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Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response
- Jianglin Zhao1Email author and
- Yong Yan1
https://doi.org/10.1186/s13662-018-1819-0
© The Author(s) 2018
- Received: 22 June 2018
- Accepted: 26 September 2018
- Published: 30 October 2018
Abstract
In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.
Keywords
- Discrete-time predator–prey system
- Flip bifurcation
- Hopf bifurcation
- Chaos
1 Introduction
The outline of this paper is as follows. In Sect. 2, we investigate in detail the existence and local stability of fixed points of model (3). In Sect. 3, we derive sufficient conditions for the existence of flip bifurcation and Hopf bifurcation. In Sect. 4, we present numerical simulations to check our results of theoretical analysis and exhibit some complex and new dynamical behaviors. In the end, we give a brief conclusion in Sect. 5.
2 Existence and stability of fixed points
From a biological point of view, system (3) must have positive values of u and v. We have the following result.
Theorem 2.1
Assume that \(\Omega = \{ (u,v)|u > 0,v > 0,r \delta u + \beta \delta v - (1 + r\delta) < 0\}\). Then Ω is an invariant set for system (3) if δ and s are enough small.
Proof
Let \(L(u_{0}) = - r^{2}\delta^{2}u_{0}^{2} + r\delta (1 + r\delta)u _{0} + \delta (1 + r\delta)H\). Then \(L(u_{0}) = \delta (1 + r\delta)H < 0\).
Therefore, \(L < 0\) if δ is small enough. Then \((u_{1},v_{1}) \in \Omega \) if δ and s are small enough, that is, Ω is an invariant set for system (3).
Lemma 2.1
Lemma 2.2
([11])
- (i)
\(\vert \lambda_{1} \vert < 1\) and \(\vert \lambda_{2} \vert < 1\) if and only if \(F( - 1) > 0\) and \(Q < 1\);
- (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 \(Q > 1\);
- (iv)
\(\lambda_{1} = - 1\) and \(\vert \lambda_{2} \vert \ne 1\) if and only if \(F( - 1) = 0\) and \(P \ne 0\), 2;
- (v)
\(\lambda_{1}\) and \(\lambda_{2}\) are the conjugate complex roots and \(\vert \lambda_{1} \vert = \vert \lambda_{2} \vert = 1\) if and only if \(P^{2} - 4Q < 0\) and \(Q = 1\).
Suppose that \(\lambda_{1}\) and \(\lambda_{2}\) are two roots of (5), which are called eigenvalues of the fixed point \((u,v)\). The point \((u,v)\) is a sink if \(\vert \lambda_{1} \vert < 1\) and \(\vert \lambda_{2} \vert < 1\). A sink is locally asymptotic stable. The point \((u,v)\) is a source if \(\vert \lambda_{1} \vert > 1\) and \(\vert \lambda_{2} \vert > 1\). A source is locally unstable. The point \((u,v)\) is a saddle if \(\vert \lambda_{1} \vert > 1\) and \(\vert \lambda_{2} \vert < 1\) (or \(\vert \lambda_{1} \vert < 1\) and \(\vert \lambda _{2} \vert > 1\)), and \((u,v)\) is nonhyperbolic if either \(\vert \lambda_{1} \vert = 1\) or \(\vert \lambda_{2} \vert = 1\).
Theorem 2.2
- (i)
\(A(1,0)\) is a saddle if \(0 < \delta < \frac{2}{r}\);
- (ii)
\(A(1,0)\) is a source if \(\delta > \frac{2}{r}\);
- (iii)
\(A(1,0)\) is nonhyperbolic if \(\delta = \frac{2}{r}\).
Theorem 2.3
- (i)\(B(u_{*},v_{*})\) is a sink if one of the following conditions holds:
- (i.1)
\(- 2\sqrt{Hs} < G < 0\) and \(0 < \delta < - \frac{G}{Hs}\);
- (i.2)
\(G < - 2\sqrt{Hs}\) and \(0 < \delta < \frac{ - G - \sqrt{G^{2} - 4Hs}}{Hs}\);
- (i.1)
- (ii)\(B(u_{*},v_{*})\) is a source if one of the following conditions holds:
- (ii.1)
\(- 2\sqrt{Hs} < G < 0\) and \(\delta > - \frac{G}{Hs}\);
- (ii.2)
\(G < - 2\sqrt{Hs}\) and \(\delta > \frac{ - G + \sqrt{G^{2} - 4Hs}}{Hs}\);
- (ii.3)
\(G \ge 0\);
- (ii.1)
- (iii)\(B(u_{*},v_{*})\) is a saddle if the following conditions hold:$$ G < - 2\sqrt{Hs} \quad \textit{and}\quad \frac{ - G - \sqrt{G^{2} - 4Hs}}{Hs} < \delta < \frac{ - G + \sqrt{G ^{2} - 4Hs}}{Hs}; $$
- (iv)\(B(u_{*},v_{*})\) is nonhyperbolic if one of the following conditions holds:
- (iv.1)
\(G < - 2\sqrt{Hs}\) and \(\delta = \frac{ - G \pm \sqrt{G^{2} - 4Hs}}{Hs}\) and \(\delta \ne - \frac{2}{G}, - \frac{4}{G}\);
- (iv.2)
\(- 2\sqrt{Hs} < G < 0\) and \(\delta = - \frac{G}{Hs}\).
- (iv.1)
From the Jury criterion and the preceding analysis it can be easily seen that one of the eigenvalues of the unique positive fixed point \(B(u_{*},v_{*})\) is −1 and the other is neither 1 nor −1 if (iv.1) of Theorem 2.3 holds. When (iv.2) of Theorem 2.3 is true, the eigenvalues of the unique positive fixed point \(B(u_{*},v_{*})\) are a pair of conjugate complex numbers with modulus one.
3 Flip bifurcation and Hopf bifurcation
Based on the previous analysis, in this section, we mainly focus on the flip bifurcation and Hopf bifurcation of the unique positive fixed point \(B(u_{*},v_{*})\). Then, we choose the integral step size δ as a bifurcation parameter for investigating the Flip bifurcation and Hopf bifurcation of \(B(u_{*},v_{*})\) by using the center manifold theorem and bifurcation theory.
We first discuss the flip bifurcation of system (3) at \(B(u_{*},v_{*})\) when the parameters vary in a small neighborhood of \(F_{B1}\). Similar arguments can be applied to the other case of \(F_{B2}\).
Let \(\alpha_{1} = ( \frac{\partial^{2}F}{\partial \tilde{u}\partial \delta_{*}} + \frac{1}{2}\frac{\partial F}{\partial \delta_{*}}\frac{\partial^{2}F}{\partial \tilde{u}^{2}} ) \vert _{(0,0)} = h_{2}\) and \(\alpha_{2} = ( \frac{1}{6}\frac{ \partial^{3}F}{\partial \tilde{u}^{3}} + ( \frac{1}{2}\frac{ \partial^{2}F}{\partial \tilde{u}^{2}} ) ^{2} ) \vert _{(0,0)} = h_{5} + h_{1}^{2}\).
Then we have the following results.
Theorem 3.1
If \(\alpha_{1} \ne 0\) and \(\alpha_{2} \ne 0\), then the map (3) undergoes a flip bifurcation at the unique positive fixed point \(B(u_{*},v_{*})\) when the parameter δ varies in a small neighborhood of \(F_{B1}\). Moreover, if \(\alpha_{2} > 0\) (resp., \(\alpha_{2} < 0\)), then the period-2 orbits that bifurcate from \(B(u_{*},v_{*})\) are stable (resp., unstable).
From the preceding analysis and theorem in [20] we have the following result.
Theorem 3.2
If condition (19) holds and \(\gamma \ne 0\), then the map (3) undergoes a Hopf bifurcation at the unique positive fixed point \(B(u_{*},v_{*})\) when the parameter δ varies in a small neighborhood of \(H_{B}\). Moreover, if \(\gamma < 0\) (resp., \(\gamma > 0\)), then an attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for \(\delta > \delta_{2}\) (resp., \(\delta > \delta_{2}\)).
4 Numerical simulations
- (i)
Varying δ in the range \(1 \le \delta \le 1.4\) and fixing \(r = 2\), \(\beta = 0.5\), \(a = 0.2\), \(m = 0.2\), \(s = 0.6\), \(h = 2\).
- (ii)
Varying δ in the range \(2 \le \delta \le 3\) and fixing \(r = 1\), \(\beta = 0.5\), \(a = 0.2\), \(m = 0.2\), \(s = 0.6\), \(h = 2\).
- (iii)
Varying r in the range \(2 \le r \le 2.99\) and fixing \(\delta = 1\), \(\beta = 0.5\), \(a = 0.2\), \(m = 0.2\), \(s = 0.6\), \(h = 2\).
(a) Flip bifurcation diagram of map (3) in the \((\delta,u)\) plane with initial value \((0.9510621, 0.29231863)\). (b) Flip bifurcation diagram of map (3) in the \((\delta,v)\) plane. (c) Maximum Lyapunov exponents corresponding to (a) and (b). (d) Local amplification of (c) for \(\delta \in [1.39,1.4]\)
Phase portraits for various values of δ corresponding to Figs. 1(a), (b)
(a) Hopf bifurcation diagram of map (3) in the \((\delta,u)\) plane with initial value \((0.8933954475, 0.2750186342)\). (b) Hopf bifurcation diagram of map (3) in the \((\delta,v)\) plane. (c) Maximum Lyapunov exponents corresponding to Figs. 3(a), (b). (d) Local amplification diagram of (a) for \(\delta \in [2.75,2.9]\)
Phase portraits for various values of δ corresponding to Figs. 3(a), (c)
5 Conclusion
In this paper, we have mainly considered the complex behaviors of a predator–prey system with modified Holling–Tanner functional response in R2. By using the center manifold theorem and bifurcation theory we prove that the unique positive fixed point of system (3) can undergo flip bifurcation and Hopf bifurcation. Most importantly, when the integral step size δ is chosen as a bifurcation parameter, numerical simulations show that system (3) shows very rich nonlinear dynamical behaviors including stable coexistence, period-doubling bifurcation leading to chaos, attracting invariant circles, and even stranger chaotic attractors. According to Figs. 1 and 2, we can observe that the small integral step size δ can stabilize the dynamical system (3), but the large integral step size may destabilize the system producing more complex dynamical behaviors. Then it reminds of us that the integral step size may play a key role in exploring the dynamical behaviors. In addition, from Fig. 5 we can see that the appropriately intrinsic growth rate r of prey can stabilize the dynamical system (3). However, the high intrinsic growth rate may destabilize system (3). From a biological point of view, when the prey population is submitted to the high intrinsic growth rate, the number of preys are abundant, and the prey consumption by predator may have a marginal effect on the dynamics of prey. Hence, the dynamical behavior of prey mainly depends on the population itself. Then, system (3) becomes the classic logistic model and exhibits the period-doubling leading to chaos.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (no. 11461058) and by Sichuan Minzu College (nos. XYZB17001 and sfkc201701).
Authors’ contributions
JZ is responsible for the model formulation and study planning. JZ and YY have done the calculation, the proof, and the simulation. Both authors have read and approved the final manuscript.
Competing interests
Both authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Berryman, A.A.: The origins and evolution of predator–prey theory. Ecology 73, 1530–1535 (1992) View ArticleGoogle Scholar
- Shi, H., Li, W., Lin, G.: Positive steady states of a diffusive predator–prey system with modified Holling–Tanner functional response. Nonlinear Anal., Real World Appl. 11, 3711–3721 (2010) MathSciNetView ArticleGoogle Scholar
- Yang, W.: Global asymptotical stability and persistent property for a diffusive predator–prey system with modified Leslie–Gower functional response. Nonlinear Anal., Real World Appl. 14, 1323–1330 (2013) MathSciNetView ArticleGoogle Scholar
- Zhang, L., Zhang, C., Zhao, M.: Dynamic complexities in a discrete predator–prey system with lower critical point for the prey. Math. Comput. Simul. 105, 119–131 (2014) MathSciNetView ArticleGoogle Scholar
- Choo, S.: Global stability in n-dimensional discrete Lotka–Volterra predator–prey models. Adv. Differ. Equ. 2014, 11 (2014) MathSciNetView ArticleGoogle Scholar
- Cao, H., Zhou, Y., Ma, Z.: Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Math. Biosci. Eng. 10, 1399–1417 (2013) MathSciNetView ArticleGoogle Scholar
- Ren, J., Yu, L., Siegmund, S.: Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response. Nonlinear Dyn. 90, 19–41 (2017) MathSciNetView ArticleGoogle Scholar
- Huang, T., Zhang, H., Yang, H., Wang, N., Zhang, F.: Complex patterns in a space- and time-discrete predator-prey model with Beddington-DeAngelis functional response. Commun. Nonlinear Sci. 43, 182–199 (2017) MathSciNetView ArticleGoogle Scholar
- Huang, J., Liu, S., Ruan, S., Xiao, D.: Bifurcations in a discrete predator–prey model with nonmonotonic functional response. J. Math. Anal. Appl. 464, 201–230 (2018) MathSciNetView ArticleGoogle Scholar
- Din, Q.: Complexity and chaos control in a discrete-time prey–predator model. Commun. Nonlinear Sci. 49, 113–134 (2017) MathSciNetView ArticleGoogle Scholar
- Cheng, L., Cao, H.: Bifurcation analysis of a discrete-time ratio-dependent predator–prey model with Allee effect. Commun. Nonlinear Sci. Numer. Simul. 38, 288–302 (2016) MathSciNetView ArticleGoogle Scholar
- Liu, X., Xiao, D.: Complex dynamic behaviors of a discrete-time predator–prey system. Chaos Solitons Fractals 32, 80–94 (2007) MathSciNetView ArticleGoogle Scholar
- Cui, Q., Zhang, Q., Qiu, Z., Hu, Z.: Complex dynamics of a discrete-time predator-prey system with Holling IV functional response. Chaos Solitons Fractals 87, 158–171 (2016) MathSciNetView ArticleGoogle Scholar
- Jana, D.: Chaotic dynamics of a discrete predator–prey system with prey refuge. Appl. Math. Comput. 224, 848–865 (2013) MathSciNetMATHGoogle Scholar
- Asheghi, R.: Bifurcations and dynamics of a discrete predator–prey system. J. Biol. Dyn. 8, 161–186 (2014) MathSciNetView ArticleGoogle Scholar
- He, Z., Lai, X.: Bifurcation and chaotic behavior of a discrete-time predator–prey system. Nonlinear Anal., Real World Appl. 12, 403–417 (2011) MathSciNetView ArticleGoogle Scholar
- Liu, X., Liu, Y., Li, Q.: Multiple bifurcations and chaos in a discrete prey–predator system with generalized Holling III functional response. Discrete Dyn. Nat. Soc. 2015, 1–10 (2015) MathSciNetGoogle Scholar
- Zhuo, X., Zhang, F.: Stability for a new discrete ratio-dependent predator–prey system. Qual. Theory Dyn. Syst. 17, 189–202 (2018) MathSciNetView ArticleGoogle Scholar
- Carr, J.: Application of Center Manifold Theorem. Springer, New York (1981) View ArticleGoogle Scholar
- Robinson, C.: Dynamical Systems, Stability, Symbolic Dynamics and Chaos, 2nd edn. CRC Press, Boca Raton (1999) MATHGoogle Scholar