Nonlinear Dynamics in Applied Sciences Systems: Advances and PerspectivesView this Special Issue
Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays
This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.
It is well known that there are many factors which can affect dynamical properties of predator-prey systems. One of the important factors is the functional response describing the number of prey consumed per predator per unit time for given quantities of prey and predators. Numerous laboratory experiment, and observations have shown that a more suitable general predator-prey system should be based on the “ratio-dependent” theory, especially when predators have to search, share, or compete for food [1–3]. And ratio-dependent predator-prey systems have been investigated by many scholars [4–11]. In , Zhang and Lu considered the following semi-ratio-dependent predator-prey system with the nonmonotonic functional response where and denote the densities of the prey and the predator, respectively. and denote the intrinsic growth rates of the prey and the predator, respectively. is the intraspecific competition rate of the prey. is the capturing rate of the predator. is a measure of the food quality that the prey provided for conversion into predator birth. The predator grows logistically with the carrying capacity proportional to the population size of the prey.
,, and ,,, are assumed to be continuously positive periodic functions with period . Zhang and Lu  established the existence of positive periodic solutions of system (1), and sufficient conditions for the uniqueness and global stability of the positive solutions of system (1) were also obtained by constructing a Lyapunov function.
Time delays of one type or another have been incorporated into predator-prey systems by many researchers since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate [5, 6, 12–15]. In , Ding et al. incorporated the time delay due to negative feedback of the prey into system (1) and got the special case () of system (1) with time delay Ding et al.  established the existence of a positive periodic solution for system (2) by using the continuation theorem of coincidence degree theory. Sufficient conditions for the permanence of system (2) were obtained by Li and Yang in . But studies on predator-prey systems not only involve the persistence and the periodic phenomenon but also involve many patterns of other behavior such as global attractivity [16, 17] and bifurcation phenomenon [18–21]. Starting from this point, we will study the bifurcation phenomenon and the properties of periodic solutions of the following predator-prey system with two delays: Unlike the assumptions in [4–6], we assume that all the parameters of system (3) are positive constants. and are the feedback delays of the prey and the predator, respectively.
The rest of this paper is organized as follows. In Section 2, sufficient conditions are obtained for the local stability of the positive equilibrium and the existence of Hopf bifurcation for possible combinations of the two delays in system (3). In Section 3, we give the formula determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, numerical simulations supporting the theoretical analysis are also included.
2. Stability of the Positive Equilibrium and Local Hopf Bifurcations
Let , . Dropping the bars for the sake of simplicity, system (2) can be rewritten in the following system: where The linearized system of (5) is The associated characteristic equation of system (7) is where
Case 2 (). Equation (8) becomes Let be the root of (11), and we have It follows that If condition holds, then (13) has only one positive root Now, we can get the critical value of by substituting into (12). After computing, we obtain Then, when , the characteristic equation (11) has a pair of purely imaginary roots .
Next, we will verify the transversality condition. Differentiating (11) with respect to , we get Thus, Obviously, if condition holds, then . Note that = .
Therefore, if condition holds, the transversality condition is satisfied. From the analysis above, we have the following results.
Theorem 1. If conditions and hold, then the positive equilibrium of system (3) is asymptotically stable for and unstable when . System (3) undergoes a Hopf bifurcation at the positive equilibrium when .
Case 3 (). Equation (8) becomes Let be the root of (18), and then we have which leads to Obviously, if the condition holds, and then (20) has only one positive root The corresponding critical value of is Similar to Case 2, differentiating (18) with respect to , we get Then we can get
Obviously, if condition holds, then . Namely, if condition holds, then the transversality condition is satisfied. Thus, we have the following results.
Case 4 (). Equation (8) can be transformed into the following form: Multiplying on both sides of (25), we get Let be the root of (26), then we obtain which follows that Then, we have where Denote , and then (29) becomes Next, we give the following assumption. Equation (31) has at least one positive real root.
For every fixed , the corresponding critical value of time delay is Let Taking the derivative of with respect to in (26), we obtain Thus, with
It is easy to see that if condition , then . Therefore, we have the following results.
Theorem 3. If conditions and holds, then the positive equilibrium of system (3) is asymptotically stable for and unstable when . System (3) undergoes a Hopf bifurcation at the positive equilibrium when .
Case 5 ( and ). Considered the following:
We consider (8) with in its stable interval and regard as a parameter.
Let () be the root of (8), and then we can obtain Then we get Define
If condition and hold, then . In addition, . Therefore, (39) has at least one positive root. We suppose that the positive roots of (39) are denoted as . For every fixed , the corresponding critical value of time delay with Let , . When , (8) has a pair of purely imaginary roots for .
Next, in order to give the main results, we give the following assumption.
Therefore, we have the following results on the stability and bifurcation in system (3).
Theorem 4. If the conditions and hold and , then the positive equilibrium of system (3) is asymptotically stable for and unstable when . System (3) undergoes a Hopf bifurcation at the positive equilibrium when .
3. Direction and Stability of the Hopf Bifurcation
In this section, we will investigate the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (3) with respect to for . We assume that where .
Let , so that is the Hopf bifurcation value of system (3). Rescale the time delay by . Let , , and then system (3) can be rewritten as an PDE in as where are given, respectively, by with By the Riesz representation theorem, there is a matrix function with bounded variation components , such that In fact, we have chosen For ,, we define Then system (43) is equivalent to the following operator equation: where for .
For , we define the adjoint operator of as and a bilinear inner product where .
Then, and are adjoint operators. From the discussion above, we can know that are the eigenvalues of and they are also eigenvalues of .
Let be the eigenvector of corresponding to , and = be the eigenvector of corresponding to . Then we have By a simple computation, we can get Then,
Following the algorithms explained in the work of Hassard et al.  and using a computation process similar to that in , we can get the following coefficients which can determine the properties of Hopf bifurcation: with where and can be determined by the following equations, respectively: with Then, we can get the following coefficients:
In conclusion, we have the following results.
Theorem 5. If (), then the Hopf bifurcation is supercritical (subcritical); if , then the bifurcating periodic solutions are stable (unstable); if (), then the period of the bifurcating periodic solutions increases (decreases).
4. Numerical Simulation and Discussions
To demonstrate the algorithm for determining the existence of Hopf bifurcation in Section 2 and the properties of Hopf bifurcation in Section 3, we give an example of system (3) in the following form: which has a positive equilibrium . By calculation, we have , , and . Namely, conditions and hold.
For , . We can get , . By Theorem 1, we know that when , the positive equilibrium is locally asymptotically stable and is unstable if . As can be seen from Figure 1, when we let , the positive equilibrium is locally asymptotically stable. However, when , the positive equilibrium is unstable and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the positive equilibrium , which can be illustrated by Figure 2. Similarly, we can get ,. The corresponding waveform and the phase plots are shown in Figures 3 and 4.
For . We can obtain ,. By Theorem 3, we know that when the time delay increases from zero to the critical value , the positive equilibrium is locally asymptotically stable. It will lose its stability and a Hopf bifurcation occurs once . This property can be illustrated by Figures 5 and 6.
In addition, from (60), we can get . Furthermore, we have , ,. Therefore, from Theorem 5, we know that the Hopf bifurcation is supercritical. The bifurcating periodic solutions are stable and increase. If the bifurcating periodic solutions are stable, then the prey and the predator may coexist in an oscillatory mode. Therefore, the two species in system (3) may coexist in an oscillatory mode under certain conditions.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.View at: Google Scholar
R. Arditi, N. Perrin, and H. Saiah, “Functional responses and heterogeneities: an experimental test with cladocerans,” Oikos, vol. 60, no. 1, pp. 69–75, 1991.View at: Google Scholar
I. Hanski, “The functional response of predators: worries about scale,” Trends in Ecology and Evolution, vol. 6, no. 5, pp. 141–142, 1991.View at: Google Scholar
X. Ding, C. Lu, and M. Liu, “Periodic solutions for a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 762–775, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Y. Wang, S. Wang, F. P. Yang, and L. R. Li, “Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects,” Applied Mathematical Modelling, vol. 34, no. 12, pp. 4278–4288, 2010.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Ferrara, L. Guerrini, and R. Mavilia, “Modified neoclassical growth models with delay: a critical survey and perspectives,” Applied Mathematical Sciences, vol. 7, pp. 4249–4257, 2013.View at: Google Scholar
J. J. Jiao, L. S. Chen, and J. J. Nieto, “Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on pr