Time delays produced by dispersal are shown to stabilize Lotka-Volterra predator-prey models. The models are formulated as integrodifferential equations that describe local predator-prey dynamics and either intrapatch or interpatch dispersal. Dispersing individuals may (or may not) differ in the duration of their trip; these differences are captured via a distributed delay in the models. Our results include those of previous studies as special cases, and show that the stabilizing effect continues to operate when the dispersal process is modelled more realistically.
Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. We study an invasion model based on an integrodifference equation with an Allee effect. We derive a general result for the sign of the speed of invasion. We then examine a special, linear-constant, Allee function and introduce a numerical scheme that allows us to estimate the speed of traveling wave solutions.
Recent advances in the mathematical theory of invasion dynamics have much to offer to biological control. Here we synthesize several results concerning the spatiotemporal dynamics that occur when a biocontrol agent spreads into a population of an invading pest species. We outline conditions under which specialist and generalist predators can influence the density and rate of spatial spread of the pest, including the rather stringent conditions under which a specialist predator can successfully reverse a pest invasion. We next discuss the connections between long distance dispersal and invasive spread, emphasizing the different consequences of fast spreading pests and predators. Recent theory has attended to the effects of population stage-structure on invasion dynamics, and we discuss how population demography can inform the biological control of invading pests. Because low population densities generally characterize early stages of an invasion, we next discuss the lessons invasion theory teaches concerning the detectability of invasions. Stochasticity and density-dependent dynamics are common features of many real invasions, influencing both the spatial character (e.g., patchiness) of pest invasions and the success of biocontrol agents. We conclude by outlining theoretical results delineating how stochastic effects and complex dynamics generated by density-dependence can facilitate or impede biological pest control.
The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.
We explore a set of simple, nonlinear, two-stage models that allow us to compare the effects of density-dependence on population dynamics among different kinds of life cycles. We characterize the behavior of these models in terms of their equilibria, bifurcations, and nonlinear dynamics, for a wide range of parameters. Our analyses lead to several generalizations about the effects of life history and density dependence on population dynamics. Among these are: (1) iteroparous life histories are more likely to be stable than semelparous life histories; (2) an increase in juvenile survivorship tends to be stabilizing; (3) density-dependent adult survival cannot control population growth when reproductive output is high; (4) density-dependent reproduction is more likely to cause chaotic dynamics than density-dependence in other vital rates; and (5) changes in development rate have only small effects on bifurcation patterns.
Biological invasions are increasingly frequent and have dramatic ecological and economic consequences. A key to coping with invasive species is our ability to predict their rates of spread. Traditional models of biological invasions assume that the environment is temporally constant. We examine the consequences for invasion speed of periodic and stochastic fluctuations in population growth rates and in dispersal distributions.
A fundamental characteristic of any biological invasion is the speed at which the geographic range of the population expands. This invasion speed is determined by both population growth and dispersal. We construct a discrete-time model for biological invasions which couples matrix population models (for population growth) with integrodifference equations (for dispersal). This model captures the important facts that individuals differ both in their vital rates and in their dispersal abilities, and that these differences are often determined by age, size, or developmental stage. For an important class of these equations, we demonstrate how to calculate the population's asymptotic invasion speed. We also derive formulae for the sensitivity and elasticity of the invasion speed to changes in demographic and dispersal parameters. These results are directly comparable to the familiar sensitivity and elasticity of population growth rate. We present illustrative examples, using published data on two plants: teasel (Dipsacus sylvestris) and Calathea ovandensis. Sensitivity and elasticity of invasion speed is highly correlated with the sensitivity and elasticity of population growth rate in both populations. We also find that when dispersal contains both long- and short-distance components it is the long-distance component that governs the invasion speed---even when long-distance dispersal is rare.
The cascade model successfuly predicts many patterns in reported food webs. A key assumption of this model is the existence of a predetermined trophic hierarchy; prey are always lower in the hierarchy than their predators. At least three studies have suggested that, in animal food webs, this hierarchy can be explained to a large extent by body size relationships. A second assumption of the standard cascade model is that trophic links not prohibited by the hierarchy occur with equal probability. Using nonparametric contingency table analyses, we tested this "equiprobability hypothesis" in 16 published animal food webs for which the adult body masses of the species had been estimated. We found that when the hierarchy was based on body size, the equiprobability hypothesis was rejected in favor of an alternative, "predator-dominance" hypothesis wherein the probability of a trophic link varies with the identity of the predator. Another alternative to equiprobabilty is that the probability of a trophic link depends upon the ratio of the body sizes of the two species. Using nonparametric regression and liklihood ratio tests, we show that a size-ratio based model represents a significant improvement over the cascade model. These results suggest that models with heterogeneous predation probabilities will fit food web data better than the homogeneous cascade model. They also suggest a new way to bridge the gap between static and dynamic food web models.
``Closure terms'' describe the mortality of top predators in plankton food chain models. Most such models use density-independent closure terms. Because they must account for mortality that might be due to higher-order predators not explicitly represented in the model, it has been suggested that density-dependent closure terms are more realistic. It has also been conjectured that density-dependent closure terms eliminate limit cycles and chaos in food chain models. Here we present a counterexample to this conjecture. We examine a well-known, chaotic, 3-species food chain model, modifying the closure term to include both density-independent and density-dependent components. Density-dependent closure terms preclude neither limit cycles nor chaos.
It has recently been discovered that basins of attraction of nonlinear systems can be ``riddled''; arbitrarily close to any point in a riddled basin are neighboring points which go to a different attractor. I present two chaotically forced single-species population models with riddled basins of attraction. As a result of this complex basin structure, the ultimate survival of these populations is effectively unpredictable. Riddled basins produce a level of unpredictability qualitatively greater than the familiar sensitive dependence on initial conditions within a single chaotic attractor, or the unpredictability caused by multiple attractors with fractal basin boundaries.
Resilience is a component of ecological stability; it is assessed as the rate at which perturbations to a stable ecological system decay. The most frequently used estimate of resilience is based on the eigenvalues of the system at its equilibrium. In most cases, this estimate describes the rate of recovery only asymptotically, as time goes to infinity. However, in the short-term, perturbations can grow significantly before they decay, and eigenvalues provide no information about this transient behavior. We present several new measures of transient response that complement resilience as a description of the response to perturbation. These indices measure the extent and duration of transient growth in models with asymptotically stable equilibria. They are the reactivity (the maximum possible growth rate immediately following the perturbation), the maximum amplification (the largest proportional deviation that can be produced by any perturbation) and the time at which this amplification occurs. We demonstrate the calculation of these indices using previously published linear compartment models (two models for phosphorus cycling through a lake ecosystem and one for the flow of elements through a tropical rain forest) and a standard nonlinear predator-prey model. Each of these models exhibits transient growth of perturbations, despite asymptotic stability. Measures of relative stability that ignore transient growth will often give a misleading picture of the response to a perturbation.
Mathematical models of marine biological populations exhibit chaotic dynamics. However, we hypothesize that in moving water, Eulerian sampling of spatially heterogeneous populations may obscure any deterministic signal beyond the resolving capabilities of presently available nonlinear signal processing techniques. To examine this hypothesis we created two spatio-temporal models of population dynamics. To caricature actual ocean sampling limitations, we sample the model outputs in two ways: random walks to simulate Eulerian sampling, and spatial averages to simulate population measurements from finite volumes. Results indicate that the ability to identify underlying nonlinear dynamics quickly degrades as the step size of a random walk sampling increases. On the other hand, the analysis techniques used are more robust in the face of spatial averaging.
We investigate the dispersal-driven instabilities that arise in a discrete-time predator-prey model formulated as a system of integrodifference equations. Integrodifference equations contain two components: (1) difference equations, which model growth and interactions during a sedentary stage, and (2) redistribution kernels, which characterize the distribution of dispersal distances that arise during a vagile stage. Redistribution kernels have been measured for a tremendous number of organisms. We derive a number of redistribution kernels from first principles. Integrodifference equations generate pattern under a far broader set of ecological conditions than do reaction-diffusion models. We delineate the necessary conditions for dispersal-driven instability for two-species systems and follow this with a detailed analysis of a particular predator-prey model.
Many discrete-time predator-prey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator's population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations, and provide examples of predator-prey systems with a subcritical flip bifurcation.
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