Dynamics of a diffusive delayed competition and cooperation system

1 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:

where x 1 ( t ) , x 2 ( t ) denote the output of enterprise x 1 and enterprise x 2 at time t, respectively; ( x 1 ( t ) , x 2 ( t ) ) ∈ ℝ 1 × ℝ 1 . The parameters r i ( i = 1 , 2 ) represent the intrinsic growth for output of two enterprises; K i ( i = 1 , 2 ) 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 ( i = 1 , 2 ) denote the initial production of them. All the parameters in the above model are positive.

Let a 1 = r 1 K 1 , a 2 = r 2 K 2 , b 1 = α r 1 K 2 , b 2 = β r 2 K 1 , d i = r i − a i c i , u ( t ) = x 1 ( t ) − c 1 , v ( t ) = x 2 ( t ) − c 2 . System (1.1) becomes

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 ( i = 1 , 2 , 3 , 4 ) ≥ 0 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 ∈ ℝ + , and Ω = ( 0 , l 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

3 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 ˆ = u − u ∗ , v ˆ = v − v ∗ and drop the hats for simplicity of notation, then system (3.13) is transformed into

In the phase space C τ ≔ C ( [ − τ , 0 ] , X ) , system (3.14) can be regarded as the following abstract functional differential equation:

where U ≔ ( u , v ) ∈ H 2 ( 0 , l π ) , e = diag ( e 1 , e 2 ) , dom ( e Δ ) = X , L : C τ ↦ X , and F : C τ ↦ X are defined by

for ϕ = ( ϕ 1 , ϕ 2 ) T ∈ C τ .

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

has eigenvalues μ n = n 2 l 2   ( n ∈ ℕ 0 ) with corresponding eigenfunctions φ n ( x ) = cos n l x .

Substituting

into the characteristic Eq. (3.17), it follows that

Therefore, the characteristic Eq. (3.17) is equivalent

where

By Eq. (3.18), we can get Δ n ( 0 , τ ) = B n − C n > 0 and obtain the following lemma.

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

where P ˜ = 4 e 1 e 2 ( u ∗ + c 1 ) ( v ∗ + c 2 ) ( a 1 a 2 − 4 b 1 b 2 u ∗ v ∗ ) , and

Then we have the following lemma.

Applying the same analytical steps as those in Ruan and Wei [27], when τ > 0 , λ = i ω ( ω > 0 ) is a root of (3.18) if and only if ω satisfies

Then we have

which leads to

Let z = ω 2 , then (3.19) can be rewritten into the following form

and its roots are given by

We obtain

and

is increasing with respect n ≥ 0 , then

Under ( H 3 ) , we have

In addition,

is increasing with respect n ≥ 0 . Hence,

Based on the discussion above, the two statements hold and ω n ± = z ± .

For simplicity, we only consider case (ii) in Lemma 3 and use τ n j to denote τ n + , j . Note that τ m j = τ n k , for some m ≠ n may occur. In this paper, we do not consider this case. In other words, we consider

Let λ n = α n ( τ ) + i ω n ( τ ) be the root of (3.18) satisfying α n ( τ n j ) = 0 and ω n ( τ n j ) = ω n when τ is close to τ n j . Then we have the following transversality condition.

Differentiating two sides of (3.18) with respect τ , we have

Then

Therefore, α n ′ ( τ n j ) < 0 .

Denote