Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points

Abstract

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species' coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.

Relative nonlinearity and permanence

“…These results on relative nonlinearity and permanence generalize the results of Kon (2004) for the case of twice differentiable f and g. We now derive special cases in the following corollary. Note that if F (x, y) and G(x, y) are both linear functions of x, y, which makes the model a discrete-time version of the LotkaVolterra model, then the second terms of (12) and (15) are zero too.…”

“…This concept allows us to fully characterize the effects of nonlinearities in population growth functions, which are of major significance in the presence of nonpoint attractors, as discussed below. We are then able to extend Kon's (2004) results beyond convexity and concavity conditions to arbitrary nonlinearities. Moreover, our sufficient condition for permanence is very easy to test for a discrete-time two-species system.…”

“…This model is studied by Kon (2004) to illustrate results on permanence from convexity and concavity conditions. Here we use the model to illustrate relative nonlinearity, but we use a different parameterization than Kon (2004) that makes the results simpler and more ecologically meaningful. Here the dynamics of x and y are given as follows:…”

“…Salceanu and Smith (2009a,b) use Lyapunov exponents to give sufficient conditions for uniform persistence for a large class of dissipative discrete-time dynamical systems on the positive orthant of R n with the property that a nontrivial compact invariant set exists on a boundary hyperplane. Kon (2004) uses an average Liapunov function to study permanence of discrete-time two-species systems. He showed that the ability of an invader to increase when the resident is at an equilibrium point is sufficient for permanence under certain convexity and concavity conditions on the growth rates of the species.…”

Permanence of two species and fixed point index

“…The problem of permanence for discrete population models has been intensively studied and several techniques have been applied. Among them are averaged Lyapunov functions, Lyapunov exponents, Conley index, Morse decompositions... See for instance [3], [4], [5], [6], [8], [12], [9], [10] and the references therein. In this paper we are going to approach the problem of permanence using the fixed point index on convex sets (see section 2 or [1]).…”