Global stability in n-dimensional discrete Lotka-Volterra predator-prey models

Sangmok Choo

doi:10.1186/1687-1847-2014-11

Abstract

There are few theoretical works on global stability of Euler difference schemes for two-dimensional Lotka-Volterra predator-prey models. Furthermore no attempt is made to show that the Euler schemes have positive solutions. In this paper, we consider Euler difference schemes for both the two-dimensional models and n-dimensional models that are a generalization of the two-dimensional models. It is first shown that the difference schemes have positive solutions and equilibrium points which are globally asymptotically stable in the two-dimensional cases. The approaches used in the two-dimensional models are extended to the n-dimensional models for obtaining the positivity and the global stability. Numerical examples are presented to verify the results.MSC: 34A34, 39A10, 40A05.

Highlights

Consider the n-dimensional system dxi dt = xi σiri + aijxj – aijxj ≤j≤i– i≤j≤n ( . )where ri >, aij > for ≤ i, j ≤ n, σ =, and σi ∈ {, } for ≤ i ≤ n
There are a number of works on investigating nonstandard finite difference schemes for the Lotka-Volterra competition models, but relatively few theoretical papers are published on discretized models of equation ( . )
We are interested in extending the method used in the two-dimensional discrete models to the n-dimensional discrete models for equation ( . )

Summary

Introduction

The solutions of equation (3.10) rotate finitely many times around the limit θ = (0.455, 10.455). The solutions of equation (3.10) rotate infinitely many times around the limit θ = (0.683, 29.317). In order to show the global asymptotic stability of the equilibrium point θ = (θ , θ ), the linearized system of equation

Results

Conclusion