Discrete Dynamics in Nature and Society

Volume 2019, Article ID 9543139, 14 pages

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

## Prey-Predator Interactions in Two and Three Species Population Models

School of Business and Economics, The Arctic University of Norway, Campus Harstad, Norway

Correspondence should be addressed to Arild Wikan; on.tiu@nakiw.dlira

Received 14 November 2018; Revised 8 January 2019; Accepted 23 January 2019; Published 17 February 2019

Academic Editor: Daniele Fournier-Prunaret

Copyright © 2019 Arild Wikan and Ørjan Kristensen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. 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’. 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. Unfortunately, the Lotka-Volterra model has an undesired property; namely, it is structurally unstable, which in turn implies that most attempts to apply the model on real world phenomena are likely to fail. Therefore, after the pioneer works in the 1920s, there has been a tremendous development of prey-predator models. At first, most of these models were formulated in continuous time; see for example the work by Rosenzweig and MacArthur 1963 [3] and Holling 1965 [4] and the study of equations of Kolmogorov type as presented by Freedman and Waltman [5]. The studies cited above, together with lots of other contributions, lead to a variety of functional responses for different species which are widely used in prey-predator interaction models.

Regarding discrete population models, 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. Later, the number of papers on discrete population models flourished (confer [7–11]), and it became clear that the dynamics found from these studies was much richer than from their continuous counterparts. 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 [15–17].

Parallel to the development of discrete age and stage structured population models, it became also customary to analyze prey-predator models formulated in discrete time. Indeed, Neubert and Kot [18] showed that the equilibrium in a two species prey-predator model may undergo a subcritical flip bifurcation with a subsequent concomitant crash of the predator population. Other excellent studies may be obtained from [19–24] and, more recently, the dynamical behaviour of fractional order Lotka-Volterra and generalized Lotka-Volterra models together with its discretizations have been scrutinized in [25, 26].

Unlike most of the papers quoted above, we shall in this paper assume interactions between the prey and predator species of exponential form, a choice which is inspired by the seminal work of Ricker [27], also cf. [21]. The purpose of this work is to analyze (A) a two species prey-predator model and (B) two versions of a three species models (two predators), where focus is on stability, nonstationary, and chaotic behaviour as well as on mechanisms which may lead to extinction of predators. Regarding the results, we prove in case (A) that the size of the region in parameter space where the equilibrium is stable strongly depends on the fecundity of the predator and moreover that the transfer from stability to instability may go through either a supercritical flip bifurcation or alternatively through a supercritical Neimark-Sacker bifurcation when the fecundity of the prey is increased. In the chaotic regime there may be two different attractors, one where both the prey and the predator coexist and another where only the prey survives. We investigate the size of basin of attraction for these possibilities. In case (B) focus is much on the same as in (A). One major result is that if the top predator preys upon both the prey and the predator or only on the predator, this has profound effects on the size of stable parameter regions and on possible nonstationary dynamics.

The plan of the paper is as follows. In Section 2 we formulate and analyze the two species prey-predator model. Section 3 deals with the case where there is one prey population and two predator populations. Finally, in Section 4 we summarize and discuss results.

#### 2. The 2-Dimensional Model

Let and be the sizes of a prey and a predator population at time , respectively. The relation between the two species at two consecutive time steps (years) is assumed to be on the formNatural restrictions to impose arewhich biologically means that intraspecific competition leads to a decrease in size of both populations while interspecific competition (predation) results in a decrease of the survival of the prey and an increase of the size of the predator population.

In this section we shall consider the model (which satisfies the restrictions above)The capital letters and denote density independent fecundity terms. and are positive interaction parameters and from a biological point of view it is natural to assume . When , (3) degenerates to a ‘pure’ prey map.

Map (3) possesses three equilibria, the trivial one (, the point where , and the nontrivial onewhere satisfies the equationand . In order to investigate stability properties we linearize about the equilibrium. This gives birth to the eigenvalue equationwhere the coefficients are is a stable equilibrium as long as all eigenvalues of (6) are located on the inside of the unit circle and according to the Jury criteria this is satisfied whenever the inequalities , , and hold. Following Murray [28], when the first of these inequalities fails (i.e., when ), it corresponds to . The second one fails when (the flip case) and the third fails when the solution of (6) is a pair of complex valued eigenvalues located on the boundary of the unit circle (i.e., , the Neimark-Sacker case). Consequently, is stable for those parameter combinations who satisfyHence, (4) will be stable wheneverwhereand in order for we must have .

If the solutions of (6) are and . If is assumed, the solutions of (6) areand these eigenvalues are indeed located on the boundary of the unit circle sinceNote that when , all eigenvalues (both real and complex) approach and the stable parameter region approaches zero.

*Example 1. *First we scrutinize the special case , i.e.,By use of the Jury criteria, it is straightforward to show that is stable for and all values of . Regarding it is stable whenever and . Note that when , then becomes arbitrary and when , then . At threshold , and which satisfies . The nontrivial equilibrium point may be expressed asNote that and in order to have a feasible equilibrium we must assume (or ).

The coefficients in (6) becomeand subsequently the Jury criteria (8a)-(8c) may be cast in the formThe stable parameter region is characterized by . Moreover, (16b) and imply . Hence, depending on , the largest interval where may be stable is .

At threshold the equilibrium point may be expressed aswhereThe solutions of (6) are and . Note that when , then and . When , the quantities , , and may be expressed by use of the Lambert function as and .

From we findwhereand the corresponding complex modulus eigenvalues becomeIn Figure 1(a) we have visualized the stable parameter regions in the plane. When , is stable for any value of . In the interval , the point is stable for , and whenever , the nontrivial equilibrium given by (14) is stable for those combinations of and which satisfy (confer (16b), (16c)); i.e., the stable parameter region is located between the curves.