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PubMed1989

Global positive coexistence of a nonlinear elliptic biological interacting model.

The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary differential equation system has positive equilibrium, the positive constant states. This result thus yields an algebraically computable criterion for the positive coexistence of predator and prey in many biological models.

Animals↗
PubMed1989

Diffusion-mediated persistence in two-species competition Lotka-Volterra model.

We consider a system composed of two Lotka-Volterra patches connected by diffusion. Each patch has two competitors. Conditions for persistence of the system are given. It is proved that the system can be made persistent under appropriate diffusion coefficients ensuring the instability of boundary equilibria, even if each species is not persistent within each patch. The choice of the coefficients depends closely on the patch dynamics without diffusion.

Animals↗
PubMed1989

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.

Animals↗
PubMed1990

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.

Animals↗
PubMed1990

The discrete Rosenzweig model.

Discrete time versions of the Rosenweig predator-prey model are studied by analytic and numerical methods. The interaction of the Hopf bifurcation leading to periodic orbits and the period-doubling bifurcation is investigated. It is shown that for certain choices of the parameters there is stable coexistence of both species together with a local attractor at which the prey is absent.

Animals↗
PubMed1990

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)].

Animals↗
PubMed1991

Diffusion-mediated persistence in three-species competition models with heteroclinic cycles.

We consider a model composed of two patches. One patch has three competing species forming a heteroclinic cycle within the path. The other is a refuge for one of the three species, which can diffuse between the two patches. The remaining two competitors are confined to the competitive patch and cannot diffuse. A new heteroclinic cycle can exist in the model, and the underlying cycle in the competitive patch cannot appear with a positive diffusion rate. It is proved that the model can be made persistent under appropriate diffusion conditions even if the underlying heteroclinic cycle is an attractor in the competitive patch and the patch is not persistent without the refuge. Further it is shown that the model with a specific structure is globally stable if the underlying cycle is a repeller.

Animals↗
PubMed1991

Disturbed nonlinear multispecies models in ecology.

We analyze a disturbed form of the general Lotka-Volterra model of an ecosystem with m interacting species. The disturbances act on the intrinsic growth rates of the species and are assumed to be bounded but otherwise unknown. We employ a Lyapunov technique and the concept of "reachable set" from control theory to estimate the set of all possible population densities that are attainable as a result of the disturbances. To calculate estimates for this reachable set, a number of numerical methods that entail the solution to one or more global optimization problems are developed. Specific examples involving two, three, and four species are solved. We also derive an explicit analytical expression that represents an estimate for the reachable set in the m-dimensional case. The estimate is conservative but can be evaluated without carrying out any optimization procedure. We show that methods developed in this paper can be applied to certain other types of nonlinear ecosystem models.

Animals↗
PubMed1992

The subcritical collapse of predator populations in discrete-time predator-prey models.

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.

Animals↗
PubMed1993

Chaos in a periodically forced predator-prey ecosystem model.

We subject to periodic forcing the classical Volterra predator-prey ecosystem model, which in its unforced state has a globally stable focus as its equilibrium. The periodic forcing is effected by assuming a periodic variation in the intrinsic growth rate of the prey. In nondimensional form the forced system contains four control parameters, including the forcing amplitude and forcing frequency. Numerical experiments carried out over sections of the parameter space reveal an abundance of steady-state chaotic solutions. We graph Poincaré maps and calculate Lyapunov exponents and fractal dimensions for a representative selection of strange attractors. The transitions to chaos were found to be either via a Feigenbaum cascade of period-doubling bifurcations or via frequency locking.

Animals↗