# Stability analysis for a time-delayed nonlinear predator–prey model

- 183 Downloads

## 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, 2, 3, 4, 5, 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, 8, 9, 10, 11, 12, 13].

*θ*-logistic growth predator–prey system with prey refuge:

*x*and

*y*respectively denote the densities of prey and predator, and

*r*,

*K*,

*θ*,

*β*,

*ε*,

*a*, and

*m*take positive values. In system (1.2), the prey species has

*θ*-logistic growth with logistic index

*θ*and intrinsic growth rate

*r*. Here,

*K*is the carrying capacity,

*β*is the predation rate of predator, and \(\varepsilon\in(0, 1)\) is the refuge rate to prey. Obviously, \(1-\varepsilon\) is the proportion of prey that is available for the predator. We use \(h_{1}\) and \(h_{2}\) to denote the rate of harvesting or the environment feedback for prey and predator, respectively. We assume that the predator has death rate

*a*. The time lag between the consumption of prey and receiving corresponding increase in predator population is denoted by

*τ*. We thus introduce a delay-dependent coefficient \(e^{-m\tau}\) to describe the probability of the predators that consume prey at time \(t-\tau\) and still remain alive at time

*t*. Such delay-dependent coefficient may have considerable influences on the dynamical behaviors of the model and it has not been investigated extensively in the literature. In the following, we compare the dynamical behaviors of the model with and without the delay-dependent coefficient.

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

## Proof

In the following subsection, we show that the solutions are ultimately bounded.

### 2.2 Boundedness of solutions

## Proof

*ρ*, there exists \(T_{1}>0\) such that if \(t>T_{1}\), \(x(t)< K+\rho\). In order to prove the boundedness of the solution, we construct the following Lyapunov function:

*V*along the trajectories of system (1.2) yields

*t*large enough. We notice that

*M*only depends on the parameters of system (1.2). The above discussion implies that \(x(t)\), \(y(t)\) is ultimately bounded. □

## 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).

*F*defined in (d), it follows from

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.

*τ*in

*I*, \(\omega(\tau)\) satisfies

## 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).

*I*.

*τ*.

The following theorem is obtained using the method proposed by Beretta and Kuang [18].

## Theorem 4.2

*If*\(\omega(\tau)\)

*is a positive root of*(4.1)

*defined for*\(\tau\in I\), \(I\subseteq R_{+0}\),

*and*\(S_{n}(\tau^{*})=0\)

*for some*\(n\in N_{0}\)

*at some*\(\tau^{*}\in I\),

*then a pair of simple conjugate pure imaginary roots*\(\lambda=\pm i\omega\)

*exist at*\(\tau=\tau^{*}\)

*and they cross the imaginary axis from left to right when*\(\delta(\tau^{*})>0\)

*and cross the imaginary axis from right to left when*\(\delta(\tau^{*})<0\).

*Here*,

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

*For system*(1.2),

*the following conclusions hold*:

- (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 \(\tau\geq0\).

*τ*yields

*τ*is increased continuously from a number less than \(\tau_{0}\) to a number greater than \(\tau_{0}\). Therefore, both the transversality condition and the conditions for Hopf bifurcation [16] are satisfied at \(\tau=\tau_{0}\). Thus, we obtain the following results for system (1.2).

## Theorem 4.4

*Let*\(m=0\), \(R_{0}^{*}>1\)

*and*\(A_{1}>0\).

*For system*(1.2),

*we have the following results*:

- (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.

*τ*, denoted by \(\tau^{*}\) and \(\tau^{**}\), respectively. Here, \(\tau^{*}\approx0.5\) and \(\tau^{**}\approx5.1\).

- (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)).

*τ*, denoted by \(\tau^{*}\). Here, \(\tau^{*}\approx2.4\).

*τ*.

*τ*, the interior equilibrium of the system is stable. The stability of the interior equilibrium changes at \(\tau \approx1\). As indicated in the figure, for large

*τ*, the interior equilibrium is no longer stable and the system displays cycling behaviors.

## 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.

## Notes

### 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.

## References

- 1.Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol.
**91**, 385–398 (1959) CrossRefGoogle Scholar - 2.Hassell, M.P., May, R.M.: Stability in insect host–parasite models. J. Anim. Ecol.
**42**, 693–726 (1973) CrossRefGoogle Scholar - 3.Smith, J.M.: Models in Ecology. Cambridge University Press, Cambridge (1974) MATHGoogle Scholar
- 4.Hassell, M.P.: The Dynamics of Arthropod Predator–Prey Systems. Princeton University Press, Princeton (1978) MATHGoogle Scholar
- 5.Ji, L.L., Wu, C.Q.: Qualitative analysis of a predator–prey model with constant-rate prey harvesting incorporating a constant prey refuge. Nonlinear Anal., Real World Appl.
**11**, 2285–2295 (2010) MathSciNetCrossRefMATHGoogle Scholar - 6.Wang, S.L., Ge, Z.H.: The Hopf bifurcation for a predator–prey system with
*θ*-logistic growth and prey refuge. Abstr. Appl. Anal.**2013**, Article ID 168340 (2013) MathSciNetMATHGoogle Scholar - 7.Sih, A.: Prey refuges and predator–prey stability. Theor. Popul. Biol.
**31**, 1–12 (1987) MathSciNetCrossRefGoogle Scholar - 8.Taylor, R.J.: Predation. Chapman & Hall, New York (1984) CrossRefGoogle Scholar
- 9.González-Olivares, E., Ramos-Jiliberto, R.: Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model.
**166**, 135–146 (2003) CrossRefGoogle Scholar - 10.Krivan, V.: Effects of optimal antipredator behavior of prey on predator–prey dynamics: the role of refuges. Theor. Popul. Biol.
**53**, 131–142 (1998) CrossRefMATHGoogle Scholar - 11.Ma, Z.H., Li, W.L., Zhao, Y., Wang, W.T., Zhang, H., Li, Z.Z.: Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges. Math. Biosci.
**218**(2), 73–79 (2009) MathSciNetCrossRefMATHGoogle Scholar - 12.Chen, L.J., Chen, F.D., Chen, L.J.: Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a constant prey refuge. Nonlinear Anal., Real World Appl.
**11**, 246–252 (2010) MathSciNetCrossRefMATHGoogle Scholar - 13.Tao, Y.D., Wang, X., Song, X.Y.: Effect of prey refuge on a harvested predator–prey model with generalized functional response. Commun. Nonlinear Sci. Numer. Simul.
**16**, 1052–1059 (2010) MathSciNetCrossRefMATHGoogle Scholar - 14.Tsoularis, A., Wallace, J.: Analysis of logistic growth models. Math. Biosci.
**179**, 21–55 (2002) MathSciNetCrossRefMATHGoogle Scholar - 15.Wonlyul, K., Kimun, R.: A qualitative study on general Gause-type predator–prey models with constant diffusion rates. J. Math. Anal. Appl.
**344**, 217–230 (2008) MathSciNetCrossRefMATHGoogle Scholar - 16.Hale, J.K.: Theory of Functional Differential Equations. Springer, Heidelberg (1977) CrossRefMATHGoogle Scholar
- 17.Wang, S.L., Wang, S.L., Song, X.Y.: Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control. Nonlinear Dyn.
**67**, 629–640 (2012) MathSciNetCrossRefMATHGoogle Scholar - 18.Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal.
**33**, 1144–1165 (2002) MathSciNetCrossRefMATHGoogle Scholar - 19.Dieuonné, J.: Foundations of Modern Analysis. Academic Press, New York (1960) Google Scholar

## Copyright information

**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.