Pattern Dynamics in a Predator-Prey Model with Diffusion Network

Diffusion plays an essential role in the distribution of predator and prey. We mainly research the diffusion network’s effect on the predator-prey model through bifurcation. First, it is found that the link probability and diffusion parameter can cause Turing instability in the network-organized predator-prey model. +en, the Turing stability region is obtained according to the sufficient condition of Turing instability and the eigenvalues’ distribution. Finally, the biological mechanism is explained through our theoretical results, which are also illustrated by numerical simulation.


Introduction
Turing instability was first investigated in the reaction-diffusion system [1], and it is constantly being promoted to explain the dynamical mechanism [2,3]. Asllani et al. pointed out that the directed network could induce Turing instability when an indirect network does not work [4,5]. Meanwhile, tuning the topology structure of the system can create or destroy patterns in a reaction-diffusion system [6]. Mimar et al. proved that the pattern formation's topological properties are determined by complex interaction [7]. Although spontaneous patterns [8] are associated with the dominance of eigenvectors and eigenvalues [9][10][11], the dynamical mechanism of the random network in pattern formation remains to be uncovered.
Since the Lotka-Volterra model was proposed in the early twentieth century, some biological mechanisms were explained in the predator-prey system [12][13][14][15][16][17][18]. Chang et al. researched the dynamics in the predator-prey system on complex networks and found that Turing instability caused by delay can generate spiral waves [19]. Liu et al. showed the effect of network and diffusion on the ecological balance of the predator-prey system [20]. Upadhyay and Bhattacharya studied the differences between the aqueous and terrestrial environments in predator-prey networks and tried to explain their biological mechanism [21]. Astarloa et al. tried to use the joint species distribution modeling to reveal the coexistence problem of prey and predator in the Bay of Biscay [22]. Although previous work shows that diffusion and randomness influence the Turing pattern significantly, the random network's effect on the distribution of predator and prey should be carefully evaluated.
To reveal the natural mechanism of biological invasion, we intend to investigate the random diffusion network's effect on the network-organized predator-prey model's stability. First, we obtain the conditions under which Turing bifurcation arises. Second, we find an estimated region of all the eigenvalues of the Laplacian matrix, the sufficient stability conditions in the network-organized predator-prey model.
ird, we explain the network-organized Turing instability by the mean-field approximation and comparison principle. Also, we estimate the Turing instability range about link probability and diffusion and try to explain the mechanism of biological invasion. Last, we illustrate our theoretical results through numerical simulation. matrix L � L ij can be treated as L ij � A ij − k i δ ij , and all the eigenvalues of L are Λ � Λ i . e eigenvectors ϕ i ∈ R n of Λ i satisfy Lϕ i � Λ i ϕ i , i � 1, . . . , n. L is real symmetric, and we choose an orthonormal basis for ϕ i ϕ j � δ ij , where the degree of node i is k i , and δ ij is the Kronecker delta function.
When r 1 a 22 − r 2 a 12 /a 11 a 22 + a 12 a 21 > 0, r 1 a 21 + r 2 a 11 / a 11 a 22 + a 12 a 21 > 0, E 3 changes its stability, and Hopf bifurcation occurs if a 12 � a * 12 . Typically, the distribution of individuals is spatially heterogeneous. So, we research (1) with a reaction-diffusion and network as follows: where d 1 , d 2 are the diffusion constants. Generally, we can regard diffusion as an interplay between network nodes. In this article, we consider a distinctive interaction between nodes. So the networkorganized system (7) is   2 Complexity e linearized network-organized system of the system (8) is e general solution of the linear network-organized system can be expressed as [3].
where all the eigenvalues of matrix L can be represented as en, system (8) has the following characteristic function: where Turing instability occurs when there is a Λ i with Reλ > 0. From (12), We can get the Turing instability's critical value about Λ Turing instability of the reaction-diffusion system is the Turing instability's prerequisite in the networkorganized system. Based on the Gershgorin circle theorem [9,10], we have.
(i) Result 1: k max � max k i , k min � min k i and Λ i is the eigenvalue of the Laplacian matrix L, then, To study the mechanism of the network-organized system's stability, we research (8) through the comparison principle: let x i � x * + εx i and y i � y * + εy i , where 0 < ε ≪ 1. Substituting x i , y i in (9), expanding in ε, we can get the linearized system: We resolve the first-order perturbations into ϕ i 's eigenfunction expansions, to consider the system's stability. Let x y

Complexity 3
Substituting (15) into (20) and noting the properties of Using the comparison principle from [26], we note where Q(t) < 0. We eliminate one of X i or Y i from (16), then where p Λ i , q Λ i are as in (12). en, we obtain the generalized condition: Assume that the instability condition (19) holds. en, the homogeneous state (x k , y k ) � (x * , y * ) is unstable under the ith Turing mode (15).
We also consider the system (8) through the mean-field theory: where H x � n j�1 A ij x j , H y � n j�1 A ij y j and k i is the network's degree. We let other nodes stay at equilibrium (x * , y * ), and rewrite the single-node system, erefore, it is easy to obtain the characteristic equation: where p k i � − (α 10 + β 01 ) + d 1 k i + d 2 k i , q k i � α 10 β 01 + (α 10 d 2 + β 01 d 1 )k i + d 1 d 2 k 2 i . Assume λ 1 and λ 2 are two eigenvalues of system (22). en system (21) is stable when d 1 � d 2 � 0, namely p k i > 0, so λ 1 + λ 2 � − p k i < 0. If q k i > 0(q k i < 0). en, the system (21) is stable(unstable). To sum up, we can use the comparison principle(the mean-field theory) to conclude that the system (8) is unstable when q Λ i < 0(q k i < 0).

Numerical Simulation
We give some numerical analysis based on the earlier theoretical analysis. From Figure 3, if we choose r 1 � 1, r 2 � − 1, a 11 � − 1.158, a 12 � 1, a 21 � 2, a 22 � 1, the equilibrium point (x * , y * ) is asymptotically stable, namely, ecological balance persists as long as there is no biological invasion. at means the predator and the prey can coexist. e equilibrium (x * c , y * c ) � (1.1326, 1.2651) is unstable, and Hopf bifurcation occurs Figure 4, when a 12 passes through the critical value a * 12 � 1.7271. Because l 1 � 8.2123 > 0, the Hopf bifurcation is subcritical, the preypredator system shows periodic changes; thus, this state is easy to destroy. e equilibrium point (x * , y * ) becomes unstable, when a 12 � 1.8271 > a * 12 . As diffusion is a vital factor in the distribution of predator and prey, we should not ignore migration. So we construct the random network and transform it into the Laplacian matrix L ij . And we consider how the random network affects the equilibrium point's stability when (7) is stable.
Based on the above theoretical analysis, the Turing instability in the reaction-diffusion equation is a precondition for Turing instability in a network-organized system. Turing instability occurs in a reaction-diffusion equation when Turing bifurcation parameters d 2 and d 1 are in a certain range Figure 1. Namely, diffusion behavior is universal and allowed, but the relative diffusion rate needs to be within a specific range; otherwise, the ecosystem will be out of balance and destroyed.
To observe Turing instability in the network-organized, we should guarantee d 1 < 0.1562d 2 while changing p. However, the Laplacian matrix eigenvalues are Λ 1 , Λ 2 , . . . , Λ N , N is the number of nodes, and k 2 is continuous in a reaction-diffusion system. In other words, the distribution of − Λ i determines the system stability. e critical point is d 1c � 0.3124 when d 2 � 2. e range of eigenvalues Λ i could be obtained by result 1. e eigenvalues of L are discrete and included in the continuous region Figure 2. e red region shows the estimated range of Turing instability about (p, d 1 ) in Figure 5. Besides, we obtain the estimated range of p, (d 2 ln N/d 1c N) > p > (1/N 2 ), which determines the system dynamical behavior and network characteristics. Moreover, p � ln N/N is the critical value between the connected network and sparse network [27]. Anyway, the invasive rate of species and p in the predatorprey network play a vital role in balancing the native biological system.
Let the value of (p, d 1 ) lie in the blue region shown in Figure 5 (here, we set p � 0.00006, d 1 � 0.1124), then,   (Figure 6(a)) and the equilibrium is stable (Figure 6(a)). We verified the correctness of the previous analysis with bifurcation diagrams (Figure 6(b)). We can conclude that with a fixed predator invasion rate and a fixed prey invasion rate, the link probability between two different systems will negatively impact the entire predator-prey network stability. Even the predator-prey network's diffusion induces the species extinction. Figure 7 shows an example of p that belongs to the red region in Figure  5, such as p � 0.08, d 1 � 0.1124 < d 1c � 0.3124, almost all of Λ i (k 2 ) in instability region Figure 7(a) left(up) and 7(a) right(up) shows that Turing instability occurs. If p � 0.08 and d 1 � 1.1124 > d 1c � 0.3124 (i.e., (p, d 1 ) in the blue region in Figure 5), there is no point Λ i (k 2 ) in the instability region (Figure 7(a), left(down)), the system is stable (Figure 7(a), right(down)). We also verified the correctness of the previous analysis with a bifurcation diagram (Figure 7(b)). Complexity e ecological interpretation is that the system is less stable when the invasion rate of the prey is lower when the link probability of predator-prey systems is fixed. Conversely, if we want to keep the predator-prey system stable, the invasion rate of the prey needs to be decreased.

Conclusion
is paper theoretically derives Turing instability conditions [28][29][30][31] in a predator-prey network and carries out a detailed numerical study. We study the effects of diffusion and link probability on pattern formation in a random system. e smaller the invasion rate d 1 of the prey undergoes, the easier the predator-prey network Turing instability. Meanwhile, Turing instability occurs when the link probability p falls in the approximate region (d 2 ln N/d 1c N) > p > (N 2 /1), which depends on the system characteristics.
Regarding an explanatory biological mechanism, the link probability and the invasion rate significantly influence the entire predator-prey network stability. e lower the invasion rate of the prey is, the less stable it is. Conversely, the prey invasion rate can be increased if we want to keep the predator-prey system stable.
at means we can find an equilibrium point between native and invasive species by adjusting the diffusion probability and species' invasion rate. Finally, we obtain the estimated region of p, (d 2 ln N/d 1c N) > p > (1/N 2 ). us, invasion rate of species and the link probability in the predator-prey network play a key role in balancing the native biological system.

Data Availability
All data and codes can be found on Baidu Wangpan (password: reff) and Baidu.

Conflicts of Interest
e authors declare that there are no conflicts of interest.