Discrete Dynamics in Nature and Society

Volume 2016, Article ID 9724139, 13 pages

http://dx.doi.org/10.1155/2016/9724139

## Border Collision Bifurcations in a Generalized Model of Population Dynamics

^{1}Department of Mathematics and Physics, University of Los Llanos, 500001 Villavicencio, Colombia^{2}Department of Economics and Law, University of Macerata, 62100 Macerata, Italy^{3}Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain

Received 21 December 2015; Revised 24 February 2016; Accepted 22 March 2016

Academic Editor: Xiaohua Ding

Copyright © 2016 Lilia M. Ladino et al. 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

We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.

#### 1. Introduction

Mathematical systems modeling natural phenomena usually depend on parameters related to their behavior. The determination of the essential parameters and their possible values is fundamental not only for the design of an adequate model, but also for the prediction of the evolution of these phenomena in the future.

The dynamics of a system can change drastically as the parameters vary, providing different kinds of evolution. Such changes in the dynamics are known as* bifurcations* and they have become a very interesting subject in the study of dynamical systems, a field in which many researchers have worked in the last years (see, e.g., Kuznetsov [1] or Balibrea et al. [2], Yuan et al. [3], Franco and Perán [4], and references therein).

Actually, these parameters can force the design of the model in order to maintain the empirical meaning, providing piecewise systems (see Simpson [5] for a wider description of piecewise smooth systems and the related bifurcations).

Piecewise smooth dynamical systems are of great interest in many areas of applied science since they show a large variety of nonlinear phenomena including chaos. While there is a complete understanding of local bifurcations for smooth dynamical systems, nonstandard bifurcations are likely to emerge in piecewise smooth dynamical systems. An analytical study regarding bifurcations in such kind of systems firstly appeared in Feigin [6]. Later, the results due to Feigin have been formalized within the context of modern bifurcation analysis in Di Bernardo et al. [7]; in that work the effects of such bifurcations are described and the related conditions are pursued. More in detail, when a piecewise smooth system is considered, the exhibited dynamics could vary when an invariant set, for example, a cycle or a fixed point, collides with a switching manifold. When these variations in the dynamics occur, it is said that the system undergoes a* border collision bifurcation*. Many authors have carried out researches on these kinds of bifurcations in the last decades (see, e.g., Nusse and Yorke [8], Brianzoni et al. [9], Simpson and Meiss [10], Agliari et al. [11], and the references therein).

In a previous work [12], we studied a discrete time continuous and differentiable dynamical system in biology, which models the population dynamics of a two-stage species with recruitment and capture. In such work, to be coherent with the biological meaning of the model, the possible values of the essential parameters are determined by two nonnegative constraints under which the dynamics of the discrete model considered coincide exactly with its continuous counterpart analyzed in Ladino and Valverde [13]. In both the discrete and continuous models, the system exhibits a* transcritical* bifurcation while one of the equilibria is a global attractor of the system (alternatively), depending on the value of a threshold parameter which is a function of the key parameters. No richer dynamics are exhibited.

Nevertheless, in literature (e.g., see CCI and INCODER [14]), one can find other approaches in which the parameters considered in the definition of the model do not satisfy the constraints given in Ladino et al. [12]. This issue has motivated us to study, in this work, what happens when the parameters are not restricted by such constraints, thus considering real data for two species. In particular, for nonrestricted parameter values, the (discrete) system can be reformulated by a piecewise nonlinear map for it to continue to be mathematically coherent.

To better explain, in the present contribution we consider the model proposed in Ladino and Valverde [13] (which is a two-dimensional model in continuous time) and we obtain its discrete time formulation by considering the variation of each state variable in a unit time. Even if this is a simplified manner to obtain the discrete counterpart of the initial model, we proceeded in such a way for the following main reasons. First of all, this contribution represents the first step in the study of the dynamics of exploited populations, when time is assumed to be discrete and real data are taken into account; hence we chose to start considering its basic initial formulation. Secondly, the main goal of the present work is to easily compare the results herewith obtained to the ones reached in the equivalent continuous time model. Finally, new and more accurate discrete time setups could be proposed in further developments and thus compared to the present one, in order to conclude about their strength and weakness points when used to describe real situations.

Once obtained the discrete time system to be studied, we explicitly take into account that nonnegativity constraints must be considered. In fact, if at a given time a state variable becomes negative, this means that the recruitment and capture processes have affected the whole related subpopulation, and hence such a subpopulation must be assumed to be equal to zero. Due to the nonnegativity constraints, the final model is described by a continuous two-dimensional piecewise smooth map. Actually, borders may appear in the phase plane where the definition of the dynamic system changes. As a consequence, the approach to the problem requires the use of new techniques from the mathematical theory of piecewise smooth dynamical systems as well as computational support (recent works of this kind are, among others, Kubin and Gardini [15], Banerjee and Grebogi [16], and Simpson [17]).

We recall that piecewise smooth systems are able to exhibit the same dynamics as those produced in smooth systems but, in addition, new phenomena related to the existence of borders may be produced (see Simpson [5]). In fact, it may occur that, when a border is crossed, a different kind of bifurcation that is not related to the eigenvalues associated with a given attractor, called border collision bifurcation, may emerge (Nusse and Yorke [8, 18]). This type of bifurcation is of great relevance from an applied point of view, since the eigenvalues of fixed or periodic points play no role and, consequently, it is more difficult to predict if a system is close to a border collision bifurcation and it is more difficult to predict what happens to the qualitative nature of the attractor after the border collision bifurcation. The latter difficulty is reinforced by the fact that, after a border collision bifurcation, coexisting additional attractors often occur, so that the related basins of attraction have to be considered.

As models in applied mathematics often consider constraints (such as capacity constraints in biology or resource constraints in economics, etc.), piecewise smooth dynamical systems emerge quite naturally in applications and consequently their study has been improved in recent years (see, e.g., Agliari et al. [11], Brianzoni et al. [9], Simpson and Meiss [10], and Sushko et al. [19]). Nevertheless, such works usually focus on the local bifurcations related to periodic points and other attractors, while the global dynamics are mainly described using numerical techniques.

We will follow this approach also in the present paper but, in addition, (1) we will be able to reach some results on the global dynamics of the system and (2) we will apply our findings to two real cases. In fact, for the numerical simulations, we will consider actual data related to the population parameters on the state of fisheries for two fish species, that is,* Prochilodus mariae* and* Prochilodus magdalenae*, which inhabit in the Orinoco and Magdalena rivers of Colombia (CCI and INCODER [14]). As far as the other parameters of the model are concerned, because of the difficulty of finding related serious research publications, we consider the values theoretically estimated in Ladino and Valverde [13]. Although using real data for the parameters would be of great interest, the numerical analysis we perform has the advantage of allowing us to simulate and analyze different scenarios of the feasible biological parametric space.

The paper is organized as follows. In Section 2, we describe the model of population dynamics; in particular, by considering nonnegativity constraints we obtain the final two-dimensional system whose evolution operator is continuous and piecewise smooth. In Section 3, we describe the structure of borders and deal with the question of the existence and local stability of fixed points. In Section 4 the global dynamics is studied. More precisely, we show that the system admits an attractor at finite distance and that the extinction equilibrium is the unique global attractor under certain parametric conditions; we also show that the system undergoes a* border collision bifurcation* in which a 2-period cycle appears and that the model also exhibits a* multistability* phenomenon which plays an important role in the study of the evolution of the system. Section 5 concludes the paper emphasizing the most important features of our research in terms of strategies to be suggested for the conservation of the species.

#### 2. The Model

In Ladino et al. [12], the population dynamics of a two-stage species with recruitment and capture is modeled by the following system of nonlinear difference equations:where all the parameters are nonnegative and verify the following two constraints:in order to have biological significance. However, empirical studies [14] estimated parameter values that do not necessarily verify the abovementioned constrains. For this reason, in this work we present a generalization of system (1), where the parameters do not necessarily verify the constraints above. In this sense, we will need to reformulate the model as a piecewise nonlinear map for it to maintain biological significance.

By taking into account system (1), the two-dimensional system that characterizes the dynamics of a two-stage species with recruitment and capture can be rewritten aswhere , , , and . System is a two-dimensional dynamical system whose iteration defines the time evolution of the prerecruit population and the exploitable population .

First of all, we observe that system (3) is biologically meaningful only when, at any time , the two states variables and belong to .

It is quite immediate to verify that not all trajectories produced by system are feasible for all parameter values. For instance, an initial condition , produces an unfeasible trajectory if or, similarly, a trajectory starting from exits from at the first iteration if .

Nevertheless, it can be observed that, if at a given time one of the two subpopulations becomes negative, that is, or , then this fact implies that, at some earlier time, the subpopulation evolved into its extinction and therefore its size must be assumed to have become equal to zero. More in detail, nonnegativity constraints must be taken into account in order to consider that the natural death rate and the capture mortality rate can affect, at most, the whole stock of a subpopulation.

As a consequence, we can define the following systems:which describe the dynamics of the two subpopulations in the case in which the prerecruit population vanishes or the exploitable population vanishes , or finally, the species becomes extinct .

System (1) can be now reformulated aswhere

Notice that system is not smooth, since its definition changes, though continuously, as any border is crossed in the phase plane , due to the nonnegativity constraints.

#### 3. Fixed Points and Local Stability

##### 3.1. Preliminary Properties

As it has been described, the phase plane is divided into several regions, , and system is defined in different ways inside each of them.

As a first step in the analysis, we want to better describe the structure of such regions on the plane , depending on the parameters of the model. The following lemma can be proved.

Lemma 1. *Let be given by (5) and , as defined in (6). Then the following statements hold: *(i)*if , then and are empty;*(ii)*if , then and are empty.*

*Proof. *(i) Consider a point . Then belongs to or to iff . Let ; then condition cannot hold. Hence we consider the case and observe that iff . Notice that ; furthermore if then and (i.e., is strictly decreasing). As a consequence and the statement is proved.

(ii) Consider a point . Then iff . Notice that and that if then . Assume and consider that . Then, after some algebra, it can be verified that, if , then is strictly decreasing and consequently condition cannot hold. Therefore, there is no point in nor in .

*With the same arguments used in the proof of Lemma 1, it can be easily demonstrated that several situations can occur, depending on the parameters of the model. In particular, the following remark can be easily verified.*

*Remark 2. *Let be given by (5) and as defined in (6). Then, (i)if and , then (see Figure 1(a));(ii)if and , then (see Figure 1(b));(iii)if and , then (see Figure 1(c)).