Journal of Applied Mathematics

Volume 2017 (2017), Article ID 8934295, 14 pages

https://doi.org/10.1155/2017/8934295

## An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities

School of Business and Economics, The Arctic University of Norway, Campus Harstad, Havnegata 5, 9480 Harstad, Norway

Correspondence should be addressed to Arild Wikan

Received 12 June 2017; Revised 23 August 2017; Accepted 24 September 2017; Published 17 December 2017

Academic Editor: Urmila Diwekar

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

A discrete age-structured semelparous Leslie matrix model where density dependence is included both in the fecundity and in the survival probabilities is analysed. Depending on strength of density dependence, we show in the precocious semelparous case that the nonstationary dynamics may indeed be rich, ranging from SYC (a dynamical state where the whole population is in one age class only) dynamics to cycles of low period where all age classes are populated. Quasiperiodic and chaotic dynamics have also been identified. Moreover, outside parameter regions where SYC dynamics dominates, we prove that the transfer from stability to instability goes through a supercritical Neimark−Sacker bifurcation, and it is further shown that when the population switches from possessing a precocious to a delayed semelparous life history both stability properties and the possibility of periodic dynamics become weaker.

#### 1. Introduction

Within the framework of nonlinear discrete age-structured population models, the dynamical properties and behaviour of a great variety of species may be explored. Such species may possess an iteroparous life history which means that individuals in several age classes of the population are fertile, or they may possess a semelparous life history which is characterized by the property that only individuals of the last age class are fertile.

Regarding iteroparity, most population models (both age-structured and stage-structured) focus on cases where nonlinearities (density dependence) are built into the fecundity elements and not into the survivals. Particularly in fishery models, this has often been motivated by the assumption that most density effects are present only in the first year of life. Examples of theoretical studies which deal with nonstationary and chaotic behaviour as well as behaviour linked to concrete species may be found in [1–7]. In case of ergodic properties we refer to [8–10]; see also [11]. Another strategy is to assume constant fecundity terms and nonlinear year-to-year survival probabilities, compare [12–16]. Depending on functional form of the nonlinear terms, a crude conclusion found in several of the papers referred to above is that, in case of small population densities, the models possess a stable nontrivial equilibrium where all age classes are populated. At higher population densities there may be nonstationary, periodic, and chaotic dynamics of stunning complexity.

Now, turning to the semelparous case with nonlinear fecundity element and constant survival probabilities a rather peculiar phenomenon has been detected, compare [4]. Here, in contrast to the iteroparous case, the nontrivial equilibrium tends to be unstable in large parameter regions, also in case of low population densities. Instead one finds, as time , that a cyclic state is attained where the whole population is in just one single age class at each time step. Such behaviour, which is called synchronization or SYC (single year class) dynamics, has been detected among insects; see [17]. Hoppensteadt and Keller [18] presented a model for the 17-year cicada (magicicada) which included both predation and intraspecific competition and in [19] cicada dynamics is further explored. Regarding biennials and possible SYC dynamics we refer to [20]. In [21] SYC phenomena and related MYC (multiple year class) dynamics are considered while Kon [22] discusses in a general context conditions for SYC dynamics to occur in matrix models, also compare [23]. However, if we take the opposite approach, constant fecundity term, and nonlinear year-to-year survivals probabilities, SYC dynamics appear to be a rare event; see [16]. Instead, the nontrivial equilibrium is stable whenever the population size is sufficiently small and the nonstationary dynamics has a strong resemblance of 4-cycles, either exact or approximate, the chaotic regime included.

In contrast to most of the papers quoted above, the purpose of this paper is to study the combined effect of nonlinear recruitment and nonlinear survival probabilities in semelparous population models. Several general results about possible SYC dynamics and stability properties of the nontrivial equilibrium in such problems may be obtained in [23] (two age classes) and [24] (three age classes). In case of the more general setting where an arbitrary number of age classes is considered we refer to [25, 26]. Regarding our study, we shall restrict the analysis to the case where both the fecundity and the survival elements depend on the total population and, moreover, the functional form of these elements is of Ricker type. As we prove, under such restrictions, it is possible to obtain explicit thresholds for secondary bifurcations of flip and Neimark−Sacker type and we may also prove (in some cases) that bifurcations involved are of supercritical nature. We may also investigate how the dynamics reported earlier will change as more density-dependent terms are included. For example, given SYC cycles, will the cycles persist if the strength of density dependence in the survival terms is included? If not, what kind of qualitative dynamics is it then possible to obtain? Assuming cycles of low period where all age classes are populated, does the inclusion of density dependence in the fecundity terms act in a stabilizing or destabilizing fashion? Such questions are difficult to address in the general models presented in [24–26].

The paper is organized in the following way. In Section 2 we present the model, compute equilibria, and derive the th order eigenvalue equation which we need in order to perform stability and bifurcation analysis. This is followed (Sections 3 and 4) by a rigorous analysis of possible dynamic outcomes in two and three age class models, respectively. Finally, in Section 5, we unify and discuss results when the number of age classes exceeds three.

#### 2. The Model, Fixed Points, and Stability

First we establish the model. At time we split the population into distinct nonoverlapping age classes where the total population is given by . Next, we introduce the transition matrixwhere is the average fecundity of a member of the last age class at time . denotes the survival probabilities, the (year-to-year) survival from age class to . In contrast to most papers the assumption here is that both the fecundity and the survivals are nonlinear terms. Thus, we write as where the constant and the survivals as where the constant satisfies . The parameters , may be regarded as parameters that measure the strength of density dependence. The relation between at two consecutive time steps is then expressed aswhich may also be formulated as a nonlinear map of the formBesides the trivial fixed point maps (2a) and (2b) also possess a unique nontrivial point . The latter may be expressed asThe quantities and are defined aswhere and is assumed throughout the paper in order to have a feasible equilibrium. The total equilibrium population is given asIn order to investigate stability we linearize about the fixed point. This gives birth to the th order eigenvalue equationwhere the coefficients satisfy is a locally stable hyperbolic fixed point as long as all eigenvalues of (6) are located inside the unit circle in the complex plane.

There are three ways in which may fail to be stable. It may lose its hyperbolicity when crosses the unit circle through 1 which in the general case leads to a saddle node bifurcation, alternatively through −1 which gives birth to a flip (period doubling) bifurcation, or it may fail to be hyperbolic as a pair of complex-valued eigenvalues cross the unit circle. Then a Neimark−Sacker bifurcation occurs. The Jury criteria, see the book by Murray [27], provide conditions for all eigenvalues to satisfy .

#### 3. Two Age Classes

Let in maps (2a) and (2b). Then we have, , and . Moreover, the fixed point becomesand the eigenvalue equation may be cast in the formwhere and .

Fixed point (9) is stable whenever the Jury criteria , , and hold, that is, as long asrespectively.

There are two cases to consider: (A) the case , which means that the strength of density dependence in the fecundity is stronger than the strength of density dependence in the survival, and (B) the case .

Considering (A), it is clear from (11b) that there does not exist any stable fixed point. Moreover, since (11b) fails as an eigenvalue crosses the unit circle through −1 it is natural to search for a 2-cycle which should be stable provided is small. Evidently, such a 2-cycle must be obtained fromand here there are two possibilities:(1) which leads to the trivial 2-cycle where the unstable fixed point is the only point the cycle.(2)The points are on the form or . In this case it follows from (12) that and must satisfy the equationsand by finding from (13b) and substitute back into (13a) we arrive atGeometrically, it is now easy to see that the graph of the left hand side of (13c) and that of the right side have a unique intersection point lying in the first (positive) quadrant. In the special case we obtain and . Hence, there exists a 2-cycle on the form (SYC form)and as shown in [4] this cycle is stable in case of small ( fixed). In Figure 1(a) we show an orbit starting at which settles on the 2-cycle (14). If we continue to increase , we find that (14) goes unstable and cycles of period 2^{k}. , are established through successive flip bifurcations. These cycles, which are all on SYC form, are stable in smaller and smaller regions as is increased. Eventually, the dynamics becomes chaotic but we emphasize that it is on SYC form also in the chaotic regime. These scenarios are demonstrated in Figures 1(b) and 1(c). Actually, we have not accounted for what happens when (or ). Here, compare (10), the eigenvalues are 1 and −1, respectively, and both the positive equilibrium and the SYC 2-cycle bifurcate forward. For proofs and details we refer to [4, 23–26].