Abstract
A delayed diffusive predator-prey model with predator interference or foraging facilitation is studied. We are interested in the existence of Turing instability, local stability, and Hopf bifurcation. We analyze the direction and stability of bifurcating periodic solutions by the normal form method. Our results suggest that diffusion can induce Turing instability and time delay can induce oscillation of prey and predator’s densities. In addition, the interference parameter has stabilizing and destabilizing effect on the positive equilibrium.
1. Introduction
The predator-prey model is one of the most important models in population dynamics. The existence of equilibrium points, periodic solutions, and bifurcation of the model are important problems in population dynamics. Many biologists and mathematicians have studied the predator-prey models [1–3]. In [4], Zhang and Zhu considered the following model:where and represent the densities of the prey and predator population, respectively, and represent the diffusion coefficient of prey and predator, respectively, is the handling time, is the average rate of growth inherent in the prey population, the environmental capacity of the prey is denoted by , represents the efficiency with which the prey is transformed into a predator’s newborn, is the death rate of predators, and is the encounter rate. If , it represents predator interference (foraging facilitation). The common forms of are as follows:where , , and are some parameters. A detailed explanation can be found in [5–7]. Zhang [4] mainly studied the nonconstant positive steady states and Hopf bifurcation.
Time delays exist widely in predator-prey models such as maturation time, capturing time, and gestation time and can induce periodic oscillations [8–16]. Motivated by this, we want to study the effect of time delay in the resource limitation of the prey on model (1). We study the following model:
The aim of this paper is to study the local stability and Hopf bifurcation induced by time delay .
The article is arranged as follows. In Section 2, we study Turing instability, local stability and the existence of Hopf bifurcation. In Section 3, we study the property of Hopf bifurcation. In Section 4, we give some numerical simulations. Finally, a summarization is given in Section 5.
2. Stability Analysis
In [4], Zhang and Zhu have studied the existence of positive equilibrium. In the following, we just suggest that system (3) has positive equilibrium and denoted as . We use the similar analysis process in our previous work [17] to analyze the stability of . It is easy to obtain that the characteristic equation iswhere
Make the following hypothesis:
2.1.
The characteristic equation of iswhere
The eigenvalues are given by
Denote
By analyzing the characteristic root, we can easily get the following theorem.
Theorem 2.1. If and , suppose hold. Then, we have the following results:(1)If , then is locally asymptotically stable(2)If and , then is locally asymptotically stable(3)If and , but (for all ), then is locally asymptotically stable(4)If and and (for some ), then is Turing unstableChoose
By direct calculation, is a unique positive equilibrium. , , , and . Then, is Turing unstable (shown in Figure 1).
2.2.
Assume and one of conditions in Theorem 2.1 hold. Suppose is a root of (4); then,
Then, we havewhich leads to
Let ; then, (16) is changed toand its roots are given by
If and one of conditions in Theorem 2.1 hold, we have
Define . For , if , then (4) has a pair of purely imaginary roots at . And if , then (4) has a pair of purely imaginary roots at , where ,
From (21), we have . For , denote
Lemma 2.1. Suppose and one of conditions in Theorem 2.1 hold. Then,
Proof. By (4), we haveThen,where . Therefore, the conclusion holds.
Theorem 2.2. For system (3), assume and one of conditions in Theorem 2.1 hold.(1) is locally asymptotically stable for when (2) is locally asymptotically stable for and unstable for with some when (3)Hopf bifurcation occurs when , for and
3. Property of Hopf Bifurcation
Similar to our previous work [17], by the standardized method, in [18–20], we give some parameters to determine the property of Hopf bifurcation. For a critical value , we denote it . Let and . Then, system (3) is (after drop tilde)where , and , for , and . Let
Then, (25) can be rewritten in an abstract form in the phase space :where and are defined bywithrespectively, for . Consider the linear equation:
According to the results in Section 2, we know that are characteristic values of system (31) and the linear functional differential equation:
There exists a matrix function , such thatfor . In fact, we can choosewhere
Define the following linear paring:for , . Choose is a basis of with and is a basis of with , where
Let and with , , for , and , , for . Then, we can compute by (36):
Define and
Then, . In addition, define , where
We also define
Thus, the center subspace of linear (31) is given by and denotes the complement subspace of in :for , , , and . Let denote the infinitesimal generator of an analytic semigroup induced by the linear system (31), and (26) can be rewritten as the following abstract form:where
By the decomposition of , the solution above can be written aswhere
In particular, the solution of (28) on the center manifold is given by
Let , and notice that . Then, we have
Hence, (48) iswhere
From [18], satisfieswhere
Let
By (51) and (55), we haveandwith
Hence,withand
Denote
Notice thatand we have
Then, by (54), (56), and (65), we have , for . If , we have the following quantities:
And for , .
From [18], we haveand satisfieswhere
Hence, we havethat is,
By (65), we have that, for ,
Therefore, by (69), for , where
By the definition of and (70), we have
That is,where
By (70), we have
As we have
That is,where
Similarly, from (72), we have
That is,
Similarly, we havewhere
Thus, we have
By [18], we have the following theorem.
Theorem 3.1. For any critical value , we have the Hopf bifurcation is forward or backward . The bifurcating periodic solutions are orbitally asymptotically stable or unstable . The period increases or decreases .
4. Numerical Simulations
Choose parameters
We give the bifurcation diagram of system (3) with parameter in Figure 2. It shows that the interference parameter has stabilizing and destabilizing effect on the positive equilibrium under parameters (90).
If we choose , then is a unique positive equilibrium. Then, . Then, is locally asymptotically stable for (Figure 3). By computation, we have
Hence, the stable bifurcating period solution exists for (Figure 4).
5. Conclusion
In the paper, we consider a delayed diffusive predator-prey model with predator interference or foraging facilitation. The Turing instability and Hopf bifurcation are analyzed. Compared with model (1), time delay can affect the coexistence mode of the prey and the predator. When the delay larger than the critical value, the prey and predator will coexist in the form of periodic oscillation. Time delay may also cause spatially inhomogeneous periodic solutions, but unfortunately, it does not appear in our numerical simulation. In addition, we observed that the interference parameter has stabilizing and destabilizing effect on the positive equilibrium.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors have no relevant financial or nonfinancial interest to disclose.
Authors’ Contributions
All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Ruizhi Yang. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by the Fundamental Research Funds for the Central Universities (no. DL13BB17) and Postdoctoral program of Heilongjiang Province (no. LBH-Q21060).