Constant-rate stocking of predator-prey systems

We examine the qualitative effects of constant-rate stocking of either or both species in a predator-prey system. The hypotheses are made as mild as possible so that several types of systems with different qualitative alternatives may be studied.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada under Grant No. A-3138.The authors wish to thank Mr. Al Mackenzie for drawing the numerous figures which appear in this paper.     

Persistence in predator-prey systems with ratio-dependent predator influence

Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823. Research carried out while visiting the University of Alberta.     

Boundedness of solutions of predator-prey systems

Under minimal assumptions, we establish boundedness of every solution of a predator-prey system with constant rate harvesting or stocking of either or both species. This leads to an extension of the classical Kolmogorov theorem on asymptotic behavior of solutions of predator-prey systems.     

Stability regions in predator-prey systems with constant-rate prey harvesting

Summary We analyze the global behavior of a predator-prey system, modelled by a pair of non-linear ordinary differential equations, under constant-rate prey harvesting. By methods analogous to those used to study predator harvesting, we characterize the theoretically possible structures and transitions. With the aid of a computer simulation we construct examples to show which of these transitions can be realized in a biologically plausible model.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada, Grant No. 67-3138.     

Global stability of Gause-type predator-prey systems

In this paper, we present some global stability results obtained from comparison analysis, Bendixson-Dulac criterion or limit cycle stability analysis for the general Gause-type predator-type systems.Research supported in part by a FGIA grant from the Arizona State University Research Fund AMS (MOS) subject classifications. Primary 34C05; secondary 34C25, 92A15.     

Interactions leading to persistence in predator-prey systems with group defence

In previous work (Freedman and Wolkowicz, 1986;Bull. math. Biol. 48, 493–508) it was shown that in a predator-prey system where the prey population exhibits group defence, it is possible that enrichment of the environment could lead to extinction of the predator population. In this paper a third population is introduced and criteria are derived under which persistence of all populations will occur. In particular, criteria for a superpredator and for a competitor to stabilize the system in the sense of persistence are analyzed. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. This research was carried out while the author was a visiting scholar at the University of Alberta.     

Qualitative dynamics of three species predator-prey systems

Summary A qualitative analysis of some two and three species predator-prey models is achieved by application of the method of averaging in conjunction with a Lyapunov function constructed from the appropriate Volterra-Lotka model. We calculate the limit cycle solution for a two-species model with a Holling type functional response of the predator to its prey by means of a time-scaled transformation. The existence of a bifurcation of steady states for a community of three species is discussed and the periodic solution around one of the unstable steady states is calculated to the lowest approximation. Several comments are made regarding the behavior of these systems under changes of some control parameters.This work was supported in parts by USERDA, Contract number E(11-1)-3001.     

Stabilizing effects of spatial heterogeneity in predator-prey systems

A general predator is assumed to divide its hunting time between two sub-habitats with different prey species, spending a larger fraction (φ) of search time in an area as the relative prey abundance there increases. This always causes switching in the model, and changes a functional response from one that imposes a risk on the average prey that decreases with prey density in the direction of one that imposes an increasing risk. I discuss the conditions for a response that is density dependent, and those predatory attributes that make such a response more likely. Transit time between subhabitats always increases the density dependent effect, and is necessary for “system stability” in a Lotka-Volterra model with two prey species. Experiments have confirmed the model's basic assumption. General predators do not fit easily into classical predator-prey models of simple “closed” communities, and then the degree of density dependence of the functional response becomes a useful measure of a predator's short-term stabilizing effect on a prey species. The model demonstrates how spatial heterogeneity can be stabilizing.     

Persistence of predator-prey systems in an uncertain environment

Summary The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.     

Consequences of behavioral dynamics for the population dynamics of predator-prey systems with switching

This article explores how different mechanisms governing the rate of change of the predators preference alter the dynamics of predator-prey systems in which the predator exhibits positive frequency-dependent predation. The models assume that individuals of the predator species adaptively adjust a trait that determines their relative capture rates of each of two prey species. The resulting switching behavior does not instantaneously attain the optimum for current prey densities, but instead lags behind it. Several mechanisms producing such lags are discussed and modeled. In all cases examined, our question is whether a realistic behavioral lag can significantly change the dynamics of the system relative to an analogous case in which the predators switching is effectively instantaneous. We also explore whether increasing the rate parameters of dynamic models of behavior results in convergence to the population dynamics of analogous models with instantaneous switching, and whether different behavioral models produce similar population dynamics. The analysis concentrates on systems that undergo endogenously generated predator-prey cycles in the absence of switching behavior. The average densities and the nature of indirect interactions are often sensitive to the rate of behavioral change, and are often qualitatively different for different classes of behavioral models. Dynamics and average densities can be very sensitive to small changes in parameters of either the prey growth or predator switching functions. These differences suggest that an understanding of switching in natural systems will require research into the behavioral mechanisms that govern lags in the response of predator preference to changes in prey density.     

Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect

Species establishment in a model system in a homogeneous environment can be dependent not only on the parameter setting, but also on the initial conditions of the system. For instance, predator invasion into an established prey population can fail and lead to system collapse, an event referred to as overexploitation. This phenomenon occurs in models with bistability properties, such as strong Allee effects. The Allee effect then prevents easy re-establishment of the prey species. In this paper, we deal with the bifurcation analyses of two previously published predator-prey models with strong Allee effects. We expand the analyses to include not only local, but also global bifurcations. We show the existence of a point-to-point heteroclinic cycle in these models, and discuss numerical techniques for continuation in parameter space. The continuation of such a cycle in two-parameter space forms the boundary of a region in parameter space where the system collapses after predator invasion, i.e. where overexploitation occurs. We argue that the detection and continuation of global bifurcations in these models are of vital importance for the understanding of the model dynamics.     

The effect of long time delays in predator-prey systems

Past studies have indicated that a time delay longer than the natural period of a system will generally cause instability; however here it is shown that including long maturational time delays in a general predator-prey model need not have this effect. In each of the three cases studied (a predator delay, a prey delay, and both), local stability can persevere despite the presence of arbitrarily long time delays. This perseverence depends upon an interaction between delayed and undelayed features of the model. Delayed processes always act to destabilize the model. For example, prey self-regulation, usually a source of stability, becomes destabilizing if subject to a long delay. However, the effect of such a delay is offset by undelayed regulatory processes, such as a stabilizing functional reponse. In addition, the adverse effects of delayed predator recruitment can be reduced by the nonreproductive component of the numerical reponse, a feature not usually involved in determining stability. Finally, it is shown that long time delays are not necessarily more disruptive than short delays; it cannot be assumed that lengthening a time delay progessively reduces stability.     

Evolutionary dynamics of predator-prey systems: an ecological perspective

 Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals – the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator–prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator–prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species. Received 10 November 1993; received in revised form 25 July 1994     

Geometric criteria for the non-existence of cycles in predator-prey systems with group defense

In this paper, we study the existence of cycles in a predator-prey system in which the prey species is equipped with the group defense capability. Some geometric criteria are developed, relating the location of the two positive equilibria on the prey isocline and the non-existence of cycles. We show that under a general geometric condition, if both positive equilibria lie on a downslope or both lie on an upslope of the prey isocline, cycles do not exist.     

Graph theoretical criteria for stability and boundedness of predator-prey systems

We introduce a graphical approach in the study of the qualitative behavior ofm species predator-prey systems. We prove that tree graphs imply global stability for Volterra models and local stability for general models; furthermore, we derive sufficient conditions so that loop graphs imply stability and boundedness of the solutions.     

Dynamics of stage-structure predator-prey systems under density-dependent effect and mortality

We develop a four dimensional predator-prey system in continuous time with stage-structure for both the communities. The reproduction rate of the prey and the transition rate for the predator, in our model, are assumed to be density-dependent. The stability results for the coexisting equilibrium are obtained by making use of Routh–Hurwitz criteria. Because of the density-dependent effects, numerical simulations are applied in complex situations. We observe that increasing values of the coefficients linked with density-dependent term promote the stability of the coexisting steady state. Our main focus is to understand the variation of stocks when mortality rates on different stage classes are increased. We verified that stable stock on mature predator increases with its increasing mortality rate in three different modeling frameworks. However, no such positive effect on the biomass of the immature predator occurs when immature predators are removed, culled or harvested. Therefore, we could conclude that the appearance of hydra effect on many unstructured predator-prey models is due to the mortality of the mature predator only. No hydra effect is also detected when mature prey is removed in several situations we discussed. Overall, the obtained results are new and could be interesting contribution in theoretical ecology.     

Handling time promotes the coevolution of aggregation in predator-prey systems

Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a Nicholson-Bailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and per-capita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predator-prey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control.     

Conditions for uniqueness of limit cycles in general predator-prey systems

Using a transformation to a generalized Lienard system, theorems are presented that give conditions under which unique limit cycles for generalized ecological systems, including those of predator-prey form, exist. The generalized systems contain those studied by Rosenzweig and MacArthur (1963); Hsu, Hubbell, and Waltman (1978); Kazarinnoff and van den Driessche (1978); Cheng (1981); Liou and Cheng (1987); and Kuang and Freedman (1988). Although very similar in approach to the result presented by Kuang and Freedman, the conditions presented here are of simpler form and in terms of the original (untransformed) functions. The results also apply to more general growth terms for the prey as shown in the examples provided. In particular, an immigration term is allowable.     

Bifurcating periodic solutions for a class of age-structured predator-prey systems

A system of mixed integrodifferential and partial differential equations for an agestructured predator-prey system is studied here. The predator eats all ages of prey, but more of the very young and very old than of the intermediate ages. The existence of periodic solutions corresponding to stable coexistence is proved for a suitable range of parameters by bifurcation theory.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.