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Stability analysis for a time-delayed nonlinear predator–prey model
- Baiyu Xie^{1} and
- Fei Xu^{2}Email author
https://doi.org/10.1186/s13662-018-1564-4
© The Author(s) 2018
Received: 3 December 2017
Accepted: 16 March 2018
Published: 3 April 2018
Abstract
In this paper, we investigate the dynamics of a time-delayed prey–predator system with θ-logistic growth. Our investigation indicates that the models based on delayed differential equations (DDEs) with and without delay-dependent coefficient both undergo Hopf bifurcation at their corresponding positive equilibria. It is shown that stability switching occurs for the interior equilibrium of the model with delay-dependent coefficient. For the DDEs model without delay-dependent coefficient, increased time delay may destabilize a stable interior equilibrium.
Keywords
- Hopf bifurcation
- Time-delay
- θ-logistic growth
- Prey refuge
1 Introduction
In biomathematics, the interaction and interplay between different species have been modeled by systems of differential equations. Such systems characterize the dynamics of a variety of ecosystems. By constructing an ecological model, the relationship between different species in the system is revealed. Analyzing such models yields the dynamics of the system and may give a precise prediction on the evolution of populations in the system. Recently, prey refuge has been integrated into ecological models to consider the effects of the refuges on the coexistence of different species and on the stability of equilibria of ecosystems [1–6]. Empirical and theoretical studies have both been carried out to illustrate the influences of prey refuge on the population dynamics of the systems. Investigations indicate that the existence of prey refuge may stabilize the system and by using such refuge, the prey population may refrain from extinction [7–13].
It follows from the fundamental theory of functional differential equations [16] that system (1.2) has a unique solution \(x(t)\), \(y(t)\) satisfying initial conditions (1.3).
This manuscript is organized as follows. In Sect. 2, we prove that solutions to system (1.2) with initial conditions (1.3) are positive and ultimately bounded. In Sect. 3, we investigate the stability of the boundary equilibria of system (1.2). In Sect. 4, we show that system (1.1) and (1.2) exhibits Hopf bifurcations at the interior equilibrium. Finally, we perform numerical analysis to illustrate the main results of this article in Sect. 5.
2 Positivity and boundedness
For model (1.2) with initial conditions (1.3), we are particularly interested in the positivity and boundedness of its solution. In this section, we prove that the solutions are positive and ultimately bounded.
2.1 Positivity of solutions
Theorem 2.1
Solutions to system (1.2) with initial conditions (1.3) are positive for all \(t\geq0\).
Proof
In the following subsection, we show that the solutions are ultimately bounded.
2.2 Boundedness of solutions
Theorem 2.2
Positive solutions of system (1.2) with initial conditions (1.3) are ultimately bounded.
Proof
3 Stability of the boundary equilibria
In the following, we consider the stability of the boundary equilibria of model (1.2) satisfying initial conditions (1.3).
The above results are summarized in the following conclusion.
Theorem 3.1
- (i)
For all \(\tau\geq0\), equilibrium \(E_{0}\) is always unstable.
- (ii)
For all \(\tau\geq0\), when \(R_{0}\leq1\), equilibrium \(E_{1}\) is stable, and when \(R_{0}>1\), \(E_{1}\) is unstable.
4 The Hopf bifurcation
Hopf bifurcations have been observed in population dynamical systems [6, 17]. In this section, we investigate the Hopf bifurcation of system (1.1).
4.1 Stability of a positive equilibrium for system (1.1)
Theorem 4.1
Example 4.1
In the following example, we choose (\(P_{2}\)) as \(r=0.11\), \(K=10\), \(\beta=0.2\), \(a=0.12\), \(h_{1}=0.01\), \(h_{2}=0.01\), \(\theta=6\), and \(\varepsilon=0.7\). It thus follows that \(R_{0}^{*}\approx5.055645375>1\) and \(A_{1}\approx-0.0153729314<0\), which guarantees that system (1.1) is unstable (see Fig. 1(b)).
4.2 The Hopf bifurcation of DDEs with delay-dependent coefficient
In this subsection, we investigate the Hopf bifurcation of the model with term \(e^{-m\tau}\). We notice that Eq. (4.1) is a second-degree exponential polynomial of λ and all the coefficients of P and Q depend on τ.
- (a)
\(P(0,\tau)+Q(0,\tau)\neq0\);
- (b)
\(P(i\omega,\tau)+Q(i\omega,\tau)\neq0\);
- (c)
\(\limsup \{|\frac{P(\lambda,\tau)}{Q(\lambda,\tau)}|:|\lambda |\rightarrow\infty, \operatorname{Re} \lambda\geq0 \}<1\);
- (d)
\(F(\omega,\tau)=|P(i\omega,\tau)|^{2}-|Q(i\omega,\tau)|^{2}\) has a finite number of zeros;
- (e)
Each positive root \(\omega(\tau)\) of \(F(\omega,\tau)=0\) is continuous and differentiable in τ whenever it exists.
Here, \(P(\lambda,\tau)\) and \(Q(\lambda,\tau)\) are defined by (4.2).
Therefore, property (d) is satisfied. Assume that \((\omega_{0}, \tau_{0})\) is a point in its domain such that \(F(\omega_{0}, \tau_{0})=0\). It is easy to see that the partial derivatives \(F_{\omega}\) and \(F_{\tau}\) exist and are continuous in a certain neighborhood of \((\omega_{0}, \tau_{0})\), and \(F_{\omega}(\omega_{0}, \tau_{0})\neq0\). Then the implicit function theorem implies that condition (e) is satisfied as well.
Proposition 4.1
If \(R_{0}>1\) and \(a_{2}(\tau)<0\), then \(F(h,\tau)=0\) has only one positive root \(h_{+}\). We also have that \(F(\omega,\tau)=0\) has a unique positive root given by \(\omega=\sqrt{h_{+}}\).
Define \(\theta(\tau)\in[0,2\pi)\), where \(\sin\theta(\tau)\) and \(\cos\theta(\tau)\) are respectively the right-hand sides of (4.7a) and (4.7b). Here, \(\theta(\tau)\) is expressed as (4.8a)–(4.8b).
The following theorem is obtained using the method proposed by Beretta and Kuang [18].
Theorem 4.2
It follows from Theorem 4.1 and the Hopf bifurcation theorem for functional differential equations [16] that there exists a Hopf bifurcation. Details are summarized in the following theorem.
Theorem 4.3
- (i)
Assume that \(R_{0}>1\), \(A_{1}>0\), and the function \(S_{0}(\tau)\) has no positive zero in I. Then equilibrium \(E^{*}\) is asymptotically stable for all \(\tau\in[0, \tau_{\max})\).
- (ii)
Assume that \(R_{0}>1\), \(A_{1}>0\), \(a_{2}(\tau)<0\), and the function \(S_{0}(\tau)\) has positive zero in I. Then there exists \(\tau^{*}\in I\) such that equilibrium \(E^{*}\) is asymptotically stable for \(\tau\in[0, \tau^{*})\), and unstable for \(\tau\in(\tau^{*}, \tau_{\max})\). A Hopf bifurcation occurs when \(\tau=\tau^{*}\).
Remark 4.1
If \(\tau\geq\frac{1}{m}[\ln\beta-\ln (a+h_{2}+\frac{a+h_{2}}{K^{2}\varepsilon^{2}(\frac{r-h_{1}}{r})^{\frac {2}{\theta}}})]:=\tau_{\mathrm{max}}\), then \(R_{0}\leq1\), \(y^{*}\leq0\) and equilibrium \(E^{*}\) converges to \(E_{1}=(K,0)\).
4.3 The Hopf bifurcation of DDEs without delay-dependent coefficient
In this section, we consider the case when \(m=0\), i.e., the DDEs has no term \(e^{-m\tau}\). Now, all the coefficients of (4.2) are not related to the delay τ.
We denote \(b_{i}=b_{i}(0)\) (\(i=1,\ldots,4\)). In this case, if \(R_{0}^{*}>1\) and \(a_{2}(0)>0\), then Eq. (4.1) has no positive root. Thus, the positive equilibrium \(E^{*}\) exists and is locally asymptotically stable for all time delay \(\tau\geq0\).
Theorem 4.4
- (i)
If \(a_{2}(0)>0\), then the positive equilibrium \(E^{*}\) of system (1.2) is asymptotically stable for all \(\tau\geq0\);
- (ii)
If \(a_{2}(0)<0\), then there exists a positive number \(\tau_{0}\) such that the positive equilibrium \(E^{*}\) of system (1.2) is asymptotically stable for \(0<\tau<\tau_{0}\) and is unstable for \(\tau>\tau_{0}\). We then obtain that system (1.2) undergoes a Hopf bifurcation at \(E^{*}\) when \(\tau=\tau_{0}\).
5 Numerical simulations
In this section, we use numerical simulations to verify the theoretical results obtained in previous sections.
The default parameters used in the simulations are as follows: \(r=0.11\), \(K=10\), \(a=0.12\), \(h_{1}=0.01\), \(h_{2}=0.01\), \(\varepsilon=0.7\), and \(\theta=6\). Here we use numerical simulations to compare the dynamical behaviors of the model with and without delay-dependent coefficient. Four groups of simulation results with different β and m are presented.
In simulation set (i), we choose \(\beta=0.3\) and \(m=0.15\) for the delay-dependent coefficient \(e^{-m\tau}\). For simulation set (ii), we choose the same \(\beta=0.3\) and consider the dynamical behaviors of the model without the delay-dependent coefficient. We then compare the simulation results (i) and (ii) to reveal the effects of the delay-dependent coefficient on the system’s dynamical behaviors. In simulation set (iii), we choose \(\beta=0.2\) and \(m=0.15\) for the delay-dependent coefficient \(e^{-m\tau}\). Then simulation results (iv) of the model for the same \(\beta=0.2\) with the absence of the delay-dependent coefficient are presented. We compare the results (iii) and (iv) to consider the effects of the delay-dependent coefficient in this scenario.
- (1a)
- (1b)
- (1c)
For \(\tau=0.6\in(\tau^{*}, \tau^{**})\), the positive equilibrium of system (1.2) is unstable and there is a Hopf bifurcation when \(\tau=\tau^{*}\) (see Fig. 2(c)).
- (1d)
For \(\tau=5.1\in(\tau^{**}, \tau_{\mathrm{max}})\), the positive equilibrium of system (1.2) is stable (see Fig. 2(d)).
- (4a)
- (4b)
6 Conclusions
In conclusion, the positive equilibrium of DDEs with delay-dependent coefficient displays stability switches and is ultimately stable under some conditions, indicating that a long delay stabilizes the interior equilibrium [18]. However, a DDEs model without delay-dependent coefficient usually behaves differently.
Declarations
Acknowledgements
This work is supported by NSFC (No. 11326200, No. 31470641), Foundation of He’nan Educational Committee (No. 15A110015), and the Grant of China Scholarship Council (No. 201408410018).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The 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
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