Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type

Zhimin He,Bo Li

doi:10.1186/1687-1847-2014-180

Abstract

In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-III type in the closed first quadrant . Firstly, the existence and stability of fixed points of the system is discussed. Secondly, it is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of by using bifurcation theory. Finally, numerical simulations including bifurcation diagrams, phase portraits, and maximum Lyapunov exponents are presented not only to explain our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, -7, -9, -15, -16, -22, -23, -32, -35 orbits, a cascade of period-doubling bifurcations in period-2, -4, -8, -16 orbits, quasi-periodic orbits, and chaotic sets. MSC: 37G05, 37G35, 39A28, 39A33.

Highlights

The Lotka-Volterra prey-predator model has become one of the fundamental population models since the theoretical works going back to Lotka ( ) [ ] and Volterra ( ) [ ] in the last century
We investigate this version as a discrete-time dynamical system in the interior of the first quadrant R + by using the normal form theory of the discrete system, and we prove that this discrete model possesses the flip bifurcation and the Neimark-Sacker bifurcation
In Section, we show that there exist some values of the parameters such that ( ) undergoes the flip bifurcation and the Neimark-Sacker bifurcation in the interior of R +

Summary

Introduction

The Lotka-Volterra prey-predator model has become one of the fundamental population models since the theoretical works going back to Lotka ( ) [ ] and Volterra ( ) [ ] in the last century. We consider the predator-prey system of Holling-III type that is given in [ ] as follows:. The predator consumes the prey according to the Holling type-III functional response x /(x + β) and contributes to its growth with rate αx /(x + β). In Section , we discuss the existence and stability of fixed points for system ( ) in the closed first quadrant R +. We consider the existence of the positive fixed points of system ( ). Using the Schur-Cohn criterion [ ], we can show the stability of the fixed points as follows. The positive fixed point (x∗ , x∗ ) of system ( ) is stable if one of the following conditions holds:.

Flip bifurcation and Neimark-Sacker bifurcation

Numerical simulations

Conclusion