Nicholson-Bailey Model Research Articles

Genetic Variation and Persistence of Predator-prey Interactions in the Nicholson–Bailey Model

The unstable Nicholson–Bailey predator-prey model is extended by assuming that the predation efficiency depends on quantitative characters in both the prey and the predator. The genetics of these characters is assumed to be determined by many diploid loci with additive effects. It is shown that in contrast to the monomorphic model, predator-prey coexistence is possible in the genetically variable system, and that even if the interactions lead to extinction, the time to extinction is typically much larger than in monomorphic model. Coexistence is mediated by a coevolutionary arms race involving oscillations of the character means in the prey and predator populations. Such an arms race can occur even if costs to increase (or decrease) the character value are introduced, and coexistence is possible if such costs are not too high. The results also hold if self-limitation of the prey is included in the model. In this case, both the basic and the genetic model show the paradox of enrichment, but whereas predator-prey coexistence is only possible for low prey carrying capacities in the monomorphic model, the population can persist in the genetically variable system even for very high prey carrying capacities. The results suggest that genetic variability is a condition which promotes predator-prey coexistence, and which should generally be taken into consideration when studying the ecology of extinctions.

Spatiotemporal Model for Studying Insect Dynamics in Large-Scale Cropping Systems

Insect ecologists have turned to models to aid them in understanding species dynamics in cropping systems. These models are used mainly to study insect-habitat relationships and to investigate factors affecting population stability in homogeneous cropping systems. The paucity of models for insect species interactions in large heterogeneous systems is very apparent. We develop and describe the framework for a spatiotemporal model that can be used to study insect dynamics in regional (heterogeneous) systems. This model is quite robust and integrates ideas from spatially explicit population models, coupled map lattices, and integrodifference equations. In spatially explicit population models, information on the environment of the organism is incorporated explicitly into the model. Our model combines an insect population simulation model with a resource (habitat) map. The population model is an age-structured Nicholson-Bailey model, whereas the resource map describes a 2-dimensional spatially heterogeneous habitat. Subpopulations in the spatial system are linked through dispersal and the pattern of dispersal is determined by the redistribution kernel used. Using a 2-dimensional discrete space counterpart of an integrodifference equation model we can couple prey reproduction with their dispersal, and we can redistribute adult natural enemies over the spatial region. The integrodifference equation format implies a mathematical convolution over space in each time step that can be solved, efficiently with Fourier transforms. At each time step, spatial movement for the insects can be solved with a simple rapid calculation. The assumptions, merits, and limitations of the model also are discussed.

Biota: An object-oriented tool for modeling complex ecological systems

Biota is an object-oriented software application for modeling, simulating, and learning about multispecies population dynamics. Biota uses a variety of mathematical models including versions of the Lotka-Volterra models and the Nicholson-Bailey model, but goes far beyond the capacity of any one model. We begin by describing the class and object hierarchies of Biota and its general functionality. Species, regions, and relations among species are all modeled as objects. This object-oriented decomposition provides Biota with the capacity to model and simulate complex natural interactions among biological populations. Moreover, Biota allows a mixing of models and the representation of causal interactions not found in standard population models. The mixing of models, however, creates problems for estimating the parameters used in complex simulations, detecting and correcting errors, and in understanding the system being simulated. These difficulties are addressed using a novel approach to creating simulations, which we call the behavior partitioning approach. We also discuss classroom experience with Biota, techniques for managing complexity, and its application to resource management.

Strange limits of stability in host-parasitoid systems

The classical Nicholson-Bailey model for a two species host-parasitoid system with discrete generations assumes random distributions of both hosts and parasitoids, randomly searching parasitoids, and random encounters between the individuals of the two species. Although unstable, this model induced many investigations into more complex host-parasitoid systems. Local linearized stability analysis shows that equilibria of host parasitoid systems within the framework of a generalized Nicholson-Bailey model are generally unstable. Stability is only possible if host fertility does not exceede4=54.5982 and if superparasitism is unsuccessful. This special situation has already been discovered by Hassell et al. (1983) in their study of the effects of variable sex ratios on host parasitoid dynamics. We discuss global behaviour of the Hassell-Waage-May model using KAM-theory and illustrate its sensitivity to small perturbations, which can give rise to radically different patterns of the population dynamics of interacting hosts and parasitoids.

The Importance of Refuges in the Interaction between Contarinia sorghicola and its Parasitic Wasp Aprostocetus diplosidis

The population dynamics and stability properties of the sorghum midge Contarinia sorghicola and its predominant parasitic wasp Aprostocetus diplosidis were explored with a modified version of the Nicholson-Bailey model using experimental data, that included age-structure effects, density dependence and a type II functional response. A Leslie matrix was used to model the non-linear transitions between age-classes within the population of each species and the interaction between these populations. The stability and persistence of the interaction between these two species depended on the presence of refuges resulting from mutual interference of the parasitoids

Stability, Variability, and Persistence in Host‐Parasitoid Systems

Insect parasitoids are ubiquitous members of natural insect communities, as well as being important agents of biological control. Because of their economic importance and relatively simple life cycles, host-parasitoid systems have received much theoretical effort. In this essay, I first review the theory available for hostparasitoid systems, and in particular explore the effects of environmental variability, spatial subdivision, and migration. These factors are often thought to influence dynamics in the field, but until recently have not received much attention in theory. I then make some recommendations for empirical work that could assess the effects of these factors in the field, and especially how they influence the persistence (through time) of host and parsitoid populations. Using the unstable Nicholson-Bailey model (Nicholson and Bailey 1935) as a starting point, theory has sought mechanisms that stabilize its dynamics; the rationale is that stability should ensure the persistence of the host and parasitoid (an apparent feature of real systems, at least on large spatial scales). Many stabilizing mechanisms have been identified, including mutual interference among parasitoids (Hassell and Varley 1969), several types of parasitoid aggregation (Hassell and May 1973, 1974, May 1978, Hassell 1984, Chesson and Murdoch 1986), density-dependent parasitoid sex ratios (Hassell et al. 1983, Comins and Wellings 1985), and competition among parasitoid larvae (Taylor 1988a). However, Morrison and Barbosa (1987) have questioned the idea that stability leads to persistence, when the population is subject to environmental variability. They used a standard host-parasitoid model, the negative-binomial model (May 1978), and simulated a fluctuating environment by allowing parameters in the model to be random variables. They found that perilously low densities could arise even when the model was strongly stable. Although stability ensures that

Aggregation by Parasitoids and Predators: Effects on Equilibrium and Stability

Predators and insect parasites (parasitoids) sometimes aggregate in patches containing high prey (host) densities because they search longer there, causing higher death (parasitism) rates in patches with more prey (hosts). This behavior can stabilize the otherwise unstable, discrete-time, Nicholson-Bailey model, typically at the cost of increasing the host equilibrium density. The key feature that produces both effects in this model appears to be the absence of within-generation dynamics; in particular, parasitoids are not able to reaggregate in response to changes in local host density within the generation. We examine the effects of aggregation on the otherwise neutrally stable Lotka-Volterra model. We allow the parasitoid to reaggregate continually as local host density changes. Aggregation appears as positive covariance between the parasitoid and host density across patches. It affects the model by altering the initially linear functional response. Aggregation always increases parasitoid efficiency and hence reduces host equilibrium density. We examine the effects of aggregation on model stability in several circumstances: (1) the parasitoid responds to a signal ranging from absolute to relative difference in local host density; (2) aggregation is linear or accelerating in response to this signal; (3) the variance of the host distribution is related in various ways to the mean host density (the host may have a Poisson, negative-binomial, or unspecified distribution). When account is taken of the probable limits set on parameter values by theoretical considerations and field conditions, it appears that aggregation is typically destabilizing. Although aggregation reduces host equilibrium density, h*, stability conditions usually require h* to be large. Stability is not affected by the magnitude of the spatial variance of host density, per se, but rather by the rate of change of the variance with respect to the mean. The chances of stability are increased if the parasitoid or predator has an accelerating aggregative response and possibly if it responds to absolute differences in local host density over a wide range of host densities, though the stabilizing effects in the latter case are likely to be weak or limited to small perturbations. We also examine aggregation that is independent of local prey (host) density. The Nicholson-Bailey model can be stabilized if, for example, the distribution of parasitism among patches is highly heterogeneous but unrelated to host density in the patch. Again, strong aggregation increases the Nicholson-Bailey host equilibrium density. Such aggregation has no effect on stability or host equilibrium in our continuous-time model. We discuss critical assumptions in the models. The results point to the need for more information about the signal to which aggregating predators actually respond and about the form of the response in the field.

Models for pest control: Complementary effects of periodic releases of sterile pests and parasitoids

Four host-parasitoid models that incorporate the simultaneous or sequential release during each generation of sterile hosts and parasitoids for control or eradication of host species are presented. The models are based on two modifications of the Nicholson-Bailey model which incorporate density regulation either in the host larvae or via parasitoid oviposition. Parasitization of host larvae and adults forms another comparison. The models indicate that the release of sterile hosts alone is more efficient than release of parasitoids alone in controlling the hosts if population regulation is in the parasitoids; otherwise, the release of parasitoids alone is more efficient. The release of both steriles and parasitoids is much more efficient than the release of either alone for either suppressing or eradicating the hosts. This greater efficiency in combination rather than separately appears to be a special case of a more general principle, which is that two pest control methods will mutually complement each other if their optimal actions in reducing host numbers are at very different host densities. This is the case for sterile releases (optimal at low host densities) and parasitoid inundation (optimal at high host densities).

Stability of Real Interacting Populations in Space and Time: Implications, Alternatives and the Negative Binomial kc

(1) Data are examined that relate to two-population interaction models based on the negative binomial parameter, k, and the resulting 'empty-cell' probability (e.g. probability of host escaping parasitoid attack). As predicted (Taylor, Woiwod & Perry 1979) estimates of k were non-linearly related to population density for samples of Drosophila melanogaster eggs, predator Phytoseiulus persimilis adults (among its prey, Tetranychus urticae), parasitoid Ooencyrtus kuvanae adults (among eggs of its host, Lymantria dispar) and parasitoid A laptusfusculus eggs (in eggs of its host Mesopsocus immunis). (2) Contagion did not increase with k, in contrast to conventional expectations, and k was highly variable as found by Elliot (1982) for the parasitoid Agriotypus armatus. There was no constant (kc) value. Hence, the criterion required for stability in May's (1978) phenomenological model did not exist. (3) This and similar two-population interaction models dependent on the 'empty-cell' probability, p0, will overestimate p0 unless they incorporate density dependence of k. They will also overestimate equilibrium densities. Stability analyses for May's phenomenological and related models, performed about these equilibrium densities, are not reliable. (4) An alternative simple host-parasitoid model is examined in which heterogeneity is built into the basic Nicholson-Bailey model by allowing parasitoid and host to respond to one another by movement; this model also applies if the host has a partial refuge from the parasitoid. (5) Models involving two interacting species should not depend upon the premise that the two species are distributed independently of one another.

Simulating conditions for the regulation of a moose population by Wolves

Computer simulations were used to explore how wolves could regulate moose populations in the presence or absence of hunting. In the model, vulnerability to predation varied with the age of the moose, and the numbers of animals killed per age-class were computed using the Nicholson-Bailey model. Vulnerability to hunting varied with sex and age of moose. Three possibilities were investigated: (a) when reproduction of both predator and prey were held constant; (b) when reproduction of wolves was directly related to winter survival; and (c) as for b, with the addition that wolves have access to garbage dumps in winter. The last set of hypotheses proved to be sufficient for the predator to regulate its prey.

Host-Parasitoid Systems in Patchy Environments: A Phenomenological Model

SUMMARY (1) A host-parasitoid model is presented which is intermediate in complexity between the Nicholson-Bailey model (in which the parasitoids search independently randomly in a homogenous environment) and complicated models for incorporating environmental patchiness (in which the overall distribution of parasitoid attacks is derived from detailed assumptions about their searching behaviour and about the spatial distribution of the hosts). The model assumes the overall distribution of parasitoid attacks per host to be of negative binomal form. There are consequently three biological parameters: two are the usual parasitoid 'area of discovery', a, and the host 'rate of increase', F; the third is the negative binomial clumping parameter, k. Such intermediate-level models have proved useful in sorting out ideas in the related disciplines of epidemiology and parasitology. (2) Empirical and theoretical arguments for using the negative binomial to give a phenomenological description of the essential consequences of spatial patchiness in models are surveyed. (3) A biological interpretation of the parameter k in host parasitoid models is offered. If the parasitoids be distributed among patches according to some arbitrary distribution which has a coefficient of variation CVp, and if the parasitoid attack distribution within a patch be Poisson, then the ensuing compound distribution can be approximated by a negative binomial which will have the same variance as the exact distribution provided k is identified as k = (I I/CVp)2. (4) Expressions are obtained for the equilibrium values of host and parasitoid popula- tions. These equilibria are stable if, and only if, k < 1; that is, provided there is sufficient clumping. (5) The dynamical effects of parasitoid aggregation in some respects mimic those introduced by mutual interference among parasitoids; the appropriate coefficient of 'psuedo-interference' is calculated.

The Effects of Temperature and Host Refuge on Insect Host-Parasite Models 1

The addition of host refuge to the Nicholson-Bailey or Hassell-Varley models was found to have a major effect on the stability of the models. If a refuge exists for a constant proportion of the host population, then stability is not affected, but if a constant number of hosts are in a refuge, then apparent limit cycles or damped oscillations are obtained. Experimental investigation suggests that the degree of host refuge in complicated laboratory “mazes” is temperature dependent at low but not at high temperatures. Cage studies in cotton fields indicate that the functional response of the parasite, Trichogramma pretiosum Riley, to the host, eggs of Trichoplusia ni Hubner, is a nearly linear function up to about 18.4 hosts/m2. In addition, there exists a nearly linear relationship between the functional and numerical responses of the parasite as determined from cage experiments. Both of these assumptions are contained in the Nicholson-Bailey and Hassell-Varley models. These cage experiments indicate that the assumptions are probably not seriously violated in the system which was studied.

Aggregation of Predators and Insect Parasites and its Effect on Stability

Searching animals, such as predators and insect parasites,t usually spend more time where their requisites are more plentiful, a behaviour that has an obvious selective advantage. Despite this, it is only from relatively recent work that aggregative responses to uneven prey distributions have been adequately quantified in terms of predator numbers, or the time spent by a predator, per unit areas of different prey density. This in turn is reflected in the relatively few predator-prey models that have allowed for such aggregative behaviour (Royama 1971; Hassell & Rogers 1972; Hassell & May 1973; Murdoch & Oaten 1974). These are in contrast to the many models (e.g. Lotka 1925; Volterra 1928; Thompson 1924; Nicholson & Bailey 1935; Watt 1959; Hassell & Varley 1969) where search is random, which effectively implies an even distribution of predators throughout the whole prey area and makes the particular types of prey distribution irrelevant to the model outcome. In an attempt to show how predator aggregation could affect stability, Hassell & May (1973) considered a simple modification of the Nicholson-Bailey model in which the prey survival was given by