Heteroclinic bifurcation in a ratio-dependent predator-prey system

In this paper we study the heteroclinic bifurcation in a general ratio-dependent predator-prey system. Based on the results of heteroclinic loop obtained in [J. Math. Biol. 43(2001): 221–246], we give parametric conditions of the existence of the heteroclinic loop analytically and describe the heteroclinic bifurcation surface in the parameter space, so as to answer further the open problem raised in [J. Math. Biol. 42(2001): 489–506].Supported by NNSFC(China) # 10171071, TRAPOYT and China MOE Research Grant # 2002061003     

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.     

Global dynamics of a ratio-dependent predator-prey system

Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey interactions where predation involves serious searching processes. However, such models have set up a challenging issue regarding their dynamics near the origin since these models are not well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey system at the origin in the interior of the first quadrant is studied. It is shown that the origin is indeed a critical point of higher order. There can exist numerous kinds of topological structures in a neighborhood of the origin including the parabolic orbits, the elliptic orbits, the hyperbolic orbits, and any combination of them. These structures have important implications for the global behavior of the model. Global qualitative analysis of the model depending on all parameters is carried out, and conditions of existence and non-existence of limit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.     

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper. Through a simple change of variable, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.     

Individual-based vs deterministic models for macroparasites: host cycles and extinction

Our understanding of the qualitative dynamics of host-macroparasite systems is mainly based on deterministic models. We study here an individual-based stochastic model that incorporates the same assumptions as the classical deterministic model. Stochastic simulations, using parameter values based on some case studies, preserve many features of the deterministic model, like the average value of the variables and the approximate length of the cycles.An important difference is that, even when deterministic models yield damped oscillations, stochastic simulations yield apparently sustained oscillations. The amplitude of such oscillations may be so large to threaten parasites' persistence.With density-dependence in parasite demographic traits, persistence increases somewhat. Allowing instead for infections from an external parasite reservoir, we found that host extinction may easily occur. However, the extinction probability is almost independent of the level of external infection over a wide intermediate parameter region.     

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

Limit cycles in predator-prey models

The general model of interaction between one predator and one prey is studied. A unimodal function of rate of growth of the prey and concave down functional response of the predator is assumed. In this work it is shown that for a given natural number n there exist models possessing at least 2n + 1 limit cycles. It is also proved, applying the Hopf bifurcation theorem, that a model exists with a logistic growth rate of the prey and concave down functional response that has at least two limit cycles.     

Multiple limit cycles in predator-prey models

A series of one-predator one-prey models are studied using two parameter Hopf bifurcation techniques which allow the determination of two periodic orbits. The biological implications of the results, in terms of domains of attraction and multiple stable states, are discussed.     

Continuous-time predator-prey models with parasites

We study a deterministic continuous-time predator-prey model with parasites, where the prey population is the intermediate host for the parasites. It is assumed that the parasites can affect the behavior of the predator-prey interaction due to infection. The asymptotic dynamics of the system are investigated. A stochastic version of the model is also presented and numerically simulated. We then compare and contrast the two types of models.     

Stabilizing dispersal delays in predator-prey metapopulation models

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 trips; 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 modeled more realistically.     

Multiple limit cycles for predator-prey models

We construct a Gause-type predator-prey model with concave prey isocline and (at least) two limit cycles. This serves as a counter-example to the global stability criterion of Hsu [Math. Biosci. 39:1-10 (1978)].     

A stochastic approach to predator-prey models

A deterministic investigation of a linear differential equation system which describes predator vs prey behavior as a function of equilibrium densities and reproductive rates is given. A more realistic structure of this model in a stochastic framework is presented. The reproductive rates and initial population sizes are considered to be random variables and their probabilistic behavior characterized by various joint probability distributions. The deterministic behaviors of the prey and predator species as functions of time are compared with the mean behaviors in the stochastic model.     

Persistence in models of three interacting predator-prey populations

This paper considers a class of deterministic models of three interacting populations with a view towards determining when all of the populations persist. In analytical terms persistence means that liminf_t→∞x(t)> 0 for each population x(t); in geometric terms, that each trajectory of the modeling system of differential equations is eventually bounded away from the coordinate planes. The class of systems considered allows three level food webs, two competing predators feeding on a single prey, or a single predator feeding on two competing prey populations. As a corollary to the last case it is shown that the addition of a predator can lead to persistence of a three population system where, without a predator, the two competing populations on the lower trophic level would have only one survivor. The basic models are of Kolmogorov type, and the results improve several previous theorems on persistence.     

Spatiotemporal dynamics of two generic predator-prey models

We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L (∞)-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L ( p )-estimate, uniform in time, for all p≥1, implies L (∞)-uniform bounds, given any nonnegative L (∞)-initial data. The applicability of the L (∞)-estimate to general reaction-diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be 'trapped' in an invariant region of phase space.     

Contact tracing in stochastic and deterministic epidemic models

We consider a simple unstructured individual based stochastic epidemic model with contact tracing. Even in the onset of the epidemic, contact tracing implies that infected individuals do not act independent of each other. Nevertheless, it is possible to analyze the embedded non-stationary Galton-Watson process. Based upon this analysis, threshold theorems and also the probability for major outbreaks can be derived. Furthermore, it is possible to obtain a deterministic model that approximates the stochastic process, and in this way, to determine the prevalence of disease in the quasi-stationary state and to investigate the dynamics of the epidemic.     

Extinction thresholds in deterministic and stochastic epidemic models

The basic reproduction number, ?(0), one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ?(0)>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ?(0)>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1-(1/?(0))( i ). With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i ( j ), j=1, …, n, and on the probability of disease extinction for each group, q ( j ). It follows from multitype branching processes that the probability of a major outbreak is approximately [Formula: see text]. In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.     

Multiparameter bifurcation diagrams in predator-prey models with time lag

A predator-prey model is considered in which prey is limited by the carrying capacity of the environment, and predator growth rate depends on past quantities of prey. Conditions for stability of an equilibrium, and its bifurcation are established taking into account all the parameters.