Boundedness and stabilization of a predator-prey model with attraction-repulsion taxis in all dimensions

: This paper establishes the existence of globally bounded classical solutions to a predator-prey model with attraction-repulsion taxis in a smooth bounded domain of any dimensions with Neumann boundary conditions. Moreover, the global stabilization of solutions with convergence rates to constant steady states is obtained. Using the local time integrability of the L 2 -norm of solutions, we build up the basic energy estimates and derive the global boundedness of solutions by the Moser iteration. The global stability of constant steady states is established based on the Lyapunov functional method.


Introduction and main results
A taxis is the movement of an organism in response to a stimulus such as chemical signal or the presence of food. Taxes can be classified based on the types of stimulus, such as chemotaxis, preytaxis, galvanotaxis, phototaxis and so on. According to the direction of movements, the taxis is said to be attractive (resp. repulsive) if the organism moves towards (resp. away from) the stimulus. In the ecosystem, a widespread phenomenon is the prey-taxis, where predators move up the prey density gradient, which is often referred to as the direct prey-taxis. However some predators may approach the prey by tracking the chemical signals released by the prey, such as the smell of blood or specific odo, and such movement is called indirect prey-taxis (cf. [1]). Since the pioneering modeling work by Kareiva and Odell [2], prey-taxis models have been widely studied in recent years (cf. [3][4][5][6][7][8][9][10][11][12]), followed by numerous extensions, such as three-species prey-taxis models (cf. [13][14][15]) and predatortaxis models (cf. [16,17]). The indirect prey-taxis models have also been well studied (cf. [18][19][20]).
Recently, a predator-prey model with attraction-repulsion taxis mechanisms was proposed by Bell and Haskell in [21] to describe the interaction between direct prey-taxis and indirect chemotaxis, where the direct prey-taxis describes the predator's directional movement towards the prey density gradient, while the indirect chemotaxis models a defense mechanism in which the prey repels the predator by releasing odour chemicals (like a fox breaking wind in order to escape from hunting dogs). The model reads as x ∈ Ω, t > 0, ∇u · ν = ∇v · ν = ∇w · ν = 0, x ∈ ∂Ω, t > 0, (u, v, w)(x, 0) = (u 0 , v 0 , w 0 )(x), x ∈ Ω, where the unknown functions u(x, t), v(x, t) and w(x, t) denote the densities of the prey, predator and prey-derived chemical repellent, respectively, at position x ∈ Ω and time t > 0. Here, Ω ⊂ R n is a bounded domain (habitat of species) with smooth boundary ∂Ω, and ν is the unit outer normal vector of ∂Ω. The parameters d, η, χ, ξ, a 1 , a 2 , a 3 , e, ρ, r, γ are all positive, where χ > 0 and ξ > 0 denote the (attractive) prey-taxis and (repulsive) chemotaxis coefficients, respectively. The predator v is assumed to be a generalist, so that it has a logistic growth term ρv(1 − v) with intrinsic growth rate ρ > 0. More modeling details with biological interpretations are referred to in [21]. We remark that the predator-prey model with attraction-repulsion taxes has some similar structures to the socalled attraction-repulsion chemotaxis model proposed originally in [22], where the species elicit both attractive and repulsive chemicals (see [23][24][25][26] and references therein for some mathematical studies). The initial data satisfy the following conditions: v 0 ∈ C 0 (Ω), u 0 , w 0 ∈ W 1,∞ (Ω), and u 0 , v 0 , w 0 0 in Ω. (1.2) In [21], the global existence of strong solutions to (1.1) was established in one dimension (n = 1), and the existence of nontrivial steady state solutions alongside pattern formation was studied by the bifurcation theory. The main purpose of this paper is to study the global dynamics of (1.1) in higher dimensional spaces, which are usually more physical in the real world. Specifically, we shall show the existence of global classical solutions in all dimensions and explore the global stability of constant steady states, by which we may see how parameter values play roles in determining these dynamical properties of solutions.
The first main result is concerned with the global existence and boundedness of solutions to (1.1). For the convenience of presentation, we let and Let Ω ⊂ R n (n 1) be a bounded domain with smooth boundary and parameters d, η, χ, ξ, a 1 , a 2 , a 3 , e, ρ, r, γ be positive. If (1.4), then for any initial data (u 0 , v 0 , w 0 ) satisfying (1.2), the system (1.1) admits a unique classical solution (u, v, w) satisfying u, v, w ∈ C 0 (Ω × [0, +∞)) ∩ C 2,1 (Ω × (0, +∞)), and u, v, w > 0 in Ω × (0, +∞). Moreover, there exists a constant C > 0 independent of t such that Our next goal is to explore the large-time behavior of solutions to (1.1). Simple calculations show the system (1.1) has four possible homogeneous equilibria as classified below: where the trivial equilibrium (0, 0, 0) is called the extinction steady state, (0, 1, 0) is the predator-only steady state, and (u * , v * , w * ) is the coexistence steady state. We shall show that if a 1 > a 3 , then the coexistence steady state is globally asymptotically stable with exponential convergence rate, provided that ξ and χ are suitably small, while if a 1 a 3 , the predator-only steady state is globally asymptotically stable with exponential or algebraic convergence rate when ξ and χ are suitably small. To state our results, we denote where K 1 is defined in (1.3). Then, the global stability result is stated in the following theorem.
Theorem 1.2 (Global stability). Let the assumptions in Theorem 1.1 hold. Then, the following results hold.
(1) Let a 1 > a 3 . If ξ and χ satisfy then there exist some constants T * , C, α > 0 such that the solution (u, v, w) obtained in Theorem 1.1 satisfies for all t T * (2) Let a 1 a 3 , If ξ and χ satisfy then there exist some constants T * , C, β > 0 such that the solution (u, v, w) obtained in Theorem 1.1 satisfies, for all t T * , Remark 1.1. In the biological view, the relative sizes of a 1 and a 2 determine the coexistence of the system. The results indicated that a large a 1 a 2 facilitates the coexistence of the species. The rest of this paper is organized as follows. In Section 2, we state the local existence of solutions to (1.1) with extensibility conditions. Then, we deduce some a priori estimates and prove Theorem 1.1 in Section 3. Finally, we show the global convergence to the constant steady states and prove Theorem 1.2 in Section 4.

Preliminary
For convenience, in what follows we shall use C i (i = 1, 2, · · · ) to denote a generic positive constant which may vary from line to line. For simplicity, we abbreviate t 0 Ω f (·, s)dxds and Ω f (·, t)dx as t 0 Ω f and Ω f , respectively. The local existence and extensibility result of problem (1.1) can be directly established by the well-known Amman's theory for triangular parabolic systems (cf. [27,28]). Below, we shall present the local existence theorem without proof for brevity, and we refer to [21] for the proof in one dimension as a reference.
. Then, the following inequality holds: for any u ∈ C 2 (Ω) satisfying ∂u ∂ν = 0 on ∂Ω, where D 2 u denotes the Hessian of u. The last one is a Gagliardo-Nirenberg type inequality shown in [31, Lemma 2.5].

Global existence
In this section, we establish the global boundedness of solutions to the system (1.1). To this end, we will proceed with several steps to derive a priori estimates for the solution of the system (1.1). The first one is the uniform-in-time L ∞ (Ω) boundedness of u.
Lemma 3.1. Let (u, v, w) be the solution of (1.1) and K 1 be as defined in (1.3). Then, we have Furthermore, there is a constant C > 0 such that for any 0 < τ < min{T max , 1}, it follows that t+τ t |∇u| 2 ≤ C for all t ∈ (0, T max − τ).
Proof. The result is a direct consequence of the maximum principle applied to the first equation in Apparently, the comparison principle of parabolic equations gives u ū on Ω × (0, T max ). Next, we multiply the first equation of (1.1) by u and integrate the result to get Then, the integration of the above inequality with respect to t over (t, t + τ) completes the proof by noting that Ω u 2 0 is bounded.
Having at hand the uniform-in-time L ∞ (Ω) boundedness of u, the a priori estimate of w follows immediately.
Lemma 3.2. Let (u, v, w) be the solution of (1.1). We can find a constant C > 0 satisfying Proof. Noting the boundedness of u L ∞ (Ω) from Lemma 3.1, we get the desired result from the third equation of (1.1) and the regularity theorem [32, Lemma 1]. Now, the a priori estimate of v can be obtained as below.
Due to the estimates of u and v obtained in Lemmas 3.1 and 3.3 respectively, we have the following improved uniform-in-time L 2 (Ω) boundedness of ∇u and the space-time L 2 boundedness of ∆u when n = 2.
Lemma 3.4. Let (u, v, w) be the solution of (1.1). If n = 2, then we can find a constant C > 0 such that and t+τ t Ω
Proof. Integrating the first equation of (1.1) by parts and using Lemma 3.1, we find a constant The Gagliardo-Nirenberg inequality in Lemma 2.4, Young's inequality and Lemma 3.1 yield some Then, applications of Lemma 2.2, 3.1 and 3.3 give (3.4). Finally, (3.5) can be obtained by integrating (3.7) over (t, t + τ). Now, the uniform-in-time boundedness of v in L 2 (Ω) can be established when n = 2.
Lemma 3.5. Let (u, v, w) be the solution of (1.1). If n = 2, then there exists a constant C > 0 such that Proof. Multiplying the second equation of (1.1) by v, integrating the result by parts and using Young's inequality, we have which along with Lemma 3.1 and Lemma 3.2 gives some constant C 1 > 0 such that Using Lemmas 3.1 and 3.3, Hölder's inequality, Lemma 2.4 and Young's inequality, we find some for all t ∈ (0, T max ) .
To get the global existence of solutions in any dimensions, we derive the following functional inequality which gives an a priori estimate on ∇u.
Lemma 3.6. Let (u, v, w) be the solution of (1.1) and q 2. If n 1, then there exists a constant C > 0 such that Proof. From the first equation of (1.1) and the fact 2∇u · ∇∆u = ∆|∇u| 2 − 2|D 2 u| 2 , it follows that Now, we estimate the right hand side of (3.11). Choosing s ∈ (0, 1 2 ) and which, along with the Gagliardo-Nirenberg inequality, Young's inequality and the embedding gives some constants C 1 , Therefore, it holds that Owning to the fact |∆u| √ n|D 2 u|, Young's inequality and Lemma 3.1, we have where K 2 is defined in (1.3). Hence, substituting the estimates I 1 and I 2 into (3.11), we finish the proof of the lemma. Now, we show the following functional inequality to derive the a priori estimate on v in any dimensions.
Lemma 3.7. Let (u, v, w) be the solution of (1.1) and q 2. If n 1, we can find a constant C > 0 such that for all t ∈ (0, T max ).
Proof. Utilizing the second equation of (1.1) and integration by parts, we get (3.12) Now, we estimate the right hand side of (3.12). An application of Young's inequality and Lemma 3.2 yields some constant C 1 > 0 such that gives a constant C 2 > 0 such that Hence, we finish the proof of the lemma.
Combining Lemmas 3.6 and 3.7, we have the following inequality which can help us to achieve the global existence of solutions in any dimensions.
Lemma 3.8. Let (u, v, w) be the solution of (1.1) and p 2. If n 1, we can find a constant C > 0 such that Proof. Combining Lemmas 3.6 and 3.7, we see for any p = q 2 there exists a constant C 1 > 0 such that for all t ∈ (0, T max ) (3.13) Now, we estimate the right hand side of the above inequality. Indeed, owing to Lemma 2.3 and Young's inequality, for all t ∈ (0, T max ), we have where K 1 and K 2 are defined in (1.3). Similarly, we can find a constant C 2 > 0 such that Substituting the above estimates into (3.13), we get where K 3 (p) is given in (1.4). Furthermore, we can use Young's inequality and Lemma 2.3 to get a constant C 3 > 0 such that and Ω |∇u| 2p d p 8 4p 2 + n u 2 which together with (3.14) finishes the proof.
Proof. We divide the proof into two steps.
Step 1: We claim that there exists a constant C 1 > 0 such that Ω v 2p 0 C 1 for all t ∈ (0, T max ) .
Indeed, due to Lemma 3.8, for any p = 2p 0 , there exists a constant C 2 > 0 such that Then, 2p 0 +1 2p 0 θ < 1 due to p 0 > n 2 . By the Gagliardo-Nirenberg inequality, Young's inequality and (3.15), we can find some constants C 3 , C 4 > 0 such that which along with (3.16) implies Therefore, the claim follows from the Grönwall inequality applied to the above inequality.
Step 2: Thanks to the regularity theorem [32, Lemma 1], we can find a constant C 5 > 0 such that ∇u L ∞ (Ω) C 5 due to 2p 0 > n. With (3.12) and Lemmas 3.1 and 3.2, we get a constant C 6 > 0 such that for any p 2 Thanks to Young's inequality, we find a constant C 7 > 0 such that which together with (3.17) implies with C 8 = C 7 + ρ + ea 3 K 1 + 1. Applying 1 + p n (1 + p) n and the following inequality [34] f 2 Then, employing the standard Moser iteration in [35] or a similar argument as in [34], we can prove that there exists a constant C 11 > 0 such that v L ∞ (Ω) C 11 for all t ∈ (0, T max ).
Thus, with the help of Lemma 3.2, we finish the proof. Now, utilizing the criterion in Lemma 3.9, we prove the global existence and boundedness of solutions for the system (1.1).
which along with the Grönwall inequality gives a constant C 2 > 0, Together with Lemma 3.9, we finish the proof by Lemma 2.1.

Stabilization
In this section, we will employ suitable Lyapunov functionals to study the large-time behavior of u, v and w. We first improve the regularity of the solution.
In particular, one can find C > 0 such that Proof. The conclusion is a consequence of the regularity of parabolic equations in [36].
We split our analysis into two cases: a 1 > a 3 and a 1 a 3 .
Proof of Theorem 1.2-(1). We complete the proof in four steps.
Step 1: The parameters ε 1 and ε 2 can be chosen in the following way. First, we recall from (1.5) and (1.6) that It is clear that f ∈ C 0 ((0, +∞)). Then, if the following holds: f (y). Then, a < ε 1 < b, or equivalently (see (4.1)) max Next, we assume χ > 0 is suitably small such that Hence, there exists a constant ε 2 > 0 such that which along with Lemma 3.1 yields Step 2: We claim Indeed, using the equations in system (1.1) along with integration by parts, we have Similarly, we obtain Then, it follows that The above results indicate that matrix S is positive definite. Using (4.3) and (4.4) again, we observe that and which imply that matrix T is positive definite. Therefore, one can choose a constant C 1 > 0 such that Integrating the above inequality with respect to time, we get a constant C 2 > 0 satisfying which together with the uniform continuity of u, v and w due to Lemma 4.1 yields By the Gagliardo-Nirenberg inequality, we can find a constant C 3 > 0 such that L 2 (Ω) (4.8) and w for all t > 1, (4.9) which along with (4.6) and Lemma 4.1 prove the claim.

Predator-only: a 1 a 3
In this case, there are three homogeneous equilibria (0, 0, 0), (0, 1, 0) and a 1 a 2 , 0, ra 1 γa 2 , and we shall show that the steady state (0, 1, 0) is global asymptotically stable, where the convergence rate is exponential if a 1 < a 3 and algebraic if a 1 = a 3 . Define an energy functional for (1.1) as follows: where ζ 1 and ζ 2 will be determined below.
Proof of Theorem 1.2-(2). We divide the proof into five steps.
Solving the above inequality directly yields a constant C 8 > 0 such that G(t) C 8 (t + 1) −1 for all t T 1 .