# Stability and bifurcation analysis for a single-species discrete model with stage structure

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## Abstract

In this paper, a single-species discrete model with stage structure is investigated. By analyzing the corresponding characteristic equations, the local asymptotic stability of non-negative equilibrium points and the existence of flip bifurcation are discussed. Using the center manifold theory, the stability of the non-hyperbolic equilibrium point is obtained. Based on bifurcation theory, we obtain the direction and the stability of a flip bifurcation at the positive equilibrium with the birth rate as the bifurcation parameter. Finally, some numerical simulations, including phase portraits, chaotic bands with period windows, and Lyapunov exponent methods, are performed to validate the theoretical results, which extends the results in previous papers.

## Keywords

Discrete model Stage structure Flip bifurcation Non-hyperbolic## 1 Introduction

It is generally known that in the theory of population dynamics, there are two kinds of mathematical models: the continuous models governed by differential equations and the discrete models governed by difference equations. When the population size is small or when births and deaths all occur at discrete times, discrete models would be more appropriate than the continuous models. Meanwhile, discrete models give rise to more efficient computational models for numerical simulations and these results exhibit rich dynamics of the discrete models compared with the continuous ones. In fact, even in one-dimensional discrete models denoted by iterated quadratic maps, like the well-known logistic map, periodic and chaotic trajectories can easily be observed. Moreover, two-dimensional discrete models can reveal a plethora of complicated asymptotic behaviors, from convergence to a fixed point or a periodic cycle until a quasi-periodic orbit along a closed invariant curve or even an erratic motion inside a bounded chaotic attractor (see e.g. [1, 2, 3, 4, 5]).

*n*, respectively. \(d_{1}\) and \(d_{2}\) (\(d_{i}>0\), \(i=1,2\)) are the death rate constants of immature and mature, respectively. The parameters

*b*and

*δ*have the same biological interpretations as those in model (1). In model (2), only the mature population have reproductive ability. Let \(\alpha=1-d_{1}-\delta,\beta=1-d_{2}\). For ecological reasons, we assume that \(0<\alpha<1\), \(0<\beta<1\), \(\delta\neq\beta\). Substituting

*α*and

*β*in model (2), we obtain the following model:

To date no paper has appeared in the literature which discusses the bifurcation problem for model (3). Our work mainly focuses on the stability of trivial equilibrium point when it is non-hyperbolic, i.e., \(b\delta=(1-\alpha)(1-\beta)\), the stability of positive equilibrium point, the existences and directions of the flip bifurcation of positive equilibrium point with the birth rate *b* as bifurcation parameter by applying the bifurcation theory.

The paper is organized as follows. In Sect. 2, we analyze the distribution of characteristic roots associated with model (3), and obtain the stability of equilibrium points, especially, the stability of non-hyperbolic equilibrium point. In Sect. 3, the direction and stability of flip bifurcations for model (3) are determined. In Sect. 4, some numerical simulations are performed to illustrate the theoretical results. A brief discussion is given in Sect. 5.

## 2 Stability of equilibrium points

In this section, we study the distribution of characteristic roots of model (3) by employing the relation between roots and coefficients of the quadratic equation, and discuss the stability of non-hyperbolic equilibrium point by center manifold theorem.

In order to analyze the stability of equilibria for model (3), we give the following lemma (see [19]), which can easily be proved by the relation between roots and coefficients of the characteristic equation (5) of model (3). Denote by \(\vert \lambda \vert \) the modulus of *λ*.

### Lemma 2.1

*Let*\(F(\lambda)=\lambda^{2}-\widehat{B}\lambda+\widehat{C}\).

*Suppose that*\(F(1)>0\), \(\lambda_{1}\)

*and*\(\lambda_{2}\)

*are the two roots of*\(F(\lambda)=0\).

*Then*

- (i)
\(\vert \lambda_{1} \vert <1\)

*and*\(\vert \lambda_{2} \vert <1\)*if and only if*\(F(-1)>0\)*and*\(\widehat{C}<1\). - (ii)
\(\vert \lambda_{1} \vert <1\)

*and*\(\vert \lambda_{2} \vert >1\) (*or*\(\vert \lambda_{1} \vert >1\)*and*\(\vert \lambda_{2} \vert <1\))*if and only if*\(F(-1)<0\). - (iii)
\(\vert \lambda_{1} \vert >1\)

*and*\(\vert \lambda_{2} \vert >1\)*if and only if*\(F(-1)>0\)*and*\(\widehat{C}>1\). - (iv)
\(\lambda_{1}=-1\)

*and*\(\vert \lambda_{2} \vert \neq1\)*if and only if*\(F(-1)=0\)*and*\(\widehat{C}\neq1\). - (v)
\(\lambda_{1}\)

*and*\(\lambda_{2}\)*are complex and*\(\vert \lambda _{1} \vert = \vert \lambda_{2} \vert =1\)*if and only if*\(\vert \widehat{B} \vert <2\)*and*\(\widehat{C}=1\).

In the following, we give the stability of trivial equilibrium point \(E_{0}\).

Note that if \(b\delta=(1-\alpha)(1-\beta)\), then \(\lambda_{1}=1\), \(\lambda _{2}=\alpha+\beta-1\). In this case, \(E_{0}\) is non-hyperbolic. However, due to \(-1<\lambda_{2}=\alpha+\beta -1<1\), it is difficult to determine the stability of \(E_{0}\). Now, by the center manifold theorem, we will analyze the stability for \(E_{0}\) when \(b\delta=(1-\alpha)(1-\beta)\).

Stability of trivial equilibrium point \(E_{0}\)

Conditions | Distribution of roots of Eq. (6) | Topological types of \(E_{0}\) |
---|---|---|

| \(-1<\lambda_{2}<\lambda_{1}<1\) | sink, stable |

| \(-1<\lambda_{2}=\alpha+\beta-1<\lambda _{1}=1\) | non-hyperbolic, stable |

(1 − | \(-1<\lambda_{2}<1<\lambda_{1}\) | saddle, unstable |

| \(-1=\lambda_{2}<1<\lambda_{1}=\alpha+\beta+1\) | non-hyperbolic, unstable |

| \(\lambda_{2}<-1<1<\lambda_{1}\) | source, unstable |

Case 1: \(\delta<\beta\). It is obvious that \(m_{2}<1<m_{1}\).

By Lemma 2.1, we have the following results as regards the stability of the equilibrium point \(E^{\ast}\).

### Theorem 2.2

*Suppose that*\(b\delta>(1-\alpha)(1-\beta)\).

- (i)
*If*\(b\delta<(1-\alpha)(1-\beta)m_{1}\),*then*\(F(-1)>0\)*and*\(\mathrm{Det}(J(E^{\ast}))<1\).*By Lemma*2.1,*we have*\(\vert \lambda_{1} \vert <1\)*and*\(\vert \lambda_{2} \vert <1\).*Therefore*, \(E^{\ast}\)*is a sink*. - (ii)
*If*\(b\delta>(1-\alpha)(1-\beta)m_{1}\),*then*\(F(-1)<0\).*By Lemma*2.1,*we have*\(\vert \lambda_{1} \vert <1\)*and*\(\vert \lambda_{2} \vert >1\) (*or*\(\vert \lambda_{1} \vert >1\)*and*\(\vert \lambda_{2} \vert <1\)).*Therefore*, \(E^{\ast}\)*is a saddle*. - (iii)
*If*\(b\delta=(1-\alpha)(1-\beta)m_{1}\)*lem*1,*we have*\(\lambda_{1}=-1\)*and*\(\vert \lambda_{2} \vert \neq1\).*Therefore*, \(E^{\ast}\)*is non*-*hyperbolic*.

Denote \(M_{1}=\{(b,\delta,\alpha,\beta)| b\delta=(1-\alpha)(1-\beta )m_{1}\}\). From the above analysis, it follows that if \((b,\delta,\alpha,\beta)\in M_{1}\), then one of the two eigenvalues of the equilibrium \(E^{\ast}\) is −1 and the other is neither 1 nor −1. Therefore, model (3) may undergo flip bifurcation at \(E^{\ast}\) if the parameters vary in the small neighborhood of \(M_{1}\).

## 3 Flip bifurcation

In this section, we choose the birth rate *b* as a bifurcation parameter to study the flip bifurcation of the equilibrium point \(E^{\ast}\) by using the bifurcation approach in [22].

*A*has a simple critical eigenvalue \(\lambda_{1}=-1\), and the corresponding critical eigenspace \(T^{C}\) is one-dimensional and spanned by an eigenvector \(q\in\mathbb{R}^{2}\) such that \(Aq=-q\). Let \(p\in\mathbb{R}^{2}\) be the adjoint eigenvector, that is, \(A^{T}p=-p\), where \(A^{T}\) is the transposed matrix. Normalize

*p*with respect to

*q*such that \(\langle p, q\rangle=1\), where \(\langle p, q\rangle =p_{1}q_{1}+p_{2}q_{2}\) is the scalar product in \(\mathbb{R}^{2}\). For satisfying the normalization \(\langle p, q\rangle=1\), we choose

*I*is the unit \(2\times2\) matrix.

Thus, we can obtain the stability and direction of flip bifurcation at the equilibrium \(E^{\ast}\) as follows.

### Theorem 3.1

*If*\({\gamma}\neq0\), *then model* (3) *undergoes a flip bifurcation at the equilibrium*\(E^{\ast}\)*when the parameter**b**varies and passes through*\(b^{\ast}\). *Moreover*, *if*\({\gamma}>0\) (*respectively*, \({\gamma}<0\)), *then the flip bifurcation of model* (3) *at*\(b=b^{\ast}\)*is supercritical* (*respectively*, *subcritical*) *and the period*-*doubling cycle is stable* (*respectively*, *unstable*).

## 4 Numerical simulations

In this section we will give examples to illustrate the analytic results.

*b*in the range \(12\leq b\leq150\). As

*b*increases beyond \(b^{\ast}\), model (3) passes through a series of bifurcations that eventually lead to chaotic dynamics. Figure 3 shows bifurcation diagrams for model (3). After the flip bifurcation, the model undergoes a series of period-doubling bifurcations wherein \(2^{k}\)-cycle loses stability and a stable \(2^{k+1}\)-cycle is born as

*b*increases. Subsequently, there is a cascade of period-doubling bifurcations leading to chaos. The Lyapunov exponent diagram corresponding to Fig. 3 is given in Fig. 4. Figure 5 shows that when \(b=120\), model (3) possesses two chaotic sets.

## 5 Conclusion and discussion

There has been much work discussing the stability and bifurcation of single-species model with stage structure, but most of them dealt with only the continuous system. In this paper, we discuss the dynamical behaviors of a single-species discrete model with the Ricker function as the birth rate and only the mature population reproducing. By analyzing the location of the roots in the characteristic equation, some conditions are derived to ensure the asymptotically stable of equilibria of model (3). In [18], the authors provided the result that \(E^{\ast}\) is asymptotically stable if \(b\delta>(1-\alpha)(1-\beta)\). However, according to Theorem 2.2, we see that \(b\delta>(1-\alpha)(1-\beta )\) is not the sufficient condition under which \(E^{\ast}\) is asymptotically stable. Moreover, by the results of [17], we find the interesting phenomenon that \(E^{\ast}\) of the continuous model corresponding to model (3) is globally asymptotically stable if \(b\delta>(1-\alpha )(1-\beta)\).

As pointed out by [17], \(R_{0}=\frac{b\delta}{(1-\alpha)(1-\beta )}\) represents the intrinsic net reproductive number, which combines the age-specific fertility rates and the age-specific survival rates and gives the expected number of offspring per individual over its life time. \(R_{0}=1\) is a threshold which controls whether or not the population will survive. From Table 1, if \(R_{0}\leq1\), then the trivial equilibrium \((0,0)\) is asymptotically stable, that is, if on average individuals do not replace themselves before they die, then the population becomes extinct. From Theorem 2.2, if \(1< R_{0}< m_{1}\), then the positive equilibrium point \(E^{\ast}\) is asymptotically stable. It means that the population can survive over the long term. If \(R_{0}=m_{1}\), the positive equilibrium point \(E^{\ast }\) is non-hyperholic, that is, it may lose stability. Moreover, as the birth rate *b* increases, the positive equilibrium of model (3) exchanges its stability and occurs flip bifurcation. In fact, we see that for the continuous model corresponding to model (3) does not occur the bifurcation at the equilibrium \(E^{\ast}\). Furthermore, the direction and stability of flip bifurcation are determined. Numerical simulations also show the rich dynamical behavior of model (3), including cascades of period-doubling bifurcations in orbit of period 2, 4, 8 and chaotic sets and invariant circles.

## Notes

### Acknowledgements

The work is partially supported by the National Natural Science Foundation of China (No. 31570417), and by the Natural Science Foundation of Anhui Province of China (Nos. 1608085MA14, 1608085MC63), and by the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (Nos. KJ2015A152, gxyqZD2016205).

### Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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