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Stability analysis of prey-predator system with Holling type functional response and prey refuge
- Zhihui Ma^{1}Email author,
- Shufan Wang^{2},
- Tingting Wang^{1} and
- Haopeng Tang^{1}
https://doi.org/10.1186/s13662-017-1301-4
© The Author(s) 2017
Received: 15 February 2017
Accepted: 2 August 2017
Published: 17 August 2017
Abstract
In this paper, a predator-prey system with Holling type function response incorporating prey refuge is presented. By applying the analytical approaches, the dynamics behavior of the considered system is investigated, including stability, limit cycle and bifurcation. The results show that the shape of the functional response plays an important role in determining the dynamics of the system. Especially, the interesting conclusion is that the prey refuge has a destabilizing effect under some certain conditions.
Keywords
- predator-prey system
- prey refuge
- stability
- limit cycle
- bifurcation
- destabilizing effect
MSC
- 37C75
- 34K18
- 92B05
1 Introduction
Researches on predation systems are always a popular issue in contemporary theoretical ecology and applied mathematics [1–12]. Results based on non-spatial systems have shown that the effect of prey refuge played an important role in determining the dynamical consequences of predator-prey systems [1, 3, 4, 7, 8, 10, 13–19]. Incorporating the effect of prey refuge into the considered predation system is initially done by modifying the originally functional response of predator to prey population, the functional response describes the per capita consumption rates of predators depending on prey density, and quantifies the energy transfer between trophic levels, like Holling I, II, III and IV functional response [20, 21]. The most widely reported conclusions are the community/interior/positive/coexistent equilibrium of the considered predation system being stabilized and the equilibrium density of prey and/or predator was enhanced by the addition of prey refuge [3, 8, 12, 19, 22, 23].
We have only three parameters, where \(A=\frac{1}{K^{n} \lambda h(1-\beta)^{n}}>0\), \(B=\frac{c}{hr}>0\), \(C=\frac{dh}{c}>0\).
2 Equilibria
If \(\beta>1-\frac{1}{K}(\frac{d}{\lambda(c-dh)})^{1/n}\), the equilibrium point \(\tilde{E}(\tilde{x},\tilde{y})\) collapses with the point \(E_{K}(K,0)\).
The equilibrium point \(\bar{E}(\bar{x},\bar{y})\) is positive if and only if \(A<\frac{1-C}{C}\).
If \(A=\frac{1-C}{C}\), the positive equilibrium point collapses with the equilibrium point \(E_{1}(1,0)\).
The equilibrium point \(\bar{E}(\bar{x},\bar{y})\) lies in the fourth quadrant when \(A>\frac{1-C}{C}\).
3 Positivity and boundedness of the solutions
Clearly, \(G\in C^{1}(R^{2}_{+})\). Thus \(\mathbf{G}:R^{2}_{+} \rightarrow R^{2}\) is locally Lipschitz on \(R^{2}_{+}=\{(x,y)|x>0,y>0\}\). Hence the fundamental theorem of existence and uniqueness ensures existence and uniqueness of solution of the system (1.5) with the given initial conditions. The state space of the system is the non-negative cone in \(R^{2}_{+}\). In the theoretical ecology, positivity and boundedness of the system establishes the biological well behaved nature of the system.
Theorem 3.1
All the solutions of the system (1.5) with the given initial conditions are always positive and bounded.
Proof
Secondly, we will prove the boundedness.
4 Stability analysis
4.1 Local stability
The two eigenvalues of matrix \(J_{1}\) are \(-(A+1)\) and \(B(1-C-AC)\).
Hence, if \(A<\frac{1-C}{C}\), the equilibrium point \(E_{1}(1,0)\) is a saddle point. Otherwise, the equilibrium point \(E_{1}(1,0)\) is locally asymptotically stable.
Clearly \(\operatorname{Det}J_{2}=-A_{12}A_{21}>0\). Therefore, the sign of the eigenvalues of Jacobian matrix \(J_{2}\) depends only on \(\operatorname{Tr}J_{2}=A_{11}\).
Case 1: If \(0< n<\frac{1}{1-C}\), then \(2-n(1-C)>0\) and \(1-n(1-C)>0\).
Hence, the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\) is locally asymptotically stable.
The results of González-Olivares et al. [6] and Huang et al. [8] are special cases of ours for \(n=1\) and \(n=2\), respectively.
Case 2: If \(\frac{1}{1-C}\leq n \leq\frac{2}{1-C}\), then the inequality (4.1) holds.
Therefore, the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\) is always asymptotically stable whenever the proportion of prey refuge is.
Case 3: If \(n>\frac{2}{1-C}\), then \(2-n(1-C)<0\) and \(1-n(1-C)<0\).
Therefore, the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\) is locally asymptotically stable under these assumptions.
4.2 Existence of limit cycle
Here \(g(x)=(1-x)(A+x^{n})\), \(p(x)=x^{n}\), \(q(x)=(1-C)p(x)\).
We present a lemma [17] regarding uniqueness of limit cycle of the above system.
Lemma
[17]
This will be equivalent to prove that \(\varphi(x)\leq0\) for all \(x>0\).
It is easy to show that \(x=0\) and \(x=\bar{x}\) are the solutions of the equation \(\varphi'(x)=0\).
Again \(\varphi''(\bar{x})=-nA\bar{x}^{n-2}[2C(1+n)+(n-1)(n-2)(1-C)]<0\).
Thus, \(x=\bar{x}\) is the maximum value point of the function \(\varphi(x)\).
Clearly, it is exactly the condition of the instability of the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\).
4.3 Global stability
In this section, we will prove the global stability of the positive equilibrium point \(\tilde{E}(\tilde{x},\tilde{y})\) of the system (1.5).
Thus, \(\frac{dW}{dt}<0\) if \(n \geq 1\). Hence, the positive equilibrium point \(\tilde{E}(\tilde{x},\tilde{y})\) of the system (1.5) is globally asymptotically stable.
5 Main results
According to the above analysis, we can obtain the following results.
Theorem 5.1
- (1)
If \(0< A<\frac{1-C}{C}[\frac{1-n(1-C)}{2-n(1-C)}]^{n}\), the system (1.4) has a unique globally stable limit cycle surrounding the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\) which is unstable.
- (2)
If \(\frac{1-C}{C}[\frac{1-n(1-C)}{2-n(1-C)}]^{n}< A<\frac{1-C}{C}\), the system (1.4) has a globally asymptotically stable equilibrium point \(\bar{E}(\bar{x},\bar{y})\) at the first quadrant.
Theorem 5.2
- (1)
If \(0< A<\frac{1-C}{C}[\frac{1-n(1-C)}{2-n(1-C)}]^{n}\), the system (1.4) has a globally asymptotically stable equilibrium point \(\bar{E}(\bar{x},\bar{y})\) in the first quadrant.
- (2)
If \(\frac{1-C}{C}[\frac{1-n(1-C)}{2-n(1-C)}]^{n}< A<\frac{1-C}{C}\), the system (1.4) has a unique globally stable limit cycle surrounding the interior equilibrium point \(\bar{E}(\bar{x},\bar{y})\) which is unstable.
In reference to the original parameters of the system (1.5), the above results can be expressed as follows.
Theorem 5.3
- (1)
If \(0<\beta<1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac {2c-n(c-dh)}{c-n(c-dh)}]\), the prey and predator populations stably oscillate around the unique interior equilibrium point.
- (2)
If \(1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac {2c-n(c-dh)}{c-n(c-dh)}]<\beta<1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}\), the two populations tend to reach a globally asymptotically stable equilibrium point at the first quadrant.
Theorem 5.4
- (1)
If \(0<\beta<1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac {2c-n(c-dh)}{c-n(c-dh)}]\), the two populations tend to reach a globally asymptotically stable equilibrium point in the first quadrant.
- (2)
If \(1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac {2c-n(c-dh)}{c-n(c-dh)}]<\beta<1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}\), the prey and predator populations stably oscillate around the unique interior equilibrium point.
6 Bifurcation analysis
To obtain a complete classification of the qualitative behavior of the system (1.5), we analyze the bifurcation pattern and illustrate the results with one parameter, saying the effect of prey refuge β. The classification requires up to two codimension-one bifurcations: (i) Hopf-bifurcation point in which the coexistence equilibrium point \(\tilde{E}(\tilde{x},\tilde{y})\) exchanges stability, (ii) the bifurcation point tracking a transcritical bifurcation between the coexistence equilibrium point \(\tilde{E}(\tilde{x},\tilde{y})\) and the prey only equilibrium \(E_{K}(K,0)\), where these two equilibria coincide and exchange their stability to each other.
6.1 Hopf bifurcation
One-dimensional bifurcation analysis reveals the behavior of the system (1.5) when a particular system parameter is varied over a long range. Here we observe the behavior of the system (1.5) when the prey refuge intensity is varied.
It can easily be observed from the characteristic equation (6.1) that the roots become purely imaginary when \(\tilde{a}_{11}=0\), i.e. \(\beta=\beta_{c}= 1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac{2c-n(c-dh)}{c-n(c-dh)}]\). In this case, \(\operatorname{Re}(\lambda)|_{\beta=\beta_{c}}=0\), \(\operatorname{Im}(\lambda)|_{\beta=\beta_{c}} \neq0\) and \(\frac{d}{d \beta}\operatorname{Re}(\lambda)|_{\beta=\beta_{c}}<0\) (we use the standard package of Mathematica to get these results) and hence the transversality condition for a Hopf bifurcation is satisfied. Therefore, there exists a Hopf bifurcation at \(\beta=\beta_{c}\). The negative sign of \(\frac{d}{d \beta}\operatorname{Re}(\lambda)|_{\beta=\beta_{c}}<0\) implies that the oscillations in the population densities dampen as the effect of prey refuge passes from lower value to higher value through \(\beta=\beta_{c}\).
Hence, we obtain the following results.
Theorem 6.1
The system (1.5) undergoes a Hopf bifurcation at \(\tilde{E}(\tilde{x},\tilde{y})\) when the effect of prey refuge β passes the threshold value \(\beta_{c}= 1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac{2c-n(c-dh)}{c-n(c-dh)}]\).
6.2 Transcritical bifurcation
In this section, we will consider the existence of a transcritical bifurcation for the system (1.5). In order to do this, we select the effect of prey refuge β as the bifurcation parameter. According to the analysis in Section 3, if \(1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}[\frac{2c-n(c-dh)}{c-n(c-dh)}] <\beta<1-\frac{1}{K}(\frac{d}{\lambda(c-dh)})^{1/n}\), the coexistence equilibrium Ẽ is stable but the axial equilibrium \(E_{K}\) is unstable. The two equilibria coincide a \(\beta=\beta_{0}=1-\frac{1}{K}(\frac{d}{\lambda(c-dh)})^{1/n}\) and exchange their stability when \(1-\frac{1}{K}(\frac{d}{\lambda(c-dh)})^{1/n}<\beta<1\).
Hence, according to Sotomayor’s theorem, the system (1.5) undergoes a transcritical bifurcation when the effect of prey refuge β passes through the threshold value \(\beta_{0}\).
Hence, we obtain the following results.
Theorem 6.2
The system (1.5) undergoes a transcritical bifurcation when the effect of prey refuge β passes the threshold value \(\beta_{0}=1-\frac{1}{K}(\frac{d}{\lambda(c-dh)})^{1/n}\).
7 Discussion
- (i)
The equilibrium point \(E_{K}(K,0)\) is locally asymptotically stable if the proportion of refuge using by prey is larger than \(1-\frac{1}{K}[\frac{d}{\lambda(c-dh)}]^{1/n}\). Therefore, when the refuge using by prey is high, the system predicts that the prey population reaches its carrying capacity and the predators go extinct, a dynamics also observed by Collings [5] for some certain parameters.
- (ii)
The shape of the functional response plays an important role in determining the dynamic behavior of the system. If \(0< n<\frac{c}{c-dh}\), the effect of prey refuge has a stabilizing effect, which is consistent with results of González-Olivares and Ramos-Jiliberto [6] who has found a clear stabilizing effect on their considered system. Here, stabilization or the increase of stability refers to cases where a community equilibrium point changes from repeller to an attractor due to changes in the value of a control parameter [6]. However, if the exponent n is larger than \(\frac{2c}{c-dh}\), the stability of the interior equilibrium point changes from the globally asymptotically stable state to the unstable state surrounding a globally stable limit cycle as the refuge using by prey increases. The prey refuge can decrease the stability of the interior equilibrium point. We call this a destabilizing effect.
Declarations
Acknowledgements
We would like to thank the editor and the anonymous referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11301238) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-166).
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
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