H. I. Freedman

Uniformly persistent systems

Conditions are given under which weak persistence of a dynamical system with respect to the boundary of a given set implies uniform persistence.

Uniform persistence and flows near a closed positively invariant set

In this paper, the behavior of a continuous flow in the vicinity of a closed positively invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider varity of ecological models.

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.

Persistence definitions and their connections

We give various definitions of types of persistence of a dynamical system and establish a hierarchy among them by proving implications and demonstrating counterexamples. Under appropriate conditions, we show that several of the definitions are equivalent.

Persistence in discrete semidynamical systems

Conditions are established for the persistence of a discrete semidynamical system formulated in terms of the global attractor of the boundary semiflow and its stable set. An application to an ecological model of a predator-prey system is given.

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

In previous work (Freedman and Wolkowicz, 1986;Bull. math. Biol.n 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.

Persistence and global stability in a predator-prey model consisting of three prey genotypes with fertility differences

A model of a predator-prey interaction, where the prey population consists of three genotypes with random mating and continuous, nonlinear birth and death processes with fertility differences, is proposed. Sufficiency conditions giving the existence of a globally stable equilibrium on one of the coordinate planes are given. This extends results of Freedman and Waltman [J. Math. Biol.n 6, 367–374 (1978) andRocky Mountain J. Math.n 12, 779–784 (1982)]. In addition, conditions are derived which guarantee the persistence of all components of the populations.