R.M. Nisbet

AN INVULNERABLE AGE CLASS AND STABILITY IN DELAY-DIFFERENTIAL PARASITOID-HOST MODELS

The models were motivated by a study of red scale, an insect pest of citrus in southern California, and its successful parasitoid, Aphytis melinus. The system appears to be stable, but the usual stabilizing mechanisms suggested appear to be absent. We hypothesized that, in combination with overlapping generations, the scales invulnerability to attack by Aphytis at several stages, particularly the long-lived reproductive adult stage, might explain the systems stability. In the model both species have a juvenile and an adult stage. Generations overlap and the host and the parasitoid are not synchronous. The model has the form of coupled, delayed differential equations representing the two populations. The key parameters determining stability are the duration of the juvenile parasitoid stage (the developmental lag) and of the invulnerable adult stage of the scale, relative to the duration of the juvenile scale stage. We conclude that an invulnerable adult stage in the scale is always stabilizing because in its presence some stable parameter space exists that does not occur in its absence. The longer the invulnerable adult stage is relative to the predator developmental lag, the more likely stability is. Stability is unlikely unless the adult scale stage is significantly longer than the juvenile stage. Stability is less likely as adult parasitoids live longer or as scale fecundity increases, but the system is less sensitive to these parameters than to the duration of the invulnerable class. An invulnerable juvenile stage can also add stability, but only for a narrow range of parameter values that are not likely in real systems. Given the parameter values in the red scale-Aphytis interaction in southern California, the invulnerable adult class is not likely to explain the observed stability, but probably contributes to it. This mechanism buys increased stability at the cost of higher pest equilibrium; we discuss the apparent tension between stability and successful pest control.

Fluctuation periodicity, generation separation, and the expression of larval competition

A suite of models has been formulated to investigate the dynamic consequences of the various routes by which uniform larval competition for food can find demographic expression. It is found that while delayed expression through the vital rates of later age classes gives rise to limit cycles containing multiple overlapping generations, immediate expression via changes in the death or growth rates of the larvae themselves leads to self-sustaining single generation limit cycles. When immediate expression of competition is combined with high adult fecundity and short reproductive lifespan the amplitude of the single generation cycles is so large that they constitute a series of evenly spaced discrete generations, which is maintained indefinitely even in the absence of external cues.

Simple mathematical models for cannibalism: A critique and a new approach

We show how to incorporate a functional response in recent models of Gurtin, Levine, and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration ϵ, we arrive at a much simpler model by passing to the limit ϵ ↓ 0. It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.

Population Regulation in Animals with Complex Life-histories: Formulation and Analysis of a Damselfly Model

Publisher Summary This chapter discusses the population regulation in animals with complex life-histories of taxa such as damselflies obscure the mechanisms of population regulation. The information suggests four plausible mechanisms of damselfly population regulation: food availability; feeding-related intraspecific interference; mortality-related intraspecific interference; and density-dependent predation. The chapter demonstrates the model represents six damselfly life-stages and their interactions with a population of aquatic prey, using coupled ordinary and delay-differential equations, which are solved numerically. Analyzing the models behavior both in steady state and dynamically with the literature-derived parameter values, and performs sensitivity analyses. The resulting larval densities, larval stage durations, emergence rates, and general emergence pattern for the standard parameter values are in good agreement with those in the literature: the generation time slightly exceeds one year, and the emergence pattern is strongly bimodal, as observed for some I. elegans populations in the British Midlands. The chapter concludes that emergence patterns produced by the model seem to reflect the balance between forces promoting and opposing the coexistence of the asynchronous subpopulations that produce separate emergence peaks; promoting coexistence are density-dependent predation and intra-stage, mortality-related larval interference, and opposing it is interstage interference.

Size-selective sex-allocation and host feeding in a parasitoid-host model

1. Hymenopterous parasitoids frequently exhibit (a) size-selective sex-allocation, laying predominantly male eggs in smaller host individuals and female eggs in larger hosts, and (b) size-selective host-feeding, i.e. feeding on and killing, but not parasitizing, smaller host individuals. We abbreviate size-selective sex-allocation and host-feeding by SSH. We analyse a parasitoid--host model incorporating SSH that recognizes the following: overlapping generations, an invulnerable adult host stage, a young immature stage, an old immature host stage, and only the female parasitoid. We assume that young immature hosts are attacked by the parasitoids, die as a consequence, but do not contribute to the juvenile female parasitoid population; each attack on an old immature host produces a juvenile female parasitoid. 2. SSH leads to delayed pseudo-density-dependence in recruitment to the parasitoid population because the current attack rate on young immatures, which is a function of parasitoid density, influences the future number of old immatures and hence the future per head rate of recruitment of searching parasitoids. SSH has two effects on the models stability properties. (a) It is potentially stabilizing because it tends to suppress the inherent long-period host-parasitoid cycles. This stabilizing propensity is enhanced when the adult host stage lasts about as long as or langer than the entire immature stage, the immature parasitoid stage is short-lived, and the young immature host stage is not too long-lived or the attack rate on it is not too high. (b) SSH can also destabilize the model by creating a region of instability in which cycles occur with a period related to the total length of the host immature stages. These shorter-period cycles are promoted by the same features that tend to suppress the underlying parasitoid--host cycles. 3. Small perturbations from equilibrium in the locally stable region of parameter space are followed by damped oscillations. However, because of the presence of a multiple attractor, in much of the locally stable region large perturbations may be followed by limit cycles, commonly of the type seen in the new unstable region.

The dynamics of population models with distributed maturation periods

An integro-differential equation for the dynamics of a subpopulation of adults in a closed system where only the adults compete and where there is a distribution of maturation periods is described. We show how the careful choice of a general weighting function based on the gamma distribution with a shift in origin enables us to characterize adequately some observed maturation-period distributions, and also makes local stability and numerical analyses straightforward. Using these results we examine the progression in the behavior of the distributed-delay model as the distribution is narrowed toward the limit of a discrete delay. We conclude that while local stability properties approach those of the limiting equation very rapidly, the persistent fluctuation behavior converges more slowly, with the dominant period and maximum amplitude being least affected by the details of the distribution, and the fine structure of solutions being most sensitive. Finally, we examine the consequences for population modeling, and using several examples of insect populations, conclude that although quite often a full maturation-period distribution should be incorporated in a given model, in many cases a discrete-delay approximation will suffice.

Instability and complex dynamic behaviour in population models with long time-delays

Some of the properties of the delay-differential equation , where R and D represent the rates of recruitment to, and death from, an adult population of size X, with maturation period τ are examined. The biological constraints upon these recruitment and death functions are specified, and they are used to establish results on stability, boundedness, and persistent fluctuations of limit cycle type. The relationship between models based on delay-differential and difference equations is then explored, and it is shown how well-established results on period-doubling and chaotic behaviour in the latter can yield insight into the qualitative dynamics of the former. Using numerical studies of two population models with differing forms of recruitment function, we show how, by making use of our results, it is possible to simplify the analysis of delay-differential equation population models.

Parameter evolution in a laboratory insect population

In this paper we analyse the measurements of population vital rates made byA. J. Nicholson (1957, Cold Spring Harbour Sympos. Quantitative Biol.22, 153–173) on laboratory cultures of the Australia sheep blowfly Lucilia cuprina (Wied.) and show that these measurements imply a progressive change in the demographic characteristics of the population during the course of the experiment. These changes, which are shown to be consistent with the changes which Nicholson observed in the stability of the population as a whole, imply evolution from instability to stability. We demonstrate that the observed changes in average population parameters are compatible with the operation of natural selection, a model of the competition between “wild-stock” and “post-experimental” type flies showing that under the conditions of the experiment a population initially containing only 1 % post-experimental type flies would change its composition to essentially 100% post-experimental type individuals over the 400- to 500-day time scale observed by Nicholson.

Population dynamics and element recycling in an aquatic plant-herbivore system

We construct a model of an aquatic plant-herbivore system which takes explicit account of the cycling of the element limiting plant growth. The work is motivated by experiments on the population dynamics of the waterflea Daphnia where the Daphnia population is regulated by the availability of algal food, but the algae are phosphorus limited. The model is used to investigate the possibility that the system may be stabilised by a mechanism in which the Daphnia constitute a temporary phosphorus “sink”. We conclude that this is unlikely if the system is closed to external input of the nutrients limiting plant growth.

Discrete generations in host-parasitoid models with contrasting life-cycles

(1) Previous theoretical studies have shown that parasitoids are capable of producing cycles in their host populations with periods of about one host generation. These studies have modelled host--parasitoid interactions where parasitoid development proceeds independently from that of its host. (2) We present experimental results of a particular host--parasitoid association, Cadra cautella--Venturia canescens, and show that the development of the parasitoid is synchronized with that of its host. (3) The empirical evidence is used to formulate an age-structured model of an idealized host--parasitoid system where the onset of parasitoid development is dependent on the state of the host. The properties of the model are described and contrasted with a model that assumes there is no developmental synchrony in host and parasitoid life cycles. (4) Our results are similar to the findings of previous studies, showing that the differences in the behaviour of population models with and without life cycle synchrony are quantitative rather than qualitative. We show that when developmental synchrony occurs, ratios of parasitoid/host development durations must be interpreted with care, when predicting whether single generation cycles are to be expected in a particular host-parasitoid system.