H.I. Freedman

A time-delay model of single-species growth with stage structure

A single-species growth model with stage structure consisting of immature and mature stages is developed using a discrete time delay. It is shown that under suitable hypotheses there exists a globally asymptotically stable positive equilibrium. Questions concerning oscillation and nonoscillation of solutions are addressed analytically and numerically. The effect of the delay on the populations at equilibrium is also considered.

Analysis of a model representing stage-structured population growth with state-dependent time delay

A stage-structured model of population growth is proposed, where the time to ma- turity is itself state dependent. It is shown that under appropriate assumptions, all solutions are positive and bounded. Criteria for the existence of positive equilibria, and further conditions for the uniqueness of the equilibria are given. The stability of the equilibria are also discussed. In addition, an attracting region is determined for solutions, such that this region collapses to the unique positive equilibrium in the state-independent case.

Persistence in models of three interacting predator-prey populations

Abstract 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 liminft→∞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.

Predator-prey populations with parasitic infection.

A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.

Uniqueness of limit cycles in Gause-type models of predator-prey systems

This paper deals with the question of uniqueness of limit cycles in predator-prey systems of Gause type. By utilizing several transformations, these systems are reduced to a generalized Lienard system as discussed by Cherkas and Zhilevich and by Zhang. As a consequence, criteria for the uniqueness of limit cycles are derived, which include results of Cheng and is related to results in Liou and Cheng. Several examples are given to illustrate our results.

Global stability and persistence of simple food chains

Abstract The main purpose of this paper is to develop criteria for which a simple food-chain model of intermediate type and of arbitrary length has a globally stable positive equilibrium and to develop criteria under which such a food chain exhibits uniform persistence. The same techniques are used to obtain conditions for a model of a predator-prey system with mutual interference of the predator to possess a globally stable positive equilibrium.

Mathematical analysis of some three-species food-chain models

Abstract The paper is basically concerned with the question of persistence of all species in a three-level food chain. A general model is introduced and the equilibria analyzed. Boundedness and stability criteria are established. Three special cases of the model are analyzed, showing the applicability of the theory, and in certain cases extensions are given. The special cases include Lotka-Volterra (where we are able to give necessary and sufficient conditions for persistence), Lotka-Volterra predation with a carrying capacity at the lowest level, and a mixed Lotka-Volterra and Holling predation (at different levels) with a carrying capacity at the lowest level.

Persistence in a model of three competitive populations

Abstract A model of three competitive populations is considered. Conditions are given for the strong persistence of the system. These conditions are in terms of the existence of equilibria on the bounding coordinate planes and their stability characteristics, and are independent, in the general case, of the existence of an interior equilibrium. Some special cases are also discussed.

Models for the effect of toxicant in single-species and predator-prey systems

Models of single-species and predator-prey systems in a polluted closed environment are developed and partially analyzed. Three cases are considered: a single influx of toxicant, a constant influx of toxicant, and a periodic pollution of the environment. In the case of single-species growth we are able to determine some local and global dynamics. In the case of predator-prey systems, we investigate the existence of steady states for a small constant influx of toxicant.On leave from Department of Mathematics, Indian Institute of Technology, Kanpur, India

A model of predator-prey dynamics as modified by the action of a parasite

A predator-prey population is described in which the prey population may be either a secondary host or a primary host to a parasite, but the predator is always a primary host. Those prey that have been invaded by the parasite have their behavior modified so as to make them more susceptible to predation. The model is described by a system of three autonomous ordinary differential equations. Conditions for persistence of all populations are given in the case that both populations are primary hosts. A brief discussion of the stability of the interior equilibrium is given.