Discrete Dynamics in Nature and Society

Volume 2018, Article ID 1613709, 12 pages

https://doi.org/10.1155/2018/1613709

## The Bifurcation of Two Invariant Closed Curves in a Discrete Model

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Correspondence should be addressed to Yicang Zhou; nc.ude.utjx@cyuohz

Received 2 March 2018; Accepted 24 April 2018; Published 30 May 2018

Academic Editor: Guang Zhang

Copyright © 2018 Yingying Zhang and Yicang Zhou. 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 population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

#### 1. Introduction

Differential equations and difference equations are generally applied in mathematical biology modeling. Compared with differential equations, there are less different equations in ecological or epidemiological modeling; the main reason is the incompleteness of theories and methodologies for dynamical studies of discrete models. However, there are increasing attentions and research results on discrete models recently. One major reason is that discrete models as computational models are more efficient for numerical simulations and research results exhibit more richer dynamics than continuous ones. Another reason is that difference equations are more realistic and intelligible to characterize the population size of a single species when it has a fixed interval between generations or measurements.

Up to now, the researches on discrete models are still focused on the dynamical behaviors (including stability, periodic solutions, bifurcations, chaos, and chaotic control; see [1–6]), and most of the scholars have studied the Neimark-Sacker bifurcation, a codimension-1 bifurcation, which has shown one invariant closed curve bifurcating from an equilibrium point. However, they do not consider further, the same as the phenomenon of limit cycles bifurcation in continuous models, the fact that the discrete models can exhibit the interesting dynamic behavior of two invariant closed curves bifurcating from an equilibrium point. One reason is the heavy computation, and another reason is the lack of proper theories and methods for the discrete models. In this paper we intend to research on this dynamic behavior in a discrete population model by using the bifurcation theory and normal form method; it is a fresh attempt different from previous related works. Meanwhile, we also propose a new research aspect by this work.

We consider a continuous of Kolmogorov model [8–10] given bywhere , represent the sizes of the prey and the predator population, respectively. It is assumed that the prey admits the Logistic growth if no predator exists. is the intrinsic growth rate; is the carrying capacity of the environment; is the minimum of viable population; is the maximal per capita consumption rate; is the number of prey necessary to achieve one-half of the maximum rate ; is the conversion efficiency of the consumed prey into new predator; is the natural death rate of predators when predator [11]. From the biological meaning all parameters are assumed to be positive and .

The change of variables and the time rescaling given by the function [12, 13]turns model (1) into a polynomial systemwhere , , for , and The new parameters satisfy the following conditions:It is clear that , and is a diffeomorphism [14].

We apply the forward Euler discrete scheme to system (3) and obtain the following discrete model:

#### 2. The Existence and Stability of Equilibria

The equilibrium point of (5) satisfies following equations:There are four equilibrium points; it is easy to get that they are , and , whereThe existence of the unique positive equilibrium point requires that . The Jacobian matrix of model (5) at equilibrium point iswhere

Let and be the two eigenvalues of matrix ; we use the following definitions and conclusions [15–17]:

If and , then is called a sink, and it is locally asymptotical stable.

If and , then is called a source, and it is unstable.

If and (or and ), then is called a saddle, and it is unstable.

If and (or and ), then is called nonhyperbolic, and the corresponding bifurcation may occur.

Lemma 1. *The eigenvalue and the stability of the trivial fixed points are as follows [1]:*(1)*The eigenvalues of are . (1.1) is a sink if and .(1.2) is a source if and .(1.3) is a saddle if and .(1.4) is nonhyperbolic, and there may be a flip bifurcation, if and (or and ). (1.5) is nonhyperbolic, and there may be a strong 1 : 2 resonance bifurcation*

*if and .*(2)

*The eigenvalues of are .(2.1)*(3)

*is a source if .*(2.2)*is a saddle if .*(2.3)*is nonhyperbolic, and there may be a flip bifurcation if .**The eigenvalues of are .(3.1)*

*is a source if .*(3.2)*is a saddle if .*(3.3)*is nonhyperbolic, and there may be a flip bifurcation if .**The Jacobian matrix of model (5) at the positive fixed point is*

*So then, the characteristic polynomial of the Jacobian matrix iswhereIt is easy to see that and *

*Lemma 2. The stability of the positive equilibrium point is as follows [1]:(1) is a sink if and (or and ).(2) is a source if one of the following conditions holds:(2.1) and .(2.2).(2.3).(3) is a saddle if and .(4) is nonhyperbolic, and there may be a flip bifurcation, if .(5) is nonhyperbolic, and there may be a N-S bifurcation if .*

*For , , and , the conclusions of Lemmas 1 and 2 are shown in plane (Figure 1).*