Nonlinear semelparous Leslie models

J M Cushing

doi:10.3934/mbe.2006.3.17

Abstract

In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at n = 1 despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at n = 1. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.

Highlights

Since their introduction by Lewis [18] and Leslie [19, 20] matrix models have found extensive use in theoretical and applied studies of the dynamics of biological populations structured by means of physiological classes [2]
Typically the positive equilibria that supercritically bifurcate at n = 1 would obtain the stability lost by the trivial equilibrium
A nongeneric bifurcation at the trivial equilibrium x = ˆ0 occurs in this model at n = 1, where n is the inherent net reproductive number

Summary

Introduction

Since their introduction by Lewis [18] and Leslie [19, 20] matrix models have found extensive use in theoretical and applied studies of the dynamics of biological populations structured by means of physiological classes [2]. An equilibrium x = α1 of the map (9) corresponds to a single-class m-cycle (consisting of the vectors (8)) of the nonlinear Leslie model (2). There exists a continuum of nonnegative single-class m-cycles (2) that bifurcates from the trivial equilibrium at n = 1.

Results

Conclusion