FILE: <BC-71

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 which 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 equilibria (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 which 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 considers 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-SPECIES POPULATIONS

Homogeneous Environments

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:

N_t+1 = Fg(N_t)N_t. (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),

A_t+1 = g1(L_t)L_t (3a)

L_t+1 = Fg_a(A_t)A_t (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).

Patchy Environments

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 which characterizes preimaginal competition. Population reproduction over the entire environment (i.e., all patches) can be characterized by the relationship by deJong (1979):

N_t+1 = jFZOE(n_t)n_tg[Fn_t] (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).

INTERSPECIFIC COMPETITION

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):

X_t+1 = Fg_x(X_t+alpha Y_t)X_t (5a)

Y_t+1 = Fg_y(Y_t+Beta X_t)Y_t (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):

X_t+1 = x_tg_xl(x_t+alpha 1y_t) (6a)

Y_t+1 = y_tg_xl(y_t+Beta 1y_t) (6b)

x_t+1 = X_tFxg_x alpha(X_t+alpha alpha Y_t) (6c)

y_t+1 = Y_tFy_gy alpha(Y_t+beta alpha X_t) (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 alpha_l and Beta_l, while adult competition is characterized by alpha_a and Beta_a.

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 more recent 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.

HOST-PARASITOID SYSTEMS

Equations

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:

N_t+1 = Fg(fN_t)N_tf(N_t,Pt) (7a)

P_t+1 = cN_t{1-f(N_t,P_t)} (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:

N_t+1 = FN_texp(-aP_t) (8a)

P_t+1 = N_t{1-exp(-aP_t)} (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 T_h. 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

N_t+1 = FN_tf(exp(-aP_t/(1+aThNt))) (9a)

P_t+1 = N_t{1-f(exp(-aP_t/(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+aT_hN)] (10)

and the simplest case with no host density dependence becomes

N_t+1 = FN_t[{1+aP_{t/(k(1+aThNt))}}^-k] (11a)

P_t+1 = cN_t{1-{1+aP_{t/(k(1+aThNt))}-k}} (11b)

Here once again a is the per capita search efficiency of parasitoid adults and T_h 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, T_h 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.

Host Density Dependence

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 (T_h=0) for function f, and the two resulting systems are:

Host density dependence acts before parasitism:

N_t+1 = F_g(N_t)N_tf(P_t), (12a)

P_t+1 = N_tg(N_t){1-f(P_t)}; (12b)

parasitism acts first:

N_t+1 = F_g(fN_t)N_tf(P_t), (13a)

P_t+1 = N_t{1-f(P_t)}. (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 equilibria 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 overcompensatory 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 alpha_i and the fraction of parasitoids in each area by Beta_i, with the condition that the entire population is represented in the environment, so that

Zalpha_i = 1, ZBeta_i = 1. [Z = summation sign]

Equation (7) can be modified to express this distribution over space,

N_t+1 = FN_tZg(falpha_iN_t)alpha_if(alpha_if(alpha_iN_t,Beta_iP_t), (14a)

P_t+1 = cN_t Z alpha_i-f(alpha_iN_tBeta_iP12_t)} (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:

N_t+1 = FN_t Z alpha_iexp(-alpha Beta_iP_t), (15a)

N_t+1 = N_t Z alpha_i{1-exp(-aBeta_iP_t). (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 (Beta_i) is dependent in some way on the host distribution (alpha_i),

Beta_i = c alpha_i. (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 ZBeta_i=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.

Age-structured Systems

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 which 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 T_h = 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 which 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 which 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.

Generalist Natural Enemies

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,

N_t+1 = Fn_t[{1+aG_t/(k(1+aT_hN_t)}^-k], (16b)

where G_t is now the number of generalist natural enemies attacking the N_t 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 (G_t₎ to rise with increasing N_t 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):

G_t = m[1-exp(-N_t/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(-N_t/b)]-k

N_t+1 = FN_t[1 + ________________] (18)

[ k(1+aT_ht)]

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.

HOST-PATHOGEN SYSTEMS

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):

N_t+1 = FN_tf(N_t), (19)

where f(N_t) now represents the fraction escaping infection. This fraction f which escapes infection as an epidemic spreads through a population density N_t is given implicitly by the Kermack-McKendrick expression, f=exp{-(1-f)N_tN_T} (Kermack & McKendrick 1927), where N_T 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 N_T 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 N_T 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 or 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 d