Prey-Predator Interactions in Two and Three Species Population Models

Arild Wikan,Ørjan Kristensen

doi:10.1155/2019/9543139

Arild Wikan, Ørjan Kristensen

Open Access

https://doi.org/10.1155/2019/9543139

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Abstract

Discrete nonlinear two and three species prey-predator models are considered. Focus is on stability and nonstationary behaviour. Regarding the two species model, depending on the fecundity of the predator, we show that the transfer from stability to instability goes through either a supercritical flip or a supercritical Neimark-Sacker bifurcation and moreover that there exist multiple attractors in the chaotic regime, one where both species coexist and another where the predator population has become extinct. Sizes of basin of attraction for these possibilities are investigated. Regarding the three species models, we show that the dynamics may differ whether both predators prey upon the prey or if the top predator preys upon the other predator only. Both the sizes of stable parameter regions as well as the qualitative structure of attractors may be different.

Highlights

In 1924 and 1926, respectively, Lotka [1] and Volterra [2] independently established a two species prey-predator model which today is known under the name ‘the Lotka-Volterra prey-predator model’
Regarding the three species models, we show that the dynamics may differ whether both predators prey upon the prey or if the top predator preys upon the other predator only
The model consists of a system of two coupled nonlinear differential equations and as it is well known; the dynamical outcome of such a system is either a stable equilibrium or a limit cycle

Summary

Introduction

In 1924 and 1926, respectively, Lotka [1] and Volterra [2] independently established a two species prey-predator model which today is known under the name ‘the Lotka-Volterra prey-predator model’. We find it fair to say that there was a major breakthrough in 1976 when Sir. Robert May [6] published his influential Nature paper where he showed that a simple one-dimensional nonlinear difference equation model could generate dynamics of stunning complexity, ranging from stable fixed points, periodic orbits of even and odd periods, and chaotic behaviour. Ergodic properties of discrete models may be obtained in [12, 13] while the question of permanence is addressed in [14] Discrete harvest models, both with or without age structure, are studied in [15,16,17]. Parallel to the development of discrete age and stage structured population models, it became customary to analyze prey-predator models formulated in discrete time.

The 2-Dimensional Model

The 3-Dimensional Models

Discussion

Proof of Theorem 2 in the Main Text

Proof of Theorem 3 in the Main Text