Discrete Dynamics in Nature and Society

Volume 2017, Article ID 9475854, 11 pages

https://doi.org/10.1155/2017/9475854

## An Analysis of Discrete Stage-Structured Prey and Prey-Predator Population Models

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

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

Received 23 February 2017; Accepted 27 March 2017; Published 16 April 2017

Academic Editor: Hammad Khalil

Copyright © 2017 Arild Wikan. 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 stage-structured prey and prey-predator models are considered. Regarding the former, we prove that the models at hand are permanent (i.e., the population will neither go extinct nor exhibit explosive oscillations) and, moreover, that the transfer from stability to nonstationary behaviour always goes through a supercritical Neimark−Sacker bifurcation. The prey model covers species that possess a wide range of different life histories. Predation pressure may both stabilize and destabilize the prey dynamics but the strength of impact is closely related to life history. Indeed, if the prey possesses a precocious semelparous life history and exhibits chaotic oscillations, it is shown that increased predation may stabilize the dynamics and also, in case of large predation pressure, transfer the population to another chaotic regime.

#### 1. Introduction

As it is well known, discrete nonlinear age- and stage-structured population models serve as excellent tools in order to study the dynamical outcomes of various ecological populations. Such models contain species which may possess different life histories ranging from biennials to species that may live for many years. Regarding the age-structured case, examples of studies on concrete species as well as pure theoretical approaches may be obtained in [1–7]. In the stage-structured case, we refer the reader to [8–13]. A comparison of dynamic outcomes from age- and stage-structured models may be obtained in [14], and the analysis of the models that also incorporate spatial structure may be found in [15, 16].

The models referred to above may also be extended to include prey-predator interactions (see [17–22]). Some of these models are two-dimensional in the sense that neither the prey nor the predator has an internal structure, while others may, for example, be four-dimensional. Considering the latter, Wikan [21] provides an analysis of several examples where both the prey and the predator populations are divided into two age classes.

The purpose of this paper is to bring out the analysis of a case where the predator preys upon newborns only. Therefore, we have developed a three-dimensional model where we split the prey population into two separate parts by use of the same strategy as in [10], while the interaction between newborn prey and the predator is modelled in the same way as in [18]. The focus is on the stability and nonlinear behaviour. In particular, we address the question how the impact from the predator acts on prey populations that possess different life histories. The results from analysis of the prey part of the model are also provided and compared with findings obtained from other models.

The structure of the paper is as follows: In Section 2, we present and analyse the various models with respect to stability and nonstationary behaviour. Section 3 provides examples of prey models as well as prey-predator interactions while in Section 4 we unify and discuss the results.

#### 2. Model(s), Fixed Points, and Stability

Let and be the immature and mature part of the prey population at time , respectively, and let be the predator population. and , such that and , are the fractions of the immature population and the mature (adult) population that survive from time to time . , with , is the fraction of the immature population that survives to become mature one time unit (year) later and , , is the density dependent fecundity. Depending on the species under consideration, may account for crowding effects, effects linked to shortage of food, and for some species it may also incorporate cannibalistic behaviour. Further, it is assumed that predation will take place only on the young of the part year, , of the prey population. This is accounted for by the term where measures the skill of predation. The constant , , may be interpreted as a conversion of prey into predator, or clutch parameter, the following year (see [18]). The relation between , and at two consecutive time steps may then be given as a map:When , (1) degenerates to the prey map:Note that (2) has a striking similarity to the general stage-structured model proposed and analysed in [10]. The difference is found in the density dependent term.

By adjusting the size of the parameters, it is easy to see that (2) covers species that possess different life histories. Indeed, following [10], if , the population is semelparous (i.e., reproducing only once). If , the population is iteroparous (repeated reproduction). The subclass , is often referred to as precocious semelparity which covers species with rapid development followed by only one reproduction, for example, biennials and annual plants (see [11]). Delayed semelparity occurs when and . Typical examples are periodical cicadas [11, 23], and several salmon species that live for many years before they become mature and reproduce only once. We may also divide the iteroparous case into two subclasses. The subclass , , is classified as precocious iteroparity and covers several small mammals species, among them small rodent species that start to reproduce at young age and may survive to reproduce for several years. The fourth subclass, delayed iteroparity, is characterized by , , which covers species which may live long before maturity and then survive to reproduce for many years. In this subclass, we find large mammals. Consequently, (2) may be used in order to capture the dynamics of a wide range of (prey) populations, and the role and impact of predation on newborns and young individuals are then analysed by (1).

We start by revealing some properties of (2). There are two fixed points, the trivial one and the nontrivial one:whereand in order for (3) to be a feasible fixed point (equilibrium), we assume . Moreover, by use of stability analysis, it is straightforward to show that is stable provided . Therefore, the restriction ensures both that the origin is a repeller and that (3) is feasible.

Model (2) is said to be permanent if there exist and such that

(cf. [24]).

Theorem 1. *Model (2) is permanent provided .*

*Proof. *See Appendix A.

Considering the stability properties of the nontrivial fixed point (3), we find that the eigenvalue equation of the linearization of (2) may be expressed aswhereand according to the Jury criteria (cf. [25]), (3) is stable provided the inequalities , , and hold; that is,Obviously, the left-hand sides, LHS, of (8a) and (8b) are positive. Therefore, there will be no transfer from stability to instability as an eigenvalue crosses the boundary of the unit circle through 1 (8a) or −1 (8b). Regarding (8c), it is clearly valid for small values of but when is increased (as a result of increasing ), LHS of (8c) eventually equals the right-hand side, RHS, and will lose its hyperbolicity through a Neimark−Sacker (Hopf) bifurcation as a pair of complex valued eigenvalues cross the unit circle. Such a bifurcation may be of supercritical or subcritical nature. In the former case, when the fixed point loses its stability, an attracting invariant curve about the unstable fixed point is established and the dynamics are restricted to that curve. In the latter case (subcritical), there is no such attracting curve. Now, considering (2), we have the following result.

Theorem 2. *Consider (2) together with the fixed point given by (3). Then, for the fixed values of and , (3) will undergo a supercritical Neimark−Sacker bifurcation at the thresholdor equivalently when*

*Proof. *See Appendix B.

Next, let us focus on the “full” prey-predator map (1). There is one obvious fixed point, namely, the trivial one . The other fixed point is in the form ofwhere must be found by means of numerical methods from the equationClearly, if , (11) implies that so one possibility is that (10) is in the form of . Depending on the value of , the same scenario persists also in the case of . Consequently, as also found in the prey-predator model analysed in [21], the interaction or skill parameter a must exceed a critical threshold in order to establish a fixed point where both species coexist. Moreover, note that implies . If , an increase of a makes larger which according to (10) leads to a reduction of and . A final observation from (10) and (11) is that a decrease of clutch parameter leads to a smaller predator equilibrium . This makes sense; the smaller the , the smaller the benefit of eating. A couple of numerical examples are presented in Tables 1 and 2. In Table 1, we have used the parameter values , , , , and which means that the prey possesses precocious semelparous life history and that the prey in the absence of the predator has a stable fixed point . In Table 2, the prey may be classified as a precocious iteroparous population. Parameter values are which means that is located at instability threshold.