Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems

Quay Van Der Hoff,Johanna C Greeff,P Hendrik Kloppers

doi:10.1590/sajs.2013/1143

Abstract

There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka–Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka–Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator–prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where it is difficult, if not impossible, to prove analytically the existence of limit cycles.

Highlights

The analytical methods used to prove the existence or non-existence of limit cycles are complex and mathematically challenging and may not be accessible to all scientists working in the field of ecological modelling
Various reports in the literature confirm the existence of a unique, stable limit cycle in a class of predator–prey models that include logistic growth and a Holling Type II functional response
To find an explicit analytical function P in continuous time is seldom possible, we propose a numerical approach, using Mathematica®,16 to generate a sequence of discrete points located on L and to observe the behaviour of the sequence in order to illustrate the possible existence of a limit cycle

Summary

Introduction

The analytical methods used to prove the existence or non-existence of limit cycles are complex and mathematically challenging and may not be accessible to all scientists working in the field of ecological modelling. Various reports in the literature confirm the existence of a unique, stable limit cycle in a class of predator–prey models that include logistic growth and a Holling Type II functional response. One such a system is given by ( ) ( ) x = rx. The latter, E*, is the only equilibrium point that could possibly lead to co-existence of both species It has been proven analytically[3,9,10,11,13] that E* exists and is unstable and that a unique, stable limit cycle will exist under the conditions. It is possible to approximate coordinates on the limit cycle and the period of the limit cycle associated with the value of K, confirming analytical calculations suggested by Kuznetsov et al.[12]

Numerical method

Conclusion