Turing Patterns in a Predator-Prey System with Self-Diffusion

and Applied Analysis 3 Then, there exists a unique positive equilibrium state for system (6), denoted by u = (u 1 , u ∗ 2 ), where


Introduction
In ecological systems, the interactions of different species indicate abundant dynamical features.It is informative to use mathematical model to study the interactions of species in these systems.Among these models, predator-prey systems, which were based on the pioneering works of Volterra [1], have been important in ecological problems.However, since we live in a spatial world, the predator-prey systems should include spatial factors.Thus, these systems should be described by using reaction-diffusion equations.As a result, it is an open problem to understand spatiotemporal behaviors of the temporal-spatial predator-prey systems [2][3][4][5][6][7][8][9][10][11].Thereinto, the formation of spatial patterns of predatorprey systems is a very active research area [12][13][14][15], which is based on the pioneering work of Turing [16] in 1952.
In recent years, there are a lot of bodies of literature to study the predator-prey system by taking into account the normal diffusion as well as cross-diffusion [17][18][19][20].Normal diffusion is a natural phenomenon of the movement of the prey or the predators from higher-density regions to lowerdensity ones.Cross-diffusion of the prey expresses a flux of the prey because of the presence of the predators and vice versa.Furthermore, in predator-prey systems, cross-diffusion can induce Turing instability to produce spatial patterns even though spatial homogeneous equilibrium states for the corresponding system in the absence of cross-diffusion are stable [8,11,[21][22][23][24][25][26][27][28].
Besides the normal diffusion and the cross-diffusion of the predator and prey in ecological systems, there exists, in fact, another diffusion form-self-diffusion for the pressure of their own species.It can describe the tendency to move along the direction of lower density of the predator's and prey's own species [29].Unfortunately, most of the studies mainly focused on well-posedness of solutions for predatorprey systems with self-diffusion [30,31].Little attention was paid to examine Turing patterns of these systems.Based on the above discussion, in this paper we mainly concentrate on Turing instability of a predator-prey system that includes a normal diffusion, cross-diffusion, and self-diffusion terms.To this end, we find a sufficient condition to generate Turing patterns.By using numerical simulation, for this system we examine parameter regions of forming patterns and show snapshots of spatial patterns.
The paper is organized as follows.In Section 2, we build the predator-prey model with nonlinear diffusion terms including normal diffusion, cross-diffusion, and selfdiffusion terms and the biological meaning of these parameters are interpreted.In Section 3, we find the sufficient condition of Turing instability.By performing a series of

A Predator-Prey Model with Nonlinear Diffusion
In this paper, we are interested in the spatiotemporal patterns of the following predator-prey system with nonlineardiffusion terms.Mathematical properties for this system have been investigated in [30,31]: where Ω is a bounded domain in R 2 with smooth boundary Ω and represents the domain that these two species inhabit.The vector  is the outward unit normal vector of Ω.In this model,  1 and  2 are the prey and the predator densities, respectively;   ,   , and   ( = 1, 2) are positive constants. 1 ,  2 are rates of the prey and predator proliferations for food source;  1 / 1 and  2 / 2 are environmental carrying capacities for prey and predator, respectively. 1 is a consumption rate;  2 is a conversion rate.In the diffusion terms, the constant   ( = 1, 2), which is usually termed as a normaldiffusion coefficient, represents the natural dispersive force of movement of a species.The positive constants  12 and  21 are referred to as cross-diffusion coefficients, which describe that the prey tends to avoid higher density of the predators and vice versa by diffusing away.In addition, for the predators and the prey, the positive constants  11 and  22 are self-diffusion rates due to pressure within their own species.Next, we want to look for the condition on the parameter values such that a positive homogeneous equilibrium state is linearly stable in the absence of the cross-diffusion and the self-diffusion (i.e., a normal reaction-diffusion system) but unstable in the present of the cross-diffusion and the selfdiffusion.
For simplicity, we set up the following notation as in [22,24].Notation 1.Let 0 =  1 <  2 < ⋅ ⋅ ⋅ → ∞ be the eigenvalues of −Δ on Ω under no-flux boundary condition, and let (  ) be the space of eigenfunction corresponding to   .We define the following space decomposition: (i) X  := {  c : c ∈ R 2 }, where   are orthonormal bases of (  ) for  = 1, . . ., dim (  ); and thus X=⊕ ∞ =1 X  , where Notation 2. For the sake of simplicity, we denote reaction terms for systems (1) by and its Jacobian matrix at the point u * is The diffusion term is denoted as and its corresponding Jacobian matrix at the point u * is ) . (5)

Linear Stability Analysis of System (1) with a Normal Diffusion
Let   = 0 for ,  = 1, 2; then the system (1) degenerates into standard reaction-diffusion equations: Through this paper, we assume the following conditions: Then, there exists a unique positive equilibrium state for system (6), denoted by u * = ( * 1 ,  * 2 ), where Theorem 1.If there are no cross-diffusion and self-diffusion, the positive equilibrium state u * of the system (6) is locally asymptotically stable when condition [1] holds.
Proof.The linearization of ( 6) around the steady state u * can be therefore expressed by where  = diag( 1 ,  2 ).Obviously, the operator Δ+G u (u * ) is invariant in the subspace X  , and   is an eigenvalue of this operator on X  , if and only if it is an eigenvalue of −   + G u (u * ).Direct calculation shows the characteristic equation where It is easy to be verified that   is negative and   positive.Thus, for each  > 1, the two roots of (10) have negative real parts.Consequently, we complete this proof.
Remark 2. According to the proof of Theorem 1, we can also calculate out the eigenvector c corresponding to the eigenvalue   for the operator −   + G u (u * ), which satisfies Furthermore, we can yield corresponding eigenspace {  c} in X  .
By Theorem 1, under condition [1] system (6) cannot destabilize u * .Next, the cross-diffusion and self-diffusion are taken into account.Proof.We first linearize the system (1) around By the same method as the proof of Theorem 1, we consider the operator Φ u (u * )Δ + G u (u * ) on the subspace X  .The eigenvalue of this operator on X  is denoted as , and then it is also an eigenvalue of the matrix −  Φ u (u * ) + G u (u * ).Thus,  satisfies the following equation: According to Notation 2, we can calculate and get det where  =  1  2 +  1 ( Let  1 (  ) and  2 (  ) be the two roots of ( 13); then we have In order to ensure that Re  1 (  ) < 0 and Re  2 (  ) > 0, a sufficient condition of Turing instability about homogeneous Next, we look for the diffusion conditions such that det(G u (u * ) −   Φ u ) < 0 holds.Furthermore, we have lim Taking condition [2] in consideration, we can obtain that lim Then, (  ) = 0 has two real roots, one being 0 and the other being positive.A continuity argument shows that there exists a positive constant  * 21 such that when  21 >  * 21 , (15) and   axis intersect on two real and positive points, denoted by  1 ,  2 ( 1 <  2 ).Hence, there exists a   ∈ ( 1 ,  2 ) such that det(G u (u * ) −   Φ u ) < 0. Remark 4. Theorem 3 is available for a case of the system (1) equipped with cross-diffusion and self-diffusion; that is,  11 ̸ = 0,  12 ̸ = 0 and  22 ̸ = 0.When  11 =  12 =  22 = 0 and  21 ̸ = 0, the system (1) possesses a diffusion term the same as in [22].

Corollary 5.
If  21 = 0, then the homogeneous steady state u * of the system (1) is always stable.

Turing Parameter Space
In this section, we will find some parameter regions of nonlinear diffusion coefficients where the equilibrium state u * is unstable.For this, according to (15) the sufficient condition of Turing instability is  < 0 and  2 − 4 > 0 besides [1] and [2].In this paper, the parameter values satisfying conditions [1] and [2] are taken as follows: Then, for these fixed parameters the homogeneous steady state u * is given by ( * 1 ,  * 2 ) = (3.4483,14.4828).In Figure 1, we examine the parameter regions where the homogeneous steady state u * is expected to be unstable.These charts are obtained by fixed parameters in (21) as well as  12 =  22 = 0.01 for Figure 1(a),  11 =  22 = 0.01 for Figure 1(b), and  11 =  12 = 0.01 for Figure 1(c).
From the mathematical viewpoint, the Turing bifurcation occurs when for the characteristic root of (13), Im(()) = 0 and Re(()) = 0 at  =   ̸ = 0. Next, we will look for the critical wave of spatial patterns and note the relationship of  and the wave number ; that is,  =  2 [26].Thus, we only need to confirm that min Then, the Turing bifurcation thresholds of parameters satisfy the following equation:  and the critical wavenumber   satisfies To well see the effect of the nonlinear diffusion, according to Turing parameter regions in Figure 1 we plot the dispersion relations in

Pattern Formation
In this section, using numerical methods, we perform numerical simulations of the system (1) in a two-dimensional space and illustrate that cross-diffusion and self-diffusion induce spatial patterns.Throughout this section, we assume that Abstract and Applied Analysis with Laplacian defined by In this paper, we set ℎ = 1,  = 0.001, and  =  = 100.
In Figure 3, we show the evolution of the spatial patterns of the prey and the predators at 2 × 10 3 , 4 × 10 4 , and 1 × 10 6 iterations for  11 = 0.05, 0.1 when we set  21 = 10,  12 = 0.01, and  22 = 0.01.One can see that the patterns arise from random initial conditions.After the cold spot patterns for the prey and the hot spot patterns for the predator arise, they turn steadily with time until these patterns are temporally independent.In addition, for  11 = 0.1 the cold spot patterns for the prey and the hot spot patterns for the predator are looser compared with those for  11 = 0.05.
In Figure 4, we fix  11 = 0.01,  22 = 0.01, and  21 = 10 and obtain the spatial patterns of species  1 ,  2 of time evolution for  12 = 0.15 and  12 = 0.3.For the case of  12 = 0.15, the random initial distribution leads to the formation of irregular patterns.After a long time evolution, we find that the cold spot-strip patterns emerge for the prey  1 and that the hot spot-strip patterns for the predator  2 arise.However, in the case of  12 = 0.3, the steady patterns of the prey  1 consist of hot spots in a bigger size, while the steady patterns of the predator  2 are in the formation of bigger cold spots.
In Figure 5, diffusion parameters are set as  11 =  12 = 0 and  21 = 5.06.We plot the patterns for  22 = 0.35 and  22 = 0.508, respectively.For both cases, one can see that as time goes on, the cold spot patterns of the prey  1 and the hot spot patterns of the predator  2 ultimately form.

Conclusion and Discussion
In this paper, we have studied the prey-predator model with the nonlinear diffusions including normal diffusion, crossdiffusion, and self-diffusion.By applying the mathematical analysis and suitable numerical simulations, we obtain the sufficient conditions of the formation of Turing patterns for this nonlinear diffusion and illustrate Turing parameter regions and Turing patterns when some parameters in system (1) are set.
In our results, we have provided Theorems 1 and 3 to demonstrate that for the nonlinear diffusion including self-diffusion and the cross-diffusion of the predator, the parameter  21 plays an important role to induce Turing instability.Furthermore, if  21 = 0, then the homogeneous equilibrium state u * is always stable; that is, the system (1) has no Turing patterns.By performing numerical simulations, we find the Turing parameter regions of the interaction between cross-diffusion  21 and other diffusion terms including cross-diffusion of the other species and self-diffusions.Besides, according to these parameter regions, we show the corresponding dispersion relations and the corresponding patterns.These results indicate that Turing patterns can emerge through the interaction between the cross-diffusion  21 and self-diffusion as well as other cross-diffusions in the system (1).
It is well known that for a prey-predator system, the formation of patterns can occur by introducing the crossdiffusion.However, our results further show that under condition (see Theorem 3), self-diffusion can produce Turing patterns.

Figure 2 .
The critical parameter values in Figure 2(a) correspond to  11 = 0.201,  12 = 0.353 in Figure 2(b), and  22 = 0.558 in Figure 2(c).In addition, we find that the lowest limit of wavenumber  corresponding to the available Turing modes Re() > 0 turns small with  11 increasing in Figure 2(a), with  12 increasing in Figure 2(b), and with  22 increasing in Figure 2(c).

Theorem 3 .
Assume that conditions [1] and [2] hold; then there exists a sufficiently large positive constant  * 21 such that the homogeneous state u * of the system (1) is unstable provided that  21 >  * 21 .