Stability and bifurcations analysis of a competition model with piecewise constant arguments

Abstract

In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equations {alignedrightleftdxdt=r1xMathClass-open(tMathClass-close)1−xMathClass-open(tMathClass-close)k1−α1xMathClass-open(tMathClass-close)yMathClass-open(MathClass-open[t−1MathClass-close]MathClass-close)+α2xMathClass-open(tMathClass-close)yMathClass-open(MathClass-open[tMathClass-close]MathClass-close),right rightleftdydt=r2yMathClass-open(tMathClass-close)1−yMathClass-open(tMathClass-close)k2+α1yMathClass-open(tMathClass-close)xMathClass-open(MathClass-open[t−1MathClass-close]MathClass-close)−α2yMathClass-open(tMathClass-close)xMathClass-open(MathClass-open[tMathClass-close]MathClass-close)−d1yMathClass-open(tMathClass-close), where t ≥ 0, the parameters r1, k1, α1, α2, r2, k2, and d1 are positive, and [t] denotes the integer part of t ∈ [0, ∞ ). x(t) and y(t) represent population density for related species. Sufficient conditions are obtained for the local and global stability of the positive equilibrium point of the corresponding difference system. We show through numerical simulations that periodic solutions arise through Neimark–Sacker bifurcation. Copyright © 2014 John Wiley & Sons, Ltd.

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