DYNAMICS AND PATTERN FORMATION IN DIFFUSIVE PREDATOR-PREY MODELS WITH PREDATOR-TAXIS

We consider a three-species predator-prey system in which the predator has a stage structure and the prey moves to avoid the mature predator, which is called the predator-taxis. We obtain the existence and uniform-intime boundedness of classical global solutions for the model in any dimensional bounded domain with the Neumann boundary conditions. If the attractive predator-taxis coefficient is under a critical value, the homogenerous positive steady state maintains its stability. Otherwise, the system may generate Hopf bifurcation solutions. Our results suggest that the predator-taxis amplifies the spatial heterogeneity of the three-species predator-prey system, which is different from the effect of that in two-species predator-prey systems.


Introduction
Predator-prey interaction is common in ecological systems. The relatively simple models which describe the behaviors of one predator and one prey have been extensively studied. Some problems of stage structures were proposed since there are always two stages in the growing process of the majority of species, such as immature and mature stages [6,8,26,27,32,33]. A reaction diffusion model with stage structure for the predator was proposed in [8], where u(x, t), v(x, t) and w(x, t) represent the densities of mature predator, immature predator and prey respectively at position x and time t; Ω is a bounded domain in R N , N ≥ 1 with smooth boundary ∂Ω and unit outer normal ν; the homogeneous Neumann boundary condition indicates that the predator-prey system is self-contained with zero population flux across the boundary. It can deem that the diffusion rates of immature and mature organisms are identical since the immature predator always follows the mature one for the same species, the constants d, d 1 (diffusion rates) and b, m, r, a, are all positive. The pursuit and evasion between predators and prey (predators chasing prey and prey evading from predators) have a strong impact on the movement pattern of predators and prey [14,22,34]. Such movement is not random but directed: predators move toward the gradient direction of prey distribution, and prey moves in the negative gradient direction of predator distribution. It is important to study such movement that describes an ecological interesting phenomenon and provides new insights into the effects of dispersal on predators and prey.
Besides the fact that predators forage prey, prey may avoid predators actively as well. Because of the great gap between the ability of the mature and immature predators to capture prey, the reality of the interaction among the prey, the mature predators and their young is that the prey tends to avoid the mature predators. We model this by the cross diffusion term α∇ · (β(w)w∇u) for the predator-taxis with predator-tactic coefficient α > 0, which implies that the prey w moves to the opposite direction of the increasing mature predators gradient u, and β(w) is the sensitivity of prey to predation risk (i.e. predator-taxis). Combined with (self-)diffusion, the prey thus diffuses with flux d 1 ∇w + α(β(w)w∇u). Thus, the cross diffusion system that we shall study is the following, (1.2) Taking into account the volume filling effect for β(w), we adopt β(w) as (e.g. see [29]): where M measures the maximum number of prey that one unit volume can be filled. Referring to [19,29], we can assume that M > a, where a represents the carrying capacity of prey. The parameters , r are both restricted in the interval (0, 1) and abr > m is set to warrant the existence of non-trivial steady states. The initial data u 0 , v 0 , w 0 are continuous functions. It is noticed that volume filling is also common in chemotaxis models. For example, Hillen and Painter in [12] considered the prevention of overcrowding in the chemotaxis model, namely there is no chemo-tactic response when the cell density is high. This phenomenon also exists in other two-species predator-prey systems (e.g. see [1,11,21,29]): so many prey occur that the volume can not accommodate, prey will not move towards the area around them which leads to nothingness of the predator-taxis term [29].
In two-species predator prey systems with prey-taxis, a large body of outcomes have been obtained. The traveling wave solutions, the pattern formation in a bounded domain under zero Neummann boundary conditions and the global existence of classical solutions were successively studied in [1,4,11,13,15,16,17,21,28,30]. These results show that prey-taxis plays a stabilization role in the dynamical behavior. Compared with prey-taxis system, predator-taxis (α > 0) systems are much less common. The global existence of positive classical solutions and stability of positive equilibria in three-species predator-prey systems with prey-taxis were studied (see [10,23,24]). As a result, they found that attractive predator-taxis (α > 0) inhibits spatial pattern formation, instead of generating that. Without predator taxis (i.e. α = 0), model (1.1) was proposed in [8], based on the classical Lotka-Volterra interaction, the authors studied the stability of nonnegative steady states of the system (1.1) and the reduced ODE system. In addition, the dynamics of the cross diffusion system were also analyzed.
Our main result in this paper is further to investigate the effect of repulsive predator-taxis on the dynamics of three-species system (1.2). It found that a strong predator-taxis can promote the spatial pattern, while the constant equilibrium regains its stability for weak predator-taxis. Moreover, attractive predator-taxis can drive the generation of spatial pattern. This provides another mechanism for spatial pattern formation: introducing an attractive predator-taxis into a reaction-diffusion system with three-species predator-prey interaction. We also obtain the existence of non-constant equilibrium of (1.2) rigorously by using the bifurcation theory. The results here differ from earlier partial results for prey-taxis systems [16,25,28] and predator-taxis systems [31] with two species.
The remainder of this paper is organized as follows: In Section 2, the global existence of the classical solutions of (1.2) is investigated; In Section 3, the effect of predator-taxis coefficient α on pattern formation is explored. Pattern formation is numerically illustrated in Section 4. We use · p as the norm of L p (Ω), 1 ≤ p ≤ ∞ through the paper.

Existence of global classical solution
In this section, the existence of global classical solutions to (1.2) will be established. First, we shall ensure that the solutions to (1.2) are classical. However, it is obvious that β(w) is not differentiable. To overcome this problem, referring to [29], we make a smooth extension of β(w) bỹ If 0 ≤ w ≤ M , we can see β(w) =β(w) in which case system (1.2) is equivalent to (2.2). Indeed, it is a fact that 0 ≤ w ≤ M (we will explain it later). Let p ∈ (n, ∞), It is easy to see that (2.2) can be written as a triangular system, then we obtain the existence of local solutions with the help of Amann's theorem [3].
Based on the second part of Lemma 2.1, it remains to derive the L ∞ -bound of u, v, w to prove the global existence of solutions.
and it exists globally in time.
Proof. Firstly, we show that w ∈ [0, M ]. We define an operator It is noticed that (2.5) satisfies the boundary condition and initial value: Thus we have that w = M is an upper solution of the w equation from (2.5) and (2.6), which implies 0 ≤ w ≤ M (2.7) from the comparison principle of parabolic equations [20]. Now we prove that the L ∞ norm of u, v are bounded. Integrating the second equation of (1.2), we obtain Similarly, integrating the first equation and the third equation of (1.2), respectively, we have Multiplying (2.10) by r and adding the resulting equation to (2.8), we obtain (2.11) In view of (2.11), it can be shown that Referring to (2.9) and (2.13), we obtain (2.14) which implies that From (2.13) and (2.15), we obtain bv + u 1 ≤ (m+1)br|Ω|(a+1)M m =: K (a finite positive constant) and sup t≥0 Ω (bv + u)dx < K + 1. Below we will illustrate that bv + u ∞ is bounded. Clearly, Therefore, by [2, Theorem 3.1], we conclude that sup t≥0 bv + u ∞ ≤ K * , where K * is a constant which depends on K and bv 0 (x) + u 0 (x) ∞ . The desired results are proved.
Theorem 2.2 indicates that the taxis terms can not give rise to blow up of solution, which is consistent with the results of many models with taxis terms introduced in volume filling effect (see [12,19,29]). Furthermore, we can obtain the boundedness of steady state solutions, which solve the elliptic system Proof. We define the operator Then we know that w = M is an upper solution in the w equation. Therefore w ≤ M by the comparison principle of elliptic equations [9], which also shows maxΩ w ≤ M . Suppose that is not true, then there exists (d n , d 1n ) satisfying d n , d 1n ≥ d * , and a corresponding positive solution (u n , v n , w n ) of (2.16) with (d, d 1 ) = (d n , d 1n ), such that max Ω u n + max Ω v n → ∞ as n → ∞. (2.20) Assume that u n (x 0 ) = max x∈Ω u n (x), then we obtain bv(x 0 ) − mu(x 0 ) ≥ 0 with the help of the maximum principle in the equation of u n , which implies that Similarly, let v n (x 1 ) = max x∈Ω v n (x). Again the maximum principle to the equation of v n , we have v n (x 1 ) ≤ ru n (x 1 )w n (x 1 ) ≤ rM u n (x 1 ) ≤ rM u n (x 0 ), which indicates that max Ω v n ≤ rM max Setũ n = un un ∞ andṽ n = vn vn ∞ , then (ũ n ,ṽ n , w n ) satisfies (2.23) In view of (2.21) and (2.22), we have A v n ∞ ≤ u n ∞ ≤ B v n ∞ , where both A and B are positive constants. Notice that 0 ≤ w n ≤ M and 0 ≤ũ n ,ṽ n ≤ 1, we can suppose that v n ∞ u n ∞ → γ (γ > 0), (2.24) and d n → d, d 1n → d 1 with d, d 1 ≥ d * ,ṽ n →ṽ strongly in L p (Ω), w n → w weakly in L p (Ω),ũ n →ũ weakly in W 2,p (Ω), and ũ ∞ = 1, where p > N . These yield u ∈ C 1+α (Ω) for some α > 0, andũ n →ũ in C 1+α (Ω).

Effect of predator-taxis on dynamical behaviors
In this section, we shall study the role that the predator-taxis plays in the dynamical behavior of (1.2). Obviously, system (1.2) has two trivial solutions (0, 0, 0), (0, 0, a), and a positive constant steady state (ū,v,w) under the condition abr > m, whereū Firstly, we consider the stability of the positive steady state (ū,v,w). For that purpose, we make a linearization of the reaction-diffusion-taxis system (1 We denote the eigenvalue of −∆ under Neumann boundary conditions and the corresponding eigenfunction by µ k and φ k (k ≥ 0). Then the stability of (ū,v,w) is determined by the eigenvalue problem where λ is an eigenvalue of D∆ + J (i.e. −µ k D + J) for each k ≥ 0. The characteristic equation for the eigenvalue λ is where A k = (2d + d 1 )µ k − (a 11 + a 22 + a 33 ) > 0, B k = (d 2 + 2dd 1 )µ k 2 − (2da 33 + (d + d 1 )(a 11 + a 22 ))µ k + a 11 a 22 + a 11 a 33 + a 22 a 33 − a 12 a 21 − a 23 a 32 > 0, Therefore, where  11 a 12 a 21 + a 23 a 32 a 33 + a 12 a 23 a 31 + a 22 a 12 a 21 + a 22 a 23 a 32 , , β(w) > 0, then α k has the minimum valueα for some k ∈ N + , i.e.α = min k∈N+ α k , where Proof. It is easy to see that α k can be reformulated as Taking the derivative of α k with respect to µ k , we obtain which indicate that α k can achieve its minimum valueα at some k.
(2) Assume that α j = α k for any j = k, then α k can derive the occurrence of periodic solutions bifurcating from (ū,v,w), where k, j ∈ N + .
Proof. According to the Routh-Hurwitz criterion [5], or [18, Corollary 2.2], we know that the constant steady state (ū,v,w) is asymptotically stable if and only if the following conditions hold: while (ū,v,w) is unstable provided that A k ≤ 0, or C k ≤ 0, or H k ≤ 0 for some k ∈ N + . Note that we always have A k > 0, C k > 0 for each k ∈ N + , thereby the stability/instability of (ū,v,w) is subject to consider the sign of H k . Setting H k = 0 and choosing α as the bifurcation point, we obtain (3.4). It is easy to check that H k > 0 as α <α and H k < 0 as α >α.
Next we demonstrate that Hopf bifurcation occurs at every α k (k ∈ N + ). It is known that H k = A k B k − C k = 0 at α k , and direct calculations show that (3.2) has one negative real root and two pure imaginary roots, i.e. λ 1 = −A k , λ 2,3 = ± √ B k i. This indicates the possibility of Hopf bifurcation and the existence of a branch of periodic solutions bifurcating from (ū,v,w) at α = α k . Denote λ 1 = ξ + ηi, λ 2 = ξ − ηi, then it remains to verify dξ dα | α=α k = 0 to ensure the occurrence of Hopf bifurcation at α k . We notice that A k B k − C k = −2ξ((ξ + λ 1 ) 2 + η 2 ), which along with (3.3) gives By the implicit function differentiability theorem, we have The proof is complete.   (2) If m > M br, then (0, 0, a) is globally attractive.

Conclusions and numerical simulations
In this paper, we propose a three species predator-prey model with stage structure for the predators. Predators are assumed to move randomly in their habitats, and prey mobiles to avoid the mature predators. Our analysis shows that the addition of repulsive predator-taxis does destroy the stability of constant steady states and induce the occurrence of spatial patterns, see Theorem 3.2. Contrast to the results, the predator-taxis induced instability can not occur for two species predator-prey systems as shown in [24,31], where both predator-taxis and preytaxis may annihilate the spatial patterns.