Abstract
A reaction-diffusion predator-prey system with two delays is investigated. It is found that the spatially homogeneous periodic solution will occur when the sum of two delays crosses some critical values and Hopf bifurcation takes place. For the fixed domain and diffusion, some numerical simulations are also given to illustrate the theoretical analysis. In addition, special attention is paid to effects of diffusion on the bifurcating periodic solution. It is found that the diffusion would lead to the bifurcating period solution to destabilize by calculating the relevant expression of the Floquet exponent.
1. Introduction
It is well known that the reaction-diffusion equations with delays are usually used to describe the biological system. Some results have proved that diffusion and delay take the very important role in the biological systems and can induce many spatiotemporal patterns (see monographs by Wu [1], Arino [2], and Murray [3]). Spatiotemporal patterns of predator-prey population models with diffusion and delay have been studied by many authors [4–9] in recent years. In particular, when the biological system with a single delay has the different boundary conditions such as Neumann boundary and Dirichlet boundary, the spatial homogeneous Hopf bifurcation and the spatially nonhomogeneous Hopf bifurcation near the spatially uniform equilibrium of the predator-prey system have been studied by some authors [10–13].
Faria [14] and Chen [15] have studied Lotka-Volterra type prey-predator model with two delays as the single delay varies where , , , , , , , , , are positive constants and have the following biological interprets, respectively. is the birth rate of the prey; is the death rate of the predator; and represent the strength of the relative effects of the interaction on the two interspecies; and denote the strength of the interaction of the intraspecies; is reaction time of the prey to the predator and is capture time of the predator.
are the diffusion coefficients of prey and predator species, respectively. The variables and are densities of population of the prey and the predator at time and space , respectively. denotes the Laplacian operator in . System (1) with a single delay varying has the existence of the spatially homogeneous and nonhomogeneous Hopf bifurcation. However, in order to investigate joint effects of the two delays, we will mainly consider effects of the sum of two delays and diffusion on the species in system (1). For convenience, the new variables are introduced as follows:System (1) can be transformed into the following form and meanwhile dropping the tildeswhich has the following initial conditions:where is a bounded open domain in with a smooth boundary and the following no flux boundary condition: where is the outward unit normal vector on the boundary.
In what follows, we investigate effects of the delay and diffusion on the dynamics of (3) with initial conditions (4) and boundary conditions (5), respectively. We also assume that and is defined by with the inner product . In addition, for convenience we restrict ourselves to the one-dimensional spatial domain throughout this paper.
In this paper, we not only consider the bifurcation phenomenon of system (1) as the sum of the two delays varies but also investigate effects of diffusion on the bifurcating periodic solutions. We find that as the sum of two delays crosses some critical values, the bifurcating periodic solution would occur through the spatially homogeneous Hopf bifurcation. In addition, once we change the value of diffusion and fix the domain, we also find that the diffusions of the species can destabilize the bifurcating stable periodic solution under certain conditions.
The rest of the paper is organized as follows. In Section 2, the local analysis is given by taking the sum of two delays as parameter to discuss stability of the positive constant equilibrium and the existence of local Hopf bifurcation. In Section 3, by employing the theory of the center manifold and normal formal theory about the partial functional differential equation developed by Wu in [1], the bifurcation direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are pointed out. In order to testify theoretical analysis results, some numerical simulation figures are also given. In Section 4, by using the Fredholm alternative theory about the periodic solution, we also investigate the effects of diffusion on the bifurcating periodic solution and obtain the conditions which determine the stability of the bifurcating periodic solution. Finally, some discussions and conclusions are drawn in Section 5.
2. Linear Stability Analysis and the Existence of Hopf Bifurcation
Obviously, if , system (1) or (3) has a positive equilibrium point , whereLet , substitute them into system (3), and meanwhile drop the bars for simplicity of notations, then system (3) can be transformed into the following vector form:whereandThe vector form of the corresponding linearization system of system (8) is as follows:The characteristic equation of linearization system (11) of system (8) at the equilibrium point has the following form:i.e.,where
Case 1. When , the associated characteristic equation (13) becomesIf , the corresponding characteristic equation (15) has two roots with negative part for any .
Case 2. When , let us get the conditions of taking place spatially homogeneous Hopf bifurcation. It is known that when the associated characteristic equation becomes the following form:Let be roots of characteristic equation (13), thenSeparating the real and imaginary parts, we havewhich leads to By the simple analysis, we can immediately obtain the following results:
(a) If or , then (20) has no positive real root.
(b) If and , then (20) has two positive roots as follows:(c) If or , and , then (20) has only one positive root .
From the direct calculations and testing the above conditions, we can get that if and , then (20) has only one positive root defined byand the corresponding where .Then when , the characteristic equation has a pair of purely imaginary roots .
Lemma 1. (1) If holds, then (15) and (17) have the same number roots with positive real parts for all for any .
(2) If and hold, then (15) and (17) have the same number roots with positive real parts for for any .
Lemma 2. Suppose that hold and , then .
Proof. Denote by the root of characteristic equation such that Substituting into the characteristic equation and taking the derivative with respect to , we have which leads to This proof is completed.
Therefore, employing Lemma 1, Lemma 2, and Hopf bifurcation theorem for partial functional differential equations in [1], we can obtain the following conclusions.
Theorem 3. (1) If holds, then the coexistence positive equilibrium of system (8) is locally asymptotically stable for all .
(2) If and hold, then the coexistence positive equilibrium of system (8) is locally asymptotically stable for and unstable when .
(3) If and hold, then system (8) undergoes the Hopf bifurcation at the coexistence positive equilibrium for .
Remark 4. For , this bifurcation is called spatially homogeneous Hopf bifurcation. For , this bifurcation is called spatially nonhomogeneous Hopf bifurcation. This spatially homogeneous Hopf bifurcation is the same as the nondispersal equation in [16] if we impose the restrictive diffusion condition as follows:
3. Direction and Stability of the Bifurcating Periodic Solution
In this section, we mainly devote our interests to the properties of the bifurcating periodic solution, including the direction of bifurcation and stability of the bifurcating periodic solution.
Let , and , substitute them into system (8), and meanwhile drop the tildes above , then system (8) can be denoted in the form of abstract partial functional differential equationswhereFor convenience, we rewrite system (29) in the following form:where And the corresponding linearization system of system (29) becomesFrom the discussion of Section 2, we know that is an equilibrium point of system (29), and the characteristic equation of (33) has a pair of simple purely imaginary eigenvalues when , .
Consider the ordinary functional differential equation:and by the Riesz representation theorem, there exists a matrix function , whose elements are of bounded variation such thatIn fact, we can chooseLet denote the infinitesimal generators of the semigroup induced by the solutions of (34) and be the formal adjoint operator of under the following bilinear functional:for , . From the discussion of Section 2, we know that and have a pair of simple pure imaginary roots . Let and be eigenvector of about eigenvalues , respectively, and and be eigenvector of about eigenvalues , respectively.
Lemma 5. thenis a basis of with andis a basis of with , where , and are the center subspace. That is to say, and are the generalized eigenspace of operators and associated with , respectively, .
Proof. According to the definition of operators and , we have the following equations: We can obtain the values of and ,Letsuch that The proof is completed.
In addition, and , for any . The center subspace of linear equation (33) is given by , whereand , where denotes the complement subspace of in .
Let be the infinitesimal generator induced by the solution of (33). Then (29) can be rewritten as the abstract form where
Using the decomposition and (45), the solution of (31) can be written aswhere and .
In particular, the solution of (31) on the center manifold is given byLet , , and notice that . Then Then system (48) can be transformed into where Moreover by [1], satisfies where From (31) and (48), it follows that We can obtain the following results:According to the following equality,we can obtain the following results:where denote the coefficients of the , , , in the polynomial about and , respectively.
Next, in order to get the value of , we need to compute the expression of . It follows from (50) that In addition, by [1], satisfies Thus from (50), (52), (58), (59), (60), and (61) we can obtain the following results:From (60)–(62), when , we haveMeanwhile when , employing the definition of operator and (60), we have Combining (63) and (64), we can get the following results:
andMeanwhile note the following equalities:Associating with (65)–(67), we can obtain the following two equalities:Thus from (68) and (69), we can get the values of and , Summarizing the above analysis, we can obtain the expression of and and get the value of . According to the normal form theory developed by Wu [1], we can get the normal form of system (8).
Theorem 6 ([1]). whereFurther, the following several terms which determine the properties of bifurcating periodic solution are given as follows: which determines the direction of bifurcation. which determines the stability of the bifurcating periodic solution. which determines the variation of period of the bifurcating periodic solution.
For system (8), employing the above discussion, we can obtain the following further results.
Theorem 7. (1) If , then the bifurcating periodic solution exists in the side of and is unstable.
(2) If , then the bifurcating periodic solution exists in the side of and is stable.
Next we give the corresponding numerical results and set the parameters as follows:We can get the equilibrium and . Figures 1 and 2 present that the equilibrium of system (3) is locally asymptotically stable when the sum of two delays is less than . But when the sum of two delays is greater than , the equilibrium point of system (3) will be unstable and lead to the spatially homogeneous periodic solution to occur; see Figures 3 and 4.
4. Effects of Diffusion on the Bifurcating Periodic Solution
In the previous section, we mainly discuss the spatially homogeneous Hopf bifurcation and the relevant properties of the bifurcating periodic solution for the fixed domain and diffusion. However, in this present section, we will employ the method based on Fredholm alternative to investigate the effects of diffusions on the bifurcating periodic solution. For simplicity, we take the same notations of [1].
From the previous section, we know that is the eigenfunction of corresponding to the simple eigenvalue , that is, ; and is the eigenfunction of corresponding to the simple eigenvalue , that is, , where
Now we define , bywhere has the formand is determined by and is a constant 2-vector satisfying With the above preparations, we can now discuss the stability of as a solution of (29). First of all, we note that the linear variation equation of (29) about is given byAdopting the new variableswe can transform (80) into where , .
Let be the mth eigenvalue of the vector and let be the eigenfunction corresponding to , i.e.,Clearly, is a Floquet exponent of the linear -periodic system (80) if and only if there exists a function and a positive integer such that satisfies (80). So calculating the Floquet exponent is reduced to finding a solution to (80) in the form (84). Substituting (84) into (80) and comparing the coefficients of on both sides of (80), we obtain Equation (85) is independent of the spatial variable .
When , (85) coincides with the one induced from the linearized equation of the diffusion-free equation, that is, From the above equations, we know (86) has Floquet exponents 0 and .
Next we mainly consider the case in (85). Take any positive integer , greater than 1, and fix it. Let us parameterize the deformation of a domain and diffusion coefficients in terms of the same parameter as above. In what follows, we assume the parameterization is taken in such a way that for some matrix . Then (85) can be written as
Now in (88), we will find a real value function and a real number in the following form:where is the Floquet exponent of the linear periodic system (80) whose sign determines the stability of the periodic solution . We regard (89) as a perturbation of (86) with perturbation term (87).
According to the theory in [1] developed by Wu, after a slightly lengthy but simple calculation, we can obtain the following form expression about : which determines the stability of the bifurcating periodic solution as the diffusion coefficients vary, whereCombining the above formulas (74), (75), (77), (78), (79), (87), we can obtain the expression of (90) whose coefficients are determined by the parameters from the original system (29).
5. Discussions
In this paper, we mainly investigate effects of the sum of two delays and diffusion on the dynamical behaviors of the Lotka-Volterra type predator-prey model with two delays. By employing theories of Hopf bifurcation for some fixed diffusions and spatial domain, we find that delays can cause the spatially homogeneous equilibrium point to destabilize and lead to the spatially homogeneous periodic solution to occur. On the other hand, we also investigate effects of diffusions on the bifurcation periodic solution and find that once the value of diffusion varies the bifurcating stable periodic solution may be unstable. As Morita [17] had pointed out, diffusion would lead to other spatiotemporal patterns to occur such as the periodic traveling wave, even spatiotemporal chaos, which will be studied in future work. So it is very important for us to consider the interaction of delay and diffusion for biological system (1) and explore the mechanism of all kinds of spatiotemporal patterns.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors have read and approved the final manuscript.
Acknowledgments
This work is supported by the Natural Science Foundation of Shaanxi Provincial Education Department (Grant no. 15JK1016), the Research Fund of Ankang University (Grant nos. 2016AYQDZR10, 2017AYJC02), in part by the National Natural Science Foundation of China (Grant nos. 11501064, 11426047), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant no. 2014-56-11), China Postdoctoral Science Foundation (Grant no. 2016M590850), Chongqing Postdoctoral Science Foundation Special Funded Project (Grant no. Xm2017174), and the Research Fund of Chongqing Technology and Business University (Grant no. KFJJ2017066).