid aggregation index (c is a normalizing constant which
permits ZBetai=1). In this way the distribution of parasitoids in
patches can vary from uniform (= 0) through distributions where parasitoids
"avoid" patches of high host density (<1), parasitoids have the
same distribution as the host population (= 1), to distributions where
parasitoids aggregate in patches of high host density (>1). In each patch
parasitoid search is random according to equation (15). In this system the dynamical behavior is now
largely determined by the host rate of increase F (as before), the number of patches, and the parameter which
determines the degree of aggregation of the natural enemy population.
Generally, conditions for stable population interactions are enhanced by
increasing the number of patches, values of >1 (aggregation of natural
enemies in patches of high host density) and low values of F. A necessity is an uneven
distribution of hosts; if the host distribution is uniform over patches the
system is equivalent to the intrinsically unstable Nicholson-Bailey
formulation of equation (8). This analysis permits some interpretation of
the circumstances under which the distributions of populations over patchy
environments may be significant in regulation of hosts by natural enemies.
First, aggregation of natural enemies is likely only to be an effective
regulatory mechanism if host distributions are non-uniform. Secondly, the
parasitoid distribution must be nonuniform, but not necessarily more so than
the host (i.e., it is not necessary that natural enemies aggregate more
intensely than their hosts). Finally, a host rate of reproduction which is
sufficiently can lead to instability. Inherent in most insect populations is the
concept of age- or stage-structure. Insects grown through distinct developmental
stages, and hence the concepts of age and stage are linked, although in some
systems more closely than others. Many of the analytical frameworks
constructed in the previous sections take such developmental stages into
account. Equation (4) is one such example, where considering dispersal to
occur prior to competition in a patchy resource implies a dispersing
reproductive stage (adults) followed by a non-dispersing stage which competes
for resources (larvae). Other examples are considerations of the interactions
of density-dependence and the action of natural enemies (equations (12) and
(13), e.g.). These implied sequences of events are for the most part easily
handled in the single-step analytical frameworks presented previously. However, there are a number of implied
assumptions in the previously presented frameworks which limit their
applications. In particular, there are several assumptions about the timing
of events (e.g., that all parasitism occurs simultaneously, that all
competition occurs either before or after parasitism, that all dispersal
occurs at once, and that host and parasitoid populations are so synchronized
that all members of the parasitoid population are able to attack hosts at the
same time that all members of the host population are in the stage
susceptible to parasitism). Systems that are characterized by biologies,
which are at significant variance to these assumptions, may not be well
characterized by these analytical frameworks. The solution to exploring the theoretical repercussions
of more complex biologies frequently has been to construct more complex
models, often called system or
simulation models, which
incorporate more biological detail at the expense of analytical tractability.
This approach has been used not only to address issues of population dynamics
but also to address matters relating to population developmental rate,
biomass and nutrient allocation, community structure and management of
ecosystems (Bellows et al. 1983). Here are considered only those features of
such systems which bear on population regulation in ways which are not
directly addressable in the simpler analytical frameworks presented above. Synchrony of Parasitoid
and Host Development.--The implied synchrony of host and parasitoid development in
the discrete-time formulations used above is one of the simplest assumptions
to relax in order to consider the implications of asynchrony. The degree of
synchrony between host and parasitoid development is a component of each of
the evaluations considered in this section. Here will begin the simplest case
followed by building upon it: Insect populations in continuously favorable
environments (e.g., laboratory populations, some tropical environments) may
develop continuously overlapping generations, but in the presence of
parasitism as a major cause of mortality they also may exhibit more or less
distinct generations (Bigger 1976, Taylor 1937, Metcalfe 1971, Notley 1955,
Utida 1957, White & Huffaker 1969, Hassell & Huffaker 1969, Banerjee
1979, Tothill 1930, van der Vecht 1954, Wood 1968, Perera 1987). Godfray
& Hassell (1987) constructed a simple system model in which they
considered an insect host population growing in a continuously favorable
environment (with no intraspecific density-dependence) which passes through
both an adult (reproductive) stage and preimaginal stages. They chose a
discrete-time-step model in which individuals progress through stages (or
ages) each time step; the adult stage reproduces for more than one time step,
thus leading eventually to overlapping generations and continuous
reproduction. The model for the host population is identical in structure to
the matrix model of unconstrained population growth of Lewis (1945) and
Leslie (1948), and left uninterrupted the host population would grow without
limit and attain a stable age-class structure with all age classes present at
all times. To this host population is added a parasitoid which also develops
through preimaginal and adult (reproductive) stages. The length of the
preimaginal developmental period was varied to examine the effect of changes
in relative developmental times in host and parasitoid populations. Attacks
by the parasitoid adult population were distributed using equation (10) with Th = 0 (May 1978). The dynamical behavior of the system was
characterized either by a stable population in which all stages were
continuously present in overlapping generations, populations which were
stable but which occurred in discrete cycles of approximately the generation
period of the host, and unstable populations. These dynamics were dependent
principally upon two parameters, the degree of contagion in parasitoid
attacks, k, and the relative
lengths of preimaginal developmental time in the host and parasitoid
population. Very low values of k
(strong contagion) promoted continuous, stable generations. Moderate values
of k (less strong contagion)
were accompanied by continuous generations when the parasitoid had
developmental times approximately the same length as the host, approximately
twice as long, or very short. When developmental times of the parasitoid were
approximately half or 1.5 times that of the host, discrete generations arose.
For even larger values of k,
unstable behavior was the result. From these examples it can be seen that asynchrony
between host and parasitoid could be an important factor affecting the
dynamical behavior of continuously breeding populations, particularly for
parasitoids which develop faster than their hosts. In particular, parasitoids
developing in approximately half the host's developmental time could promote
discrete (and stable) generations. Parasitism and Competition
in Asynchronous Systems.--Utida (1953) reported the dynamics of a host-parasitoid
system which had unusual dynamical behavior characterized by bounded, but
aperiodic, cyclic oscillations. These oscillations appear chaotic in nature
but are not typified by the dynamics of any of the discrete systems
considered earlier. The laboratory system consisted of a regularly renewed
food source, a phytophagous weevil, and a hymenopteran parasitoid. Important
characteristics of the system were host-parasitoid asynchrony (the parasitoid
developed in 2/3rds of the weevil developmental time), host density
dependence (the weevil adults competed for oviposition sites and larvae for
food resources), and age-specificity in the parasitoid-host relationship
(parasitoids could attack and kill three larval weevil stages and pupae, but
could only produce female progeny on the last larval stage and pupae). A system model of this system was
constructed by Bellows & Hassell (1988), which incorporated detailed
age-structured host and parasitoid populations, intraspecific competition
among host larvae and among host adults, and age-specific interactions between
host and parasitoid. The dynamics of the model had characteristics similar to
those exhibited by the experimental population and distinct from those of any
simpler model. Important features contributing to the observed dynamics were
host-parasitoid asynchronous development, the attack by the parasitoid of
young hosts (on which reproduction was limited to male offspring), and
intraspecific competition by the host. The interaction of these three factors
caused continual changes in both host density and age-class structure. In
generations where parasitoid emergence was contemporaneous with the presence
of late larval hosts, there was substantial host mortality and parasitoid
reproduction. This produced a large parasitoid population in the succeeding
generation which, emerging coincident with young host larvae, killed many
host larvae but produced few female parasitoids. The reduced host larval
population suffered little competition (because of reduced density). This
continual change in intensity of competition and parasitism contributed
significantly to the cyclic behavior of the system; simpler models without
this age-class structure would not account for these important aspects of
this host-parasitoid interaction. Invulnerable Age-classes.--The two previous models both incorporated susceptible and
unsusceptible stages, ideas which are inherent to any stage-specific
modelling construction for insects where the parasitoid attacks a specific
stage such as egg, larvae or pupae. The consequences of the presence of
invulnerable stages in a population has been considered analytically by
Murdoch et al (1987) in a consideration of the interaction between California
red scale, Aonidiella aurantii (Maskell), and its
external parasitoid Aphytis melinus (DeBach). They
constructed a system model which includes invulnerable host stages, a
vulnerable host stage, juvenile parasitoids and adult parasitoids. This model
contains no explicit density dependence in any of the vital rates or attack
parameters, but does contain time-delays in the form of developmental times
from juvenile to adult stages of both populations. Murdoch et al (1987) developed two models,
one in which the adult hosts are invulnerable and one in which the juvenile
hosts are invulnerable. The particular frameworks that were constructed
permitted analytical solutions regarding the dynamical behavior of the
systems. In particular, it was found that the model could portray stable
equilibria (approached either monotonically or via damped oscillations),
stable cyclic behavior or chaotic behavior. The realm of parameter space
which permitted stable populations was substantially larger for the model in
which the adult was invulnerable than for the model when the juvenile was
invulnerable. The overall conclusion is that an invulnerable age class can
contribute to the stability of the system. Whether this contribution is
sufficient to overcome the destabilizing influence of parasitoid
developmental delay depends on the relative values of parameters, but short
adult parasitoid lifespan, low host fecundity and long adult invulnerable age
class all promote stability. Many insect parasitoids attack only one or
few stages of a host population (although predators may be more general), and
hence many populations possess potentially unattacked stages. In addition,
however, many insect populations host more than one natural enemy, and
general statements concerning the aggregate effect of a complex of natural
enemies attacking different stages of a continuously developing host
population are not yet possible. Nonetheless, it appears that in at least the
California red scale--A. melinus system the combination
of an invulnerable adult stage and overlapping generations is likely a factor
contributing to the observed stability of the system (Reeve & Murdoch
1985, Murdoch et al. 1987). Spatial Complexity
and Asynchrony.--In predator-prey or parasitoid-host systems which occur in
a patchy heterogeneous environment, there is a distinction between dynamics
which occur between the species within a patch and the dynamics of the
regional or global system. Here there is a distinction between
"local" dynamics (those within a patch) and "global"
dynamics (the characteristics of the system as a whole). Also, while still
interested in such dynamical behavior as stability of the equilibrium, there
is also a desire to understand what features of the system might lead to
global persistence (the maintenance of the interacting populations) in the
face of unstable dynamical behavior at the local level. One set of theories
concerned with the global persistence of predator-prey systems emphasizes the
importance of asynchrony of local predator-prey cycles (those occurring
within patches) (e.g., den Boer 1968, Reddingius & den Boer 1970, Reddingius
1971, Maynard Smith 1974, Levin 1974, 1976; Crowley 1977, 1978, 1981). In
this context, asynchrony among patches implies that, on a regional basis,
unstable predator-prey cycles may be occurring in each patch at the local
scale but they will be occurring out of phase with one another (prey
populations my be increasing in some fraction of the environment while they
are being driven to extinction by predators in another); such asynchrony may
reduce the likelihood of global extinction and thus promote the persistence
of the populations. An example of one such system is the model
of interacting populations of the spider mite Tetranychus urticae
Kock and the predatory mite Phytoseiulus
persimilis Athias-Henriot
constructed by Sebalis & Laane (1986). This is a regional model of a
plant-phytophage-predator system that incorporates patches of plant resource
that may be colonized by dispersing spider mites; colonies of spider mites
may in turn be discovered by dispersing predators. The dynamics of the
populations within the patch are unstable (Sebalis 1981, Sebalis et al. 1983,
Sebalis & van der Meer 1986), with overexploitation of the plant by the
spider mite leading to decline of the spider mite population in the absence
of predators, and when predators are present in a patch they consume prey at
a rate sufficient to cause local (patch) extinction of the prey and
subsequent extinction of the predator. In contrast to the local dynamics of the
system, the regional or global dynamics of the system was characterized by two
stages, one in which the plant and spider mite coexisted but exhibited stable
cycles (driven by the intraspecific depletion of plant resource in each patch
and the time delay of plant regeneration), and one in which all three species
coexisted. This latter case was also characterized by stable cycles, but
these were primarily the result of predator-prey dynamics; the average number
of plant patches occupied by mites in the three-species system was less than
0.01 times the average number occupied by spider mites in the absence of
predators. Thus in this system consisting of a region of patches
characterized by unstable dynamics, the system persists. Principal among the models features, which
contributed to global persistence, was asynchrony of local cycles. Because of
this it was unlikely that prey could be eliminated in all patches at the same
time, and hence the global persistence. This asynchrony could be disturbed
when the predators became so numerous that the likelihood of all prey patches
being discovered would rise toward unity, a circumstance which could
eventually lead to global extinction of both prey and predator. Other
features of the system were also explored by Sebalis & Laane (1986). If a
small number of prey were able to avoid predation in each patch (a prey
"refuge" effect), the system reached a stable equilibrium, while
other parameter changes led to unstable cycles of increasing amplitude. The results of this exercise accord with
certain experiments reported in the literature. Huffaker (1958) found
self-perpetuating cycles of predator and prey in spatially complex
environments, and Huffaker et al. (1963) found that increasing spatial
heterogeneity enhanced population persistence. Three features of these
experiments were in accord with the behavior of the model of Sebalis &
Laane (1986): (1) overall population numbers in the environment did not
converge to an equilibrium value but oscillated with a more or less constant
period and amplitude; (2) facilitation of prey dispersal relative to predator
dispersal enhanced the persistence of the populations (Huffaker 1958); (3)
increase in the amount of food available per prey patch resulted in the
generation of abundant predators at times of high prey density, and the areas
were subsequently searched sufficiently well that synchronization of the
local cycles resulted, leading to regional extinction (Huffaker et al. 1963). Results reported in larger-scale systems,
particularly glasshouses, include reports of elimination of prey and
subsequently of predator (Chang 1961, Bravenboer & Dosse 1962, Laing
& Huffaker 1969, Takafuji 1977, Takafuji et al. 1981), perpetual
fluctuations of varying amplitude (Hamai & Huffaker 1978), and wide
fluctuations of increasing amplitude (Burnett 1979, Nachman 1981). Specific
interpretation of these results relative to any particular model must be made
with caution because of differences in scale, relation of the experimental
period to the period of the local cycles, and relative differences in ease of
prey and predator redistribution in different systems. Nonetheless, it is
clear that asynchrony among local patches can play an important role in
conferring global stability or persistence to a system composed of locally
unstable population interactions. The preceding has focused on natural enemies
whose population dynamics have been intimately related to that of their
hosts. Such systems might be considered typical of specialist natural enemies, parasitoids whose reproduction
depends primarily on a specific host species or population. Many species of
natural enemies, however, feed or reproduce on a variety of different hosts,
and in such cases their population dynamics may be more independent of a
particular host population. These may be considered under the term generalist
natural enemies, which are characterized by populations which have densities
independent of and relatively constant over many generations of their hosts,
as distinguished from the specialist whose dynamics is integrally bound to the dynamics
of the host. Equation (11) may be modified to represent a
host population subject to a generalist natural enemy, Nt+1 = Fnt[{1+aGt/(k(1+aThNt)}-k], (16b) where Gt
is now the number of generalist natural enemies attacking the Nt hosts, and the other
parameters have the same meaning as before. This equation includes a type II
functional response for a generalist whose interactions with the host
population may be aggregated or independently distributed (depending on the
value of k). One further
important feature, the numerical response of the generalist, may now also be
considered. Where such responses have been considered in the literature, the
data to show a tendency for the density of generalists (Gt) to rise
with increasing Nt
to an upper asymptote (Holling 1959a, Mook 1963, Kowalski 1976). This simple
relationship may be described by a formula derived from Southwood &
Comins (1976) and Hassell & May (1986): Gt = m[1-exp(-Nt/b)]. (17) Here m
is the saturation number of predators and b determines the prey density at which the number of predators
reaches a maximum. Such a numerical response implies that the generalist
population responds to changes in host density quickly relative to the
generation time of the host, as might occur from rapid reproduction relative
to the time scale of the host or by switching from feeding on other prey to
feeding more prominently on the host in question (Murdoch 1969, Royama 1979).
The complete model for this host-generalist interaction (incorporating (17) into
(16) becomes: am[1-exp(-Nt/b)]-k Nt+1 = FNt[1 +
________________] (18) [ k(1+aTht) ] This equation represents a reproduction
curve with implicit density dependence. Hassell & May (1986) present an
analysis of this interaction and present the following conclusions: At first
the action of the generalist reduces the growth rate of the host population
(which in the absence of the natural enemy grows without limit in this case).
Whether the growth rate has been reduced sufficiently to produce a new equilibrium
depends upon the attack rate and the maximum number of generalists being
sufficiently large relative to the host fecundity F. The host equilibrium falls as predation by the generalist
becomes less clumped, as the combined effect of search efficiency and maximum
number of generalists (the overall measure of natural enemy efficiency ah) increases, and as the host
fecundity (F) decreases. A new
equilibrium may be stable or unstable (in which case populations will show
limit cycle or chaotic dynamics). These latter persistent but non-steady
state interactions can arise when the generalists cause sufficiently severe
density-dependent mortality, promoted by low degrees of aggregation (high
values for k), large ah, and intermediate values of host
fecundity F. Insect populations can be subject to
infection by viruses, bacteria, Protozoa and fungi, the effects of which may
vary from reduced fertility to death. In many cases these have been
intentionally manipulated against insect populations; reviews of case studies
have been presented by Tinsley & Entwhistle (1974), Tinsley (1979) and
Falcon (1982). Much of this early work was largely
empirical, and a theoretical analysis for interactions among insect populations
and insect pathogens was until recently lacking. An analysis of underlying
dynamical processes in such systems has recently been developed by Anderson
& May 9181) (also see May & Hassell 1988). The principal features of
this framework are as follows: Considering first a host population with
discrete, non-overlapping generations (envisaging perhaps such univoltine
temperate Lepidoptera as the gypsy moth, Lymantria
dispar, and its nuclear polyhedrosis
virus disease) which is affected by a lethal pathogen which is spread in an
epidemic manner via contact between infected and healthy individuals in the
population each generation prior to reproduction. A variant of equation (5)
may be applied to describe the dynamics of such a population (where g=1 so that there is no other
density-dependent mortality): Nt+1 = FNtf(Nt), (19) where f(Nt)
now represents the fraction escaping infection. This fraction f which escapes infection as an
epidemic spreads through a population density Nt is given implicitly by the Kermack-McKendrick
expression, f=exp{-(1-f)NtNT}
(Kermack & McKendrick 1927), where NT is the threshold host density (which depends on
the virulence and transmissibility of the pathogen) below which the pathogen
cannot maintain itself in the population. For populations of size N less than NT the epidemic cannot spread (f=1) and the population consequently
grows geometrically while the infected fraction f decreases to ever smaller values. As the population continues
to grow it eventually exceeds NT
and the epidemic can again spread. This very simple system has very
complicated dynamical behavior; it is completely deterministic yet has
neither a stable equilibrium nor stable cycles, but exhibits completely
chaotic behavior (where the population fluctuates between relatively high and
low densities) in an apparently random sequence. May (1985) has reported in
more detail on this model and its behavior. Many insect host-pathogen systems which have
been studied differ from equation (19) in that transmission is via
free-living stages of the pathogen (rather than direct contact between
diseased and healthy individuals). Additionally, many such populations may
have generations which overlap to a sufficient degree that differential,
rather than difference, equations are a more appropriate framework for their
analysis. Primarily for these reasons the study of many insect host-pathogen
systems have been framed in differential equations. To construct a simple differential
framework, it is first assumed that the host population has constant per
capita birth rates a and death
rates (from sources other than the pathogen) b. The host population N(t)
is divided into uninfected (X(t))
and infected (Y(t))
individuals, N=X+Y. For
consideration of insect systems the model does not require the separate class
of individuals which have recovered from infection and are immune, as may be
required in vertebrate systems, because current evidence does not indicate
that insects are able to acquire immunity to infective agents. This basic
model further assumes that infection is transmitted directly from infected to
uninfected hosts as a rate characterized by the parameter B, so that the rate at which new
infections arise is BXY (Anderson
& May 1981). Infected hosts either recover at rate a or die at rate b. Both infected and healthy hosts
continue to reproduce at rate a
and be subject to other causes of death at rate b. The dynamics of the infected and healthy
portions of the population are now characterized by dX/dt = a(X+Y)-bX-BXY+Y, (20a) dY/dt = BXY-(alpha+b+)Y. (20b) The healthy host population increases from
both births and recovery of infected individuals. Infected individuals appear
at rate BXY and remain infectious for average time 1/(alpha+b+) before they
die from disease or other causes or recover. The dynamics of the entire
population are characterized by: dN/dt = rN-alphaY, (21) where r=a-b is
the per capita growth rate of the population in the absence of the pathogen.
There is no intraspecific density dependence or self-limiting feature in the
host population, so that in the absence of the pathogen the population will
grow exponentially at rate r. Considering now a global feature of the
system what the consequences are of introducing a few infectious individuals
into a population previously free from disease. The disease will spread and
establish itself provided the right-hind side of equation (20b) is positive.
This will occur if the population is sufficiently large relatively to a
threshold density, N>NT,
where NT is defined by: NT = (alpha + b + C)/B (22) Because the population in this simple
analysis increases exponentially in the absence of the disease, the
population will eventually increase beyond the threshold. In a more general
situation where other density-dependent factors may regulate the population
around some long-term equilibrium level K (in the absence of disease), the pathogen can only establish
in the population if K>NT Once established in the host population, the
disease can (in the absence of other density-dependent factors) regulate the
population so long as it is sufficiently pathogenic, with alpha > r. In such cases, the
population of equation (20) will be regulated at a constant equilibrium level
N*=[alpha/{alpha-r)]NT.
The proportion of the host population infected is simply Y*/N*=r/a. Hence the equilibrium
fraction infected is inversely proportional to disease virulence, and so
decreases with increasing virulence of the pathogen. If the disease is
insufficiently pathogenic to regulate the host (A < r), the host population will increase exponentially at
the reduced per capita rate r'=r-A
(until other limiting factors affect the population). The relatively simple system envisaged by
equation (20) permits some additional instructive analysis. First, pathogens
cannot in general drive their hosts to extinction, because the declining host
populations eventually fall below the threshold for maintenance of the
pathogen. Additionally, the features of a pathogen, which might be implicated
in maximal reduction of pest density to an equilibrium regulated by the
disease, should be considered. In particular what degree of pathogenicity
produces optimal host population suppression. Pathogens with low or high virulence lead to high equilibrium host
populations, while pathogens with intermediate virulence lead to optimal
suppression (Anderson & May 1981) This is a vital point because many control
programs (and indeed many genetic engineering programs) often begin with an
assumption that high degrees of virulence are desirable qualities. While this
may be true in some special cases of inundation, it is not true for systems
which rely on any degree of perpetual host-pathogen interaction (May &
Hassell 1988). A number of potentially important biological
features are not considered explicitly in the basic representation of
equation (20) (Anderson & May 1981). Several of these have fairly simple
impacts on the general conclusions presented above. Pathogens may reduce the
reproductive output of infected hosts prior to their death (which renders the
conditions for regulation of the host population by the pathogen less
restrictive). Pathogens may be transmitted between generations
("vertically") from parent to unborn offspring (which reduces NT and thus permits
maintenance of the pathogen in a lower density host population). The pathogen
may have a latency period where infected individuals are not yet infectious
(which increases NT
and also makes population regulation by the pathogen less likely). The
pathogenicity of the infection may depend on the nutritional state of the
host, and hence indirectly on host density. Under these conditions the host
population may alternate discontinuously between two stable equilibria.
Anderson & May (1981) give further attention to these cases. A more serious complication arises when the
free-living transmission stage of the pathogen is long-lived relative to the
host species. Such is the case with the spores of many bacteria, protozoa and
fungi and the encapsulated forms of many viruses (Tinsley 1979). Most of the
analytical conclusions for equations (20) still hold, but the regulated state
of the system may not be either a stable point or a stable cycle with period
of greater than two generations. Anderson & May (1981) show that the
cyclic solution is more likely for organisms of high pathogenicity (and many
insect pathogens are highly pathogenic--Anderson & May 1981, Ewald 1987)
and which produce large numbers of long-lived infective stages. The cyclic
behavior results from the time-delay introduced into the system by the pool
of long-lived infectious stages. Such cyclic behavior appears characteristic
of populations of several forest Lepidoptera and their associated diseases
(Anderson & May 1981). In one case where sufficient data were available
to estimate the parameters required by the analytical framework, thee was
substantial agreement between the expected and observed period of population
oscillation (Anderson & May 1981, McNamee et al. 1981). This field of
endeavor will benefit from additional work relating actual populations and
relevant analytical development. The analysis of the simple, two species
interactions considered thus far have focused primarily on single- or
two-factor systems, where the principal features acting on the population
where either intraspecific competition, interspecific competition in the
absence of natural enemies, the action of a natural enemy, or (in some cases)
the action of a natural enemy together with intraspecific competition. In
many populations there may be more than two species interacting, and such
systems would necessarily involve additional interactions, such as herbivores
competing in the presence of a natural enemy or different natural enemies
competing for the same host population. Four such cases are now considered,
with an examination of their dynamical behavior and the relative role the
different interactions may play in population regulation. [ Please also see Cichlid
Research ] In many natural systems phytophagous species
are attacked by a entourage of natural enemies, and plants are often attended
by a complex of herbivores. In biological control programs attempts to
reconstruct such multiple-species systems have often met with some debate in
spite of their ubiquitous occurrence. Some researchers have suggested that
interspecific competition among multiple natural enemies will tend to reduce
the overall level of host suppression (Turnbull & Chant 1961, Watt 1965,
Kakehashi et al. 1984). Others view multiple introductions as a potential
means to increase host suppression with no risk of diminished control (van
den Bosch & Messenger 1973, Huffaker et al. 1971, May & Hassell 1981,
Waage & Hassell 1982). The significance of this issue probably varies in
different systems, but the basic principles may be addressed analytically. The dynamics of a system with a single host
and two parasitoids may be addressed by extending the single host-single
parasitoid model of equation (7) to include an additional parasitoid. One
possibility is the case described by May & Hassell (1981): N+1 = FNth(Qt)f(Pt) (23a) Qt+1 = Nt{1-g(Qt)}, (23b) Pt+1 = Nth(Qt){1-f(Pt)} (23c) Here the host is attacked sequentially by
parasitoids Q and P. the functions h and f represent the fractions of the host population surviving
attack from Q and P, respectively, and are described
by equation (10); the distribution of attacks by one species is independent
of attacks by the other. Variations on this theme have also been considered,
such as when P and Q attack the same stage
simultaneously (May & Hassell 1981); the general qualitative conclusions
are the same. Three general conclusions arise from an
examination of this system. First, the coexistence of the two species of
parasitoids is more likely if both contribute some measure of stability to
the interaction (e.g., the attacks of both species are aggregated: they both
have values of k<1 in
equation (10)). Secondly, if in the system the host and
parasitoid P already coexist
and an attempt is made to introduce parasitoid Q, then coexistence is more likely if Q has a searching efficiency higher than P. If Q has too
low a searching efficiency it will fail to become established, precluding
coexistence. If the search efficiency of Q is sufficiently high, it may suppress the host population
below the point at which P can
continue to persist, thus leading to a new single host-single parasitoid
system. Examples of such competitive displacement include the successive
introductions of Opius spp.
against Dacus dorsalis in Hawaii and the
displacement of Aphytis lingnanensis by A. melinus in interior southern California (Luck &
Podoler 1985). Third, and finally, the successful
establishment of a second parasitoid species (Q) will in almost every case further reduce the equilibrium host
population. For certain parameter values, it can be shown that the equilibrium
might have been lower still if only the host and parasitoid Q were present, but this additional
depression is slight. In general, the analysis points to multiple
introductions as a sound
biological strategy. Kakehashi et al. (1984) have considered a
case similar to equation (23) but where the distributions of attacks by the
two parasitoid species are not independent but rather are identical,
indicative of the extreme
hypothetical case where two species of parasitoids respond in the same
way to environmental cues, and in locating hosts they have exactly the same
distribution of attacks among the host population. This alteration does not
change appreciably the stability properties of equation (23), but does change
the equilibrium properties. In particular, a single host-single parasitoid
system with the superior parasitoid now has a greater host population
depression than does the three-species system. In natural systems complete
covariance between species of distribution of parasitism may be less likely
than more independent distributions (Hassell & Waage 1984) and the
conclusions regarding this extreme case may be less applicable. Nevertheless,
this is an example where general, tactical predictions can be affected by
changes in detailed model assumptions, emphasizing the importance of a
critical review of the biological implications underlying them. Generalist
and Specialist Natural Enemies The preceding discussion on competing
natural enemies concerns those whose dynamics are inherently related to the
dynamics of their hosts, as is appropriate for such fairly specific natural
enemies as many insect parasitoids. Alternatively, natural enemies with more
generalist prey habits are considered whose dynamics may be more independent
of a particular host species, and turn now to interactions between
populations of specialist and generalist natural enemies. Starting with an
analytical framework, the biological implications are considered with respect
to coexistence of the natural enemies and the effect on the host population
equilibrium and stability. As mentioned earlier in the section of
natural enemies and host density dependence, discrete systems with more than
one mortality factor may have different dynamics depending on the sequence of
mortalities in the hosts life cycle. A situation is presented where the
specialist natural enemy acts first, followed by the generalist, both
preceding reproduction of the host adults. The general framework for this
sequence of events is equation (13), which can now be employed to explore the
particular case of specialist natural enemy followed by generalist (Hassell
& May 1986): Nt+1 = FNtf(Pt)g[Ntf(Pt)], (24a) Pt+1 = Nt{1-f(Pt)}. (24b) Here g(Nt)
is the effect of the generalist which, following developments earlier, incorporates
a numerical response together with the negative binomial distribution of
attacks (which allows for independently random to contagious dispersion of
attacks). If it is assumed that handling time is small relative to the total
searching time available, so Th=0
: [ am[1-exp(-N/b)]-k g(N) = [1 +
_____________]. (25) [ k ] The function f(P) is the proportion surviving parasitism and, similarly
incorporating the negative binomial distribution of attacks (and allowing Th=0), is given by: f(P) = [1+a'P/k]-k. (26) Other formulations of these ideas are
possible, in particular structuring equation (24) after (12) to represent the
situation where the specialist natural enemy follows the generalist in the
life history of the host, but the conclusions regarding roles and regulation
are similar. It might now be asked under what
circumstances the generalist and specialist can exist together and what their
combined effect on the host population will be. In particular, a specialist
natural enemy can coexist with the host and generalist most easily if the
effect of the generalist is small (k
and am are small, indicating
low levels of highly aggregated attacks) and the efficiency of the
specialists is high and their is low density dependence in the numerical
response of the generalist (Hassell & May 1986). Simply, if the effect of
the generalist is small in terms of the proportions of the population subject
to it and in its regulatory effect, there is greater potential that the host
population can support an additional natural enemy (the specialist). On the
other hand if the host rate of increase F is low or the efficiency of the generalist population (am) too high, then a specialist is
unlikely to be able to coexist in the host-generalist system. Generally, the
parameter values indicating coexistence of the specialist and generalist are
somewhat more relaxed for the case of the specialist acting before the
generalist in the host life history, because there are more hosts present on
which reproduction of the specialist can take place. In each case the
equilibrium population of the host if further reduced in the three-species
system than in either two-species system. Further details are presented by
Hassell & May (1986). Parasitoid-Pathogen-Host
Systems Another type of system in which there occur
more than one type of natural enemy is that where a host is subject to both a
parasitoid (or predator) and a pathogen (Carpenter 1981, Anderson & May
1986, May & Hassell 1988). These systems may be considered cases of
two-species competition, where the natural enemies compete for the resource
represented by the host population. As in the case for interspecific
competition they are characterized by four possible outcomes: (1) the
parasitoid and pathogen may coexist with the host, (2) either parasitoid or
pathogen may regulate the population at a density below the threshold for
maintenance of the other agent, (3) there may be two alternative stable
stages (one with host and parasitoid and one with host and pathogen), with
the outcome of any particular situation depending on the initial condition of
the system, and (4) the dynamical properties of the component systems may
each be represented in the joint system and additionally may interact and
thereby lead to behavior not present in each individual system. Consequently,
any of the four possible outcomes of the interaction may be characterized by
a steady equilibrium, stable cycles or chaos (May & Hassell 1988). The complex effects of a
host-pathogen-parasitoid system may be illustrated with reference to a simple
model of their combined interactions. The models of equations (7) and (19)
are combined to represent a population which is first attacked by a lethal
pathogen (spread by direct contact) with the survivors then being attacked by
parasitoids: Nt+1 = FNtS(Nt)f(Pt), (27a) Pt+1 = cNtS(Nt){1-f(Pt)}. (27b) Here S(N)
is the fraction surviving the epidemic given earlier (equation (19)) by the
implicit relation S=ext[-(1-S)Nt/Nt],
and f has the Nicholson-Bailey
form f(P)=exp(-aP)
representing independent, random search by parasitoids. The dynamical character of this system has
been summarized by May & Hassell (1988). For acNT(lnF)/(F-1)<1 the pathogen excludes the
parasitoid by maintaining the host population at levels too low to sustain
the parasitoid. For parasitoids with greater searching efficiency, or greater
degrees of gregariousness, or for systems with higher thresholds (NT), so that acNT(lnF)/(F-1)>1, a
linear analysis would suggest that the parasitoid would exclude the pathogen
in a similar manner. However, the diverging oscillations of the
Nicholson-Bailey system eventually lead to densities higher than NT and the pathogen can
repeatedly invade the system as the host population cycles to high densities.
The resulting dynamics can be quite complex, even from the simple and purely
deterministic interactions of equation (27). Here the basic period of the
oscillation is driven by the Nicholson-Bailey model, with the additional
effects of the (chaotic) pathogen-host interaction leading to stable (rather
than diverging) oscillations. As May & Hassell (1988) discuss, in such
complex interactions it can be relatively meaningless to ask whether the
dynamics of the system are determined mainly by the parasitoid or by the
pathogen. Both contribute significantly to the dynamical behavior, the
parasitoid by setting the average host abundance and the period of the
oscillations, and the pathogen providing long term "stability" in
the sense of limiting the amplitude of the fluctuations and thereby
preventing catastrophic overcompensation and population "crash." Competing
Herbivores and Natural Enemies The presence of polyphagous predators in
communities on interspecific competitors can have profound effects on the
number of species in the community and in the relative roles which predation
and competition play in population dynamics. Classic experiments by Paine
(1966, 1974) demonstrated that communities of shellfish contain more species
when subject to predation by the predatory starfish Pisaster ochraceus
than when the starfish is absent, and since that time considerable attention
has been devoted to theoretical considerations of the relative roles of
predation and competition in multispecies communities. Much of this work has
dealt with interactions in homogeneous environments (Parrish & Saila
1970, Cramer & May 1972, Steele 1974, van Valen 1974, Murdoch & Oaten
1975, Roughgarden & Feldman 1975, Comins & Hassell 1976, Fujii 1977,
Hassell 1978, 1979; Hanski 1981). One general conclusion of this work is that
the regulating influence of natural enemies can, under certain conditions,
enable competing species to coexist where they otherwise could not. This
effect is enhanced if the natural enemy shows some preference for the dominant
competitors or switch between prey species as one becomes more abundant than
the other. This work has also been extended to the case
of competing prey and natural enemies existing in a patchy environment
(Comins & Hassell 1987), where the work of Atkinson & Shorrocks
(1981) on two-species competition was used as a foundation. Comins &
Hassell considered the cases for competing preys which are distributed in
patches and either a generalist natural enemy (whose dynamics were unrelated
to the dynamics of the prey community) and for a natural enemy whose
population dynamics was intrinsically related to the prey community (a
"specialist", but polyphagous on the members of the competition
community). For both cases the findings generally supported the earlier results
that the action of natural enemy populations can, in certain cases, add
stability to an otherwise unstable competition community. This is more
readily done by the generalist than the specialist by virtue of the assumed
stability of the generalist population. In all cases aggregation by the
natural enemy in patches of high prey density (which leads to a
"switching" effect) is an important attribute for a natural enemy
to be able to stabilize an otherwise unstable system. Predation which is independently
random across patches is destabilizing for both the generalist and specialist
cases. Coexistence of competing prey species is possible in this spatially
heterogeneous model even when the distributions of the prey species in the
environment are correlated, and when interspecific competition is extreme. An examination of the problem of searching
in animals shows that it is fundamentally very simple, provided the searching
within a population is random. It is important to realize that
we are not concerned with the searching of individuals, but with that of whole populations. Many individual animals follow a definite
plan when searching (e.g., a fox follows the scent of a rabbit, or a bee
moves systematically from flower to flower without returning on its course).
However, there is nothing to prevent an area that has been searched by an
individual from again being searched systematically by another, or even the
same individual. If individuals, or groups of individuals, search independently of one
another, the searching within the population
is unorganized and therefore random.
Systematic searching by individuals improves the efficiency of the
individuals, but otherwise the character of the searching within a population
remains unaltered. Therefore, in competition, it may safely be assumed that
the searching is random. The area searched by animals may be measured
in two distinct ways: (1) we may follow the animals through the whole of
their wanderings and measure the area they search, without reference to
whether any portions have already been searched, and so measured, or not:
this is called the area traversed.
Or, we may measure only the previously unsearched area the animals search:
this is called the area covered.
Thus, the area traversed
represents the total amount of searching carried out by the animals, while
the area covered represents
their successful searching,
i.e., the area within which the objects sought have been found. Competition Curve.--Nicholson (1933) gave an example of
this process. Suppose we take a unit of area, say a square mile, and consider
what happens at each step when animals traverse a further tenth of that area.
When the animals begin to traverse the first tenth of the area, no part of
the area has already been searched, so that in traversing one-tenth the animals also cover one-tenth of the area. At the beginning of the next step
only 9/10th os the area remains unsearched, so as the animals search at random (= their populations now), only 9/10ths of
the second 10th of the area they search is previously unsearched area.
Consequently, after traversing 2/10th of the area the animals have covered only 2.71 tenths. At each
step of 1/10th of area traversed, the animals cover a smaller fraction of the area than in the preceding step.
Because at each step the animals cover only 1/10th of the previously
unsearched area, the whole area can never be completely searched. This is
true only if the total area
occupied by the animals is very large (not one square mile as suggested here,
necessarily). The results of this progressive calculation approximates
Nicholson's competition curve. Although the competition curve gives the
general character of the effects produced by progressively increasing
competition, it actually only approximates the true form. When the animals
have nearly completed their search of the first 10th of the area, only
slightly more than 9/10ths of the area remains unsearched. This is because
even while traversing the first 10th of the area the animals spend some small
part of the time searching over areas that have already been searched, and
the same type of effort runs through the remainder of the curve. The curve
would become more accurate as its calculations were based on indefinitely
smaller and smaller steps. Bailey (1931, p. 69) gives a formula for this
curve which is the most accurate of all. Examination of the competition curve shows
that as the area traversed increases there is a progressive slowing down in
the rate of increase of the area covered. The searching animals have
progressively increasing difficulty in finding the things they seek. With
random searching, this relation is independent of the properties of the
animals and those of their environments. Because the competition curve represents a probability,
if small numbers of animals and small areas are taken, it is likely that the
relation between the area traversed and that covered will not be found to be
exactly as shown on the curve. This does not mean that there is anything
wrong with the curve, but it does mean that small samples of a statistical
population are not good representatives of the large population from which
they area taken. The Limitation
of Animal Density.--Necessary considerations in the limitation of animal density determined
from the competition curve are the power of increase and the area of
discovery. The power of
increase is the number of times a population of animals would be
multiplied in each generation if unchecked. This value is fixed for a given
set of conditions (eg., temperature, RH, host distribution including pattern,
etc.). It determines the fraction of the animals that needs to be destroyed
in each generation in order to prevent increase in density. The area of discovery is the area effectively
traversed by an average individual during its lifetime. Area of
discovery is also a fixed value for a given set of conditions (e.g.,
temperature, terrain, etc.). If an average individual fails to capture, e.g.,
one-half the objects of the required kind it meets, then the area of
discovery is 1/2 the area traversed. The value of the area of discovery is
determined partly by the properties of the searching animals, and partly by
the properties of the objects sought. Thus, it is dependent upon the
movement, the keenness of the senses and the efficiency of capture of an
average individual when searching. It is also dependent upon the movement,
size, appearance, smell, etc. and the dodging or resistance of the average
object that is being sought. Therefore, under given conditions, a species has
a different area of discovery for each kind of object it seeks. The value of the area of discovery defines
the efficiency of a species in discovering and utilizing objects of a given
kind under given conditions. It determines the density of animals necessary
in order to cause any given degree of intraspecific competition. The power of
increase and the area of discovery together embrace all those things that
influence the possible rate of increase of the animals and all those that influence
the efficiency of the animals in searching (Nicholson 1933, Nicholson &
Bailey 1935). They are not merely properties of species, but properties of
species when living under given conditions. The same species may have different properties in different places,
or in the same place at different
times. It is also important to notice that climatic conditions and
other environmental factors play their part in determining the values of
these properties, for they influence the vitality and activity of animals.
Therefore, although such environmental factors may not be specifically
mentioned, they appear implicitly in all investigations in which values are
given to the powers of increase and areas of discovery of animals. STEADY
DENSITIES (Steady State) The concept of a steady density has led to
much debate over the years, but in general is misunderstood, for in reality
there is no steady
density possible in animals. It is a mathematical concept, which is useful in showing population
trends. Nicholson (1933) summarized the concept of steady density. He considered it to be the point where further
increase of a population is prevented when all the surplus animals are
destroyed, or when the animals are prevented from producing any surplus. When
this happens, the animals are in a state of stationary balance with their environments, and maintain their
population densities unchanged from generation to generation under constant
conditions. Because constant conditions are not possible, the actual steady
state is never reached, however. Whenever the animals' densities reach the
mathematical calculation of zero population growth, this is referred to as the steady state:
the densities of animals when at this position of balance area their steady
densities under the given conditions. The steady densities of animals are
determined from the values of their areas of discovery and powers of
increase. An example was given in Nicholson (1933) as follows: An entomophagous parasitoid attacks a
certain species of host. One host individual provides sufficient food for the
full development of one parasitoid. The area of discovery of the parasitoid
is 0.04. The power of increase of the host is 50. There are no factors
operating other than the above. The steady state will be reached when the
parasitoids are sufficiently numerous to destroy 49 out of every 50 hosts,
and when there are sufficient hosts to maintain this density of parasitoids.
The parasitoids are required to destroy 98% of the hosts and so to cover
0.98 of the area occupied by the animals. To do this it is necessary for the
parasitoids to traverse an
area of 3.91, as can be seen from the competition curve. The required density
of parasitoids, therefore, is 3.91 / 0.04, i.e., 98 approximately. But in
order that the density of the parasitoids may be maintained exactly, each
parasitoid is required to find on the average one host. Therefore, the
parasitoids are required to find 98 hosts in the area of 0.98 they cover, so
that the steady density is 98 / 0.98, i.e., 100. Of course the steady densities calculated
are the numbers of animals per unit of area. It is always convenient to
choose a large unit for the measurement of area, so that the areas of
discovery of the animals are represented by fractions, for the densities of
animals can then be given in whole numbers. If small units of measurement are
used, the character of the results obtained is actually unaffected, but the
densities calculated have to be expressed as small fractions of an animal per
unit of area, which is not desirable. It should also be noticed that the
densities calculated are those within the areas in which the animals
interact, and not necessarily within the whole countryside. Thus, if the
animals can live only in areas containing a certain kind of vegetation, then
the calculated densities are those within such areas, while the intervening
area in which the vegetation is unsuitable for animals are ignored. Other things being
equal, the density of species within the whole countryside varies directly
with the fraction of the countryside that provides suitable conditions for the species. In this considering this
further, Nicholson (1933) concluded that this is however only approximately
true. GENERALITIES
ON MODELING ARTHROPOD POPULATIONS The subject of modeling of arthropod
populations has been recently reexamined by A. P. Gutierrez (personal
commun.). It was concluded that modeling should be regarded as but another
tool in an increasing arsenal of methods for examining prey-predator
interactions. The strength of the method lies in the ease with which one can
capture the relevant biology in a mathematically simple form, and the utility
of the model for examining field problems and theory (Gutierrez 1992). The
major deficiencies are the possible lack of mathematical rigor in the
formulation of many simulation models and the tendency to add too much
detail, both of which may impair utility for examining population theory. The
question posed may not have a simple answer, as many factors may affect the
outcome making interpretation of the results difficult. For example, the
cassava mealybug model has age structure, invulnerable age classes, age and
time varying fecundity and death rates, relationships to higher and lower
trophic levels, and other factors which interact. Gutierrez (1992) states that simulation models, however,
provide good summaries of our current knowledge of a system, and furnish a
mechanism for examining this knowledge in a dynamic manner. This capability
may stimulate further questions and help guide research. At their best,
simulation models are good tools for explaining components of interactions
not readily amenable to field experimentation and for the development of
simpler models designed to answer specific questions, including those
concerning theory. Most important, model predictions may be compared with
field data and may be used to help evaluate the economic impact of pests and
of introduced natural enemies. We might even be able to evaluate possible
candidate biological control agents before they are introduced. However,
Gutierrez (1992) stresses that only the introduction and release of a species
will provide the definitive answer concerning its potential as a biological
control agent. REFERENCES: <bc-71.ref.htm> [ Additional
references may be found at MELVYL
Library ] |