Dynamics of a diffusive delayed competition and cooperation system

Abstract In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.


Introduction
In the past few decades, delay differential equations that change in time and involve delays have become a hot research topic. Many complicated and large-scale systems in nature and society can be modeled as delay differential systems, due to their flexibility and generality for representing virtually any natural and man-made structure. They have received much attention in interdisciplinary subjects including natural science [1][2][3], engineering [4], life sciences, and others [5,6]. In particular, many scientists paid their attention to the stability and bifurcation phenomena of the predator-prey system with multiple delays (see, for example, [7][8][9][10][11][12][13]). When the delay is continuous and modeled by a convolution, the problem on the periodic phenomenon can be restricted on the critical manifold, and the limit cycle can be detected by the zeros of Melnikov function, see [14][15][16]. In fact, much commonness is reflected between species that co-evolve in nature and different enterprises that co-exist in economic society, so numerous researchers have widely presented the competition and cooperation model of the enterprises [17,18], which are governed by the following ordinary differential equation: 1 , where x t x t , 1 2 ( ) ( ) denote the output of enterprise x 1 and enterprise x 2 at time t, respectively; x t x t , 1 2 1 1 ( ( ) ( ))∈ × . The parameters r i 1, 2 i ( = ) represent the intrinsic growth for output of two enterprises; K i 1, 2 i ( = ) measure the load capacity of two enterprises in an unrestricted natural market; α β , stand for the coefficient of competition of enterprise x 1 and x 2 ; c i 1, 2 i ( = ) denote the initial production of them. All the parameters in the above model are positive. Taking into account the influence of history, the researchers introduce the time delay τ to the feedback in model (1.2), which is a more realistic approach to the understanding of competition and cooperation dynamics. Delays can induce various oscillations and periodic solutions through bifurcations as the delay is increasing. Therefore, it is interesting to investigate the following delayed model: where τ i 1, 2, 3, 4 0 i ( = ) ≥ represent the time delay, system (1.3) had been studied extensively by many researchers and some interesting conclusions have also been obtained in [17][18][19].
In the natural economical environment, due to limited customer resources, enterprises are not evenly distributed in the space, and in order to survive, enterprises will look for customers everywhere, which will lead to migration and diffusion. Therefore, considering the heterogeneity of enterprise spatial distribution, motivated by the present situation stated above, we take the inhomogeneity of the spatial distribution into account and obtain the following competition and cooperation system incorporating diffusion and delay subject to Neumann boundary conditions where e 1 and e 1 are the diffusion coefficients of competition and cooperation enterprises, l ∈ + , and l Ω 0, π = ( ) is a bounded domain with a smooth boundary Ω ∂ . Note that the homogeneous Neumann boundary condition means that neither enterprise can cross the boundary. The appearance of the spatial dispersal makes the dynamics and behaviors of the organisms even more complicated [20][21][22]. The investigation in connection with the dynamics of the diffusive competition and cooperation system (1.4) will make significant economic implications. It is worth mentioning that in the study of systems with diffusion terms, different boundary conditions represent different practical meanings. For instance, for the predator-prey system, homogeneous Dirichlet boundary conditions are imposed so that both species die out on the boundary [23,24], and the homogeneous Robin boundary condition means that the prey or predator can cross the boundary [25].
This paper aims to investigate the stability of equilibria and the properties of Hopf bifurcation at the unique positive constant equilibrium of system (1.4). The rest of this paper is organized as follows. In Section 2, the stability properties of the equilibria are studied for system (1.4) with τ 0 = . In Section 3, as for system (1.4), the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter, and the direction and stability of spatial Hopf bifurcating periodic solutions are determined. Finally, some numerical simulations and summarizations are given in Sections 4 and 5.
2 Stability analysis of equilibria for the diffusive system In this section, we only consider the diffusive competition and cooperation system without delay terms subject to Neumann boundary conditions Proof. The proof is similar to that in [17], we omit it here. Suppose H 1 ( ) holds, then E u v , = ( ) * * * is the unique positive constant equilibrium of system (2.5). We discuss the stability of the unique positive constant equilibrium. First, we transform E u v , = ( ) * * * of system (2.5) to the origin via the translation u u u v v v ,= − = − * * and drop the hats for simplicity of notation, Define the real-valued Sobolev space : , 0, π , , 0,0 and the complexification of X to be X X iX x ix x x X , .  Clearly, the roots of (2.10) are given by Based on the aforementioned statements, we have the following conclusions. □ Theorem 1. Suppose H 1 ( ) holds, then T 0 n < and D 0 n > , for n 0 ∈ , that is, all the roots of Eq. (2.12) have negative real parts. More precisely, the unique positive constant equilibrium is locally asymptotically stable.    [17], assume that H 1 ( ) holds, and the positive equilibrium E * of system (1.2) is asymptotically stable. In this paper, we show that the positive equilibrium of system (2.5) is also asymptotically stable. That is, the diffusive term has no influence on the dynamical behavior of system (1.2).

Dynamical analysis of the diffusive delayed system
In this section, we consider the diffusive delayed competition and cooperation system subject to Neumann boundary conditions where τ 0 > stands for the feedback delay of the one enterprise to the other one. Suppose H 1 ( ) holds and τ 0 ≠ , and we discuss the existence of local Hopf bifurcations occurring at the unique constant positive equilibrium E u v , = ( ) * * * by regarding delay term τ as the parameter. We first transform E u v , = ( ) * * * of (3.13) to the origin via the translation u u u v v v ,= − = − * * and drop the hats for simplicity of notation, then system (3.13) is transformed into In the phase space , system (3.14) can be regarded as the following abstract functional differential equation: Then the linearization of system (3.15) at the origin is given by According to [26], we obtain that the characteristic equation for linear system (3.16) is given by It is well known that the eigenvalue problem Therefore, the characteristic Eq.
We make the following hypotheses From the result of [27], the sum of the multiplicities of the roots of (3.18) in the open right-half plane changes only if a root appears on or crosses the imaginary axis. In the following, we will derive the conditions under which the aforementioned cases occur. Denote Then we have the following lemma. Here, Applying the same analytical steps as those in Ruan and Wei [27],     = ± , respectively. By Lemma 4, the transversality condition is also satisfied. According to Ruan and Wei [27], we have the following theorem. In this part, we shall study the direction of Hopf bifurcation and stability of the bifurcating periodic solution of system (3.13) by applying center manifold theorem and normal form theorem of partial functional differential equations [26][27][28].   ). A τ ( ) has a pair of simple purely imaginary eigenvalues iω τ n ± , and they are also eigenvalues of A * . Let P and P * be the center subspace, that is, the generalized eigenspace of A τ ( ) and A * associated with Λ n , respectively. Then P * is the adjoint space of P and P P dim dim 2 = = * . It can be verified that p θ ξ e θ p θ p θ 1, 1, 0 , is a basis of A * with Λ n and q r η e r q r q r 1, 0, 1 , ]. Then we can compute by (3.26) By the decomposition of P P CN S 1 1 1 = ⊕ , the solution above can be written as and h x x μ P h Dh , , , 0, 0, 0 0, 0, 0, 0 0.
In particular, the solution of (3.21) on the center manifold is given by

27)
Let z x ix 1 2 = − , and note that p i Φ Φ and h x x h z z i z z , , 02 ,2 , 0 .

Numerical simulations
Through the previous discussion, we conclude that the delay term plays an important role in the diffusive delayed system, which can let the stable equilibrium unstable. In this section, we shall give some numerical simulations and actual conclusions to support the theoretical analysis discussed in the previous section.
Dynamics of a diffusive delayed competition and cooperation system  1245 First, we consider the following diffusive model contains no delay term  which satisfies H 1 ( ), by computing, the positive equilibrium E 1, 1 = ( ) * . By Theorem 1, we get that E * is asymptotically stable as demonstrated in Figure 1(a) and (b) in the u x t − − space and v x t − − space, respectively. That is as time increases, the numerical solution tends to the positive equilibrium E * .
Furthermore, we study the following specific diffusive model with delay term x v x x x = . However, when τ crosses the critical value τ 0 0 , a family of inhomogeneous periodic solutions is bifurcated from E * . We choose τ 2 = , E * loses its stability and Hopf bifurcation occurs when τ crosses τ 0 0 as illustrated in Figure 3(a) and (b), respectively.

Conclusions
In this paper, a diffusive competition and cooperation system subject to local delayed feedback control under Neumann boundary value conditions has been studied in detail to show its rich spatial-temporal patterns. From the economical aspect, the most interesting results are the following: under certain hypotheses, the patterns caused by the Turing instability can be expected for the competition and cooperation model. In particular, it is interesting that delayed feedback control can break the stability of the system and stabilize the unstable oscillation in an originally spatially stable domain. With the increase of delay, the constant equilibrium may switch finite times from stability to instability to stability and become unstable, and a sequence of inhomogeneous periodic solutions bifurcates from the equilibrium eventually. That is, delayed feedback control plays an essential role in destabilizing the spatially extended system. Moreover, the short-term data observed in nature may be misleading to make predictions due to complex dynamical behaviors. The analysis and interesting observations in this paper may be useful both in the mathematical and economical research areas.   Dynamics of a diffusive delayed competition and cooperation system  1247