Delay-induced self-organization dynamics in a prey-predator network with diffusion

Considering that time delay (delay) is a common phenomenon in biological systems, reaction-diffusion equations with delay are widely used to study the dynamic mechanism of those systems, in which delay can induce the loss of stability and degradation of performance. In this paper, taking into account the inhomogeneous distribution of species in space and this can be considered as a random network, the pattern dynamics of a prey-predator network system with diffusion and delay are investigated. The effect of delay and diffusion on the network system is obtained by linear stability analysis, including the stability and Hopf bifurcation as well as Turing pattern. Our results show that the stability of the system changes with the value of delay. Moreover, we obtain Turing pattern related to the network connection probability and diffusion. Finally, the numerical simulation verifies our results.

model, there are many variations in this model; for example, Aziz-Alaoui and Daher [9] proposed a modified predator-prey model, and Leslie [10] studied the boundedness and global stability of the model.
For a reaction-diffusion system, Turing [1] first discovered that diffusion could cause the system to be unstable under certain conditions and obtained Turing pattern phenomena. Then, many scholars began to study the pattern formations of the prey-predator model by using the reaction-diffusion equation [11,12,13,14,15]. Camara and Aziz-Alaoui [16] investigated a modified Holling-Tanner model and found Turing and Hopf bifurcations; they also derived the bifurcation conditions. Then, for a diffusive predator-prey model with the Holling-Tanner functional response, Shi et al. [17] investigated the stability and the existence of a positive solution.
By studying diffusion, Othmer and Scriven [18] proposed a general method for analyzing the instability of the reaction-diffusion model in small networks, and they also obtained the effect of network topology changes on the stability and pattern. Then, Jansen and Lloyd [19] extended the method to large random networks and gave the corresponding Turing patterns. Aly and Farkas [20] studied the effect of diffusion in a small network on the Turing pattern of the prey-predator system, where the migration of populations among patches causes diffusion. Subsequently, Turing patterns in complex organized networks were studied in [21,22,23]. For the FitzHugh-Nagumo model in a networkorganized system, Zheng and Shen [24] recently studied Turing instability, where the network structure and diffusion lead to Turing instability. Liu et al. [25] placed the prey-predator model into complex networks and found that the network topology and diffusion rate can affect the Turing pattern formation. Considering the inhomogeneous distribution of prey and predators in communities and the fact that communities in real life can be abstracted as a complex network structure, we can investigate a prey-predator diffusive system functional response in the random network.
Time delay is a universal phenomenon in biological systems [26,27,28]. For the prey-predator system, delay generally represents the gestation or maturity period of population reproduction; it can destroy the stability and lead to Hopf bifurcation as well as Turing bifurcation. Ruan and Sen [29,30] provided two methods to investigate the time delay. In [31], Yang and Zhang considered a diffusive prey-predator system with delay, and they obtained Turing instability and Hopf bifurcation caused by the delay. Recently, Chang et al. [32] investigated a predator-prey diffusive model in complex networks and found that delay can induce pattern dynamics. Wang et al. [33] found that the network structure hardly has an impact on the stability of this model with delay and that a large time delay can induce the thick-tailed phenomenon of evolution patterns.
However, there are few studies on reaction-diffusion systems with delays in diffusion networks. In this paper, for a prey-predator diffusion network with Holling type II and modified Leslie-Gower functional response, we will investigate the dynamical behaviors emerging from the time delay caused by predator gestation. For the random migration and inhomogeneous spatial distribution of prey and predators among different communities, we consider them through a complex network and derive the stability condition of the positive equilibrium by linear stability analysis in Section 2. Then, in Section 3, we use two methods to study Hopf and Turing bifurcations caused by delay and network diffusion. In Section 4, consistent results are obtained by numerical simulation. Finally, the results are summarized in Section 5.

Stability analysis of the prey-predator diffusion network without delay
The modified Leslie-Gower prey-predator diffusion network with a Holling type II functional response can be written as follows: where v u+b is the modified Leslie-Gower functional response. Considering the inhomogeneous distribution of species and the effect of delay τ (τ > 0) caused by predator gestation, model (1) with delay in the random network is where k i is the degree of node i (we assume that k i is a decreasing sequence, that is, k 1 ≥ k 2 ≥ · · · ≥ k N ) and A ij is the adjacency matrix's element. N nodes make up the random network, and the nodes are connected with the connection probability p [24]. If τ = 0, the system (2) becomes: Clearly, the equilibrium points of system (3) are derived by the following equations: Then, when a − δ β b > 0, we obtain only one positive equilibrium point E (u * , v * ): . We assume that system (3) has only a positive equilibrium point E(u * , v * ) in the following, namely, a − δ β b > 0.

Stability analysis of the model without delay and diffusion
By linear stability analysis, the Jacobian matrix of (3) without diffusion at Then, the corresponding characteristic equation is where According to the Routh-Hurwitz Criterion, Eq. (5) has roots with negative real parts when trJ < 0 and detJ > 0, namely, Lemma 1 Supposing that condition (7) holds, for system (3) without diffusion, E(u * , v * ) is asymptotically stable [9].

Stability analysis of the network diffusion model without delay
For system (3), the eigenvalues Λ α and their corresponding eigenvectors has the general solution of the following form: Substituting Eq. (8) into Eq. (3) give the following Thus, the characteristic equation of system (3) is where The roots of Eq.

Theorem 1 Supposing that condition
is asymptotically stable if the diffusion coefficients d 1 and d 2 satisfy where > 0 always holds, then q 1 > 0. In addition, it is evident that p 1 < 0 (due to that Λ α ≤ 0 always holds). The proof is completed.

Stability analysis of the predator-prey diffusion network with delay
For system (2), the equilibrium point is clearly E(u * , v * ). Here, we use two methods [29,30] to study the bifurcation dynamics property of system (2).

Method I
In this section, according to the method in [29] and regarding the delay τ as the bifurcation parameter, we analyze the local stability of E(u * , v * ) and Hopf bifurcation.
By linear stability analysis and substituting Eq. (8) into Eq. (2), we can obtain Therefore, the corresponding characteristic equation is where When τ = 0, Eq. (12) becomes (10). (7) and (11) hold and regardless of what the network connection probability p is, then Equation (12) has at least a simple pair of complex roots with zero real part ±iω α+ when τ = τ α j , where

Method II
In this section, according to the method in [30] and assuming the delay τ to be small, we consider the Turing instability of system (2).
Because τ is small enough, we replace v i (t − τ ) = v i (t) − τ dvi dt , and then, system (2) can be written as Expanding (16) in the Taylor series and neglecting the higher order non- According to the research method in Section 2.2, we obtain the corresponding characteristic equation of the system at E(u * , v * ) as where a22 . Turing instability means that system (2) without diffusion is stable (namely, when (7) holds), but unstable for diffusion system (2). According to the Routh-Hurwitz criterion, system (2) is stable when p 2 < 0 and q 2 > 0 always hold. When at least one of p 2 < 0 and q 2 > 0 is violated, Turing instability occurs. Therefore, we have the following theoretical results. (7) hold, for system (2)

Numerical results
In this section, based on the above theoretical results, we present some numerical simulations. Here, we construct a random network with N = 100 nodes and consider the parameters as: a = 0.08, b = 0.04, δ = 0.6, β = 0.6, d 1 = 0.02.

Conclusion
In this paper, a prey-predator diffusion network with delay is studied, and the stability of the positive equilibrium point E(u * , v * ) is investigated. From the linear stability analysis, we find that delay can change the stability and produce Hopf and Turing bifurcations. Then, the numerical simulations are given.
First, for the network model without delay, we give the conditions of asymptotic stability, and the corresponding figures are shown in Figures 1 and 2. Then, for the network system with delay, we use two methods to analyze the influences of delay and diffusion on the stability (Theorems 2 and 3). Hopf and Turing bifurcations are obtained. When condition (11) holds, the network system always undergoes Hopf bifurcation and Turing bifurcation regardless of the network connection probability p [Figures 3, 4, 5, 6, and 11]. Namely, regardless of how the network nodes are connected, prey and predators will coexist, and the system will oscillate periodically. If condition (11) is not held, Turing instability occurs when p approximately satisfies 1 Figures 7-10 show the numerical simulations. For this Turing instability caused by network diffusion, the network structure affects the stability. However, there are some problems that we have not studied, such as the stability and period as well as the direction of the bifurcating periodic solution. Thus, we will study these problems in the future.