Steady state for a predator–prey cross-diffusion system with the Beddington–DeAngelis and Tanner functional response

where a, b, α, β , γ are positive constants. Here u and v stand for the population densities of the prey and predator, respectively. The bounded domain ∈ Rn (n ≥ 1) possesses a smooth boundary ∂ ; ν is the outward directional derivative normal to ∂ . The diffusion rates are described by d1 > 0, d2 > 0, d3 > 0 and d4 > 0; a and b are the death rate of the predator and the intrinsic growth rate of the prey, respectively. The smooth functions u0 and v0 on represent the initial population densities of prey and predator. The corresponding ordinary differential system (ODS) of (1.1) is performed as


Introduction
We propose and study a predator-prey cross-diffusion system with the Beddington-DeAngelis and Tanner functional response where a, b, α, β, γ are positive constants. Here u and v stand for the population densities of the prey and predator, respectively. The bounded domain ∈ R n (n ≥ 1) possesses a smooth boundary ∂ ; ν is the outward directional derivative normal to ∂ . The diffusion rates are described by d 1 > 0, d 2 > 0, d 3 > 0 and d 4 > 0; a and b are the death rate of the predator and the intrinsic growth rate of the prey, respectively. The smooth functions u 0 and v 0 on represent the initial population densities of prey and predator. The corresponding ordinary differential system (ODS) of (1.1) is performed as which was introduced by Luo [1] in recent years, and the globally asymptotical stability of (1.2) has been discussed by constructing a feasible Lyapunov function. Recently, with rapid development of biotechnology, we need to accurately illustrate the interaction between prey and predator in a real ecological environment such as hare and lynx, sparrow hawk and sparrow, spider mite and mite, and so on. Such interesting natural phenomena as the existence and global stability of interior periodic solutions and the global stability of reaction-diffusion system have been described by this ODE model [1] or the corresponding PDE system [2]. By utilizing the iteration method, Liu and Li [2] proved the globally asymptotic stability of the corresponding parabolic PDE system.
The impact of prey-taxis leads to the linear cross-diffusion in model (1.1), which means that the prey attraction to the predator may give rise to the migration of predator following the gradient of the density function of prey. By utilizing -d 3 u to describe this impact we will incorporate d 2 v into the first equation of system (1.1) to indicate the repelling effect from predator to prey. The message about the location of predator can guide the prey escape from them (see [3][4][5]).
This term illustrates the interaction effect among predators and provides a more general situation to describe the interaction between prey and predator.
As far as we can survey, this is the first paper studying the steady state for a predatorprey cross-diffusion system with the Beddington-DeAngelis and Tanner functional response. For notational simplicity, we introduce some notations used in the rest of this paper.

Notation
(1) We denote the eigenvalues sequence by 0 = λ 0 ≤ λ 1 ≤ · · · ≤ λ k ≤ · · · for the elliptic operator on with homogeneous Neumann boundary condition; (2) The multiplicity of the eigenvalue λ k is denoted by M k ; (3) The corresponding normalized eigenfunctions of λ k are represented by the set The rest part of this paper is organized as follows. In Sect. 2, we provide the study of the local stability of semitrivial or interior equilibrium of model (1.1). In Sect. 3, by utilizing the fixed point index theory we investigate the existence of the nonconstant positive steady state of system (1.1). Finally, in Sect. 4, we offer some conclusions on our main theoretical results.

Local stability
It is obvious via a direct calculation that model (1.1) possesses one semitrivial equilibrium (a, 0) and one unique interior equilibrium (u * , v * ), where We discuss the local stability of the semitrivial equilibrium (a, 0) and the interior equilibrium (u * , v * ) of system (1.1) as follows.
Proof At a constant solution E * = (u, v), the linearized issue of model (1.1) can be proposed by For notational simplicity, we denote In view of Theorem 5.1.1 of [10], it is obvious that E * = (u, v) is locally asymptotically stable when all the eigenvalues of the operator L possess negative real parts. On the contrary, E * = (u, v) is unstable. Thus the stability of the equilibrium is based on the study of the characteristic equation Here (1) When E * = (a, 0), we get If b > a, then for i = 0, we obtain trace J i = -a + b > 0. Therefore (a, 0) is unstable.
Together with the hypothesis of (2), we can observe that and then In view of the hypothesis of (2), we can observe that We can find a sufficiently large eigenvalue d 2 to guarantee that Thus the interior equilibrium (u * , v * ) is locally asymptotically stable.
Therefore (u * , v * ) is unstable. This ends the proof of the lemma.

Nonconstant positive steady states
In this section, we offer a rigorous study of interior solutions of the corresponding strongly coupled elliptic model Meanwhile, we will also investigate the nonconstant positive steady states of model (1.1). The existence of an interior solution of a linear cross-diffusion model has been studied by utilizing the approach proposed by [11] (upper and lower solutions) and many others.
Proof In view of the equation of u for model (3.1), we obtain By the maximum principle proposed by [12] we have u(au) ≥ 0. Hence u ≤ a. Together with the equation for u and the equation for v for system (3.1), we get Applying the maximum principle in [12], we have bv -v 2 aγ ≥ 0. Thus we obtain v ≤ abγ . This ends the proof of the lemma.
Next, we introduce the compact map CM ∈ C 2 ( )⊕C 2 ( ) → C 1 ( )⊕C 1 ( ) as follows: Here we can choose a sufficiently large constant c > 0 guaranteeing that the functions auu 2 -uv 1+αu+βv + cu and bv -v 2 γ u + cv are increasing for u and v, respectively. We can easily observe that system Proof We introduce the homotopic map, for η ∈ (0, 1), Utilizing a similar discussion as in Lemma 3.1, we can easily obtain that any interior fixed point (u, v) for HM fulfills u(x), v(x) ≤ max{a, abγ } in . Thus we can easily draw a conclusion that each fixed point for HM belongs to Int D . In addition, the corresponding index of CM η over the interior of D regarding Q is well defined. In view of the homotopy invariance theorem, we can obtain that index Q (CM 0 , Based on the above discussion, we argue that index S (B, Int D) = 1. To prove this, we propose the homotopic map Then we can directly obtain that index S (B 0 , Int D) = index S (B 1 , Int D). Our concern is the eigenvalue issue for ρ > 0, where The corresponding eigenfunction formulas for and can be performed as follows: where φ kj , ψ kj ∈ R. We easily rewrite system (3.2) as follows: As we know, {E kj : 1 ≤ j ≤ M k } in L 2 ( ) stands for a complete orthonormal basis. By multiplying the above two equations by ϕ kj and integrating over we get the equation , where = c(d 1 λ k + d 4 λ k + 2c), we have  Set CM (a, 0)( , ) T = ( , ) T ∈ Q (a,0) . Then we get In the rest of the paper, we consider index Q (CM, (u * , v * )) in the following three situations: (1) For all λ > 0, Det(λ) > 0; (2) There exists, with multiplicity one, precisely a simple interior solution for Det(λ) = 0; (3) There exist, with multiplicity one, two interior solutions for Det(λ) = 0. Now we present our main theoretical result. (2) If Det(λ) = 0 has exactly a multiplicity-one simple positive solution λ * in an open interval (λ k * , λ k * +1 ) for some nonnegative integer k * , then In addition, when k * k=0 M k is odd, system (3.1) possesses at least one nonconstant interior solution; (3) Suppose that Det(λ) = 0 has two interior solutions λ * + and λ * -in two open intervals In addition, when In addition, when k * k=0 M k is odd, system (3.1) possesses at least one nonconstant interior solution.
Proof By applying the elliptic PDE theory and the eigenfunction theory, we prove that index Q (CM, (u * , v * )) = 1.
(2) In view of the assumption introduced by (2), we can observe that for a nonnegative integer k, Det(λ k ) = 0. Clearly, I -CM (u * , v * ) is an invertible matrix on Q (u * ,v * ) . Meanwhile, we can conclude that CM (u * , v * ) cannot possess property γ on Q (u * ,v * ) . Next, we focus on the total algebraic multiplicity δ for each eigenvalue of CM (u * , v * ) -I, which is greater than zero. When λ 0 = 0 (k = 0), we get Det (Q(ρ, 0) Obviously, Det(Q(ρ, 0)) = 0 has exactly a simple interior solution. Together with 1 ≤ k ≤ k * and the hypothesis in (2), wee get Det(λ k ) < 0, and hence Det(Q(ρ, λ k )) = 0 possesses precisely one positive simple solution. If k ≥ k * + 1, then Det(λ k ) > 0, and hence Det(Q(ρ, λ k )) = 0 may have either precisely one multiplicity-two simple interior solution, or two multiplicity-one interior solutions, or no interior solution. This discussion yields that k * k=0 M k +t = δ, where t is an even number or 0. In addition, suppose on the contrary that model (3.1) possesses no nonconstant interior solution. By Lemma 3.2, Lemma 3.3, and the last discussion we get This contradiction tells us that system (3.1) possesses at least one nonconstant interior solution. The above argument derives the ideal outcome.

Conclusion
In this paper, we introduce a complicated predator-prey linear cross-diffusion system with the Beddington-DeAngelis and Tanner functional response and applied an effective rigorous approach to obtain the existence of nonconstant positive steady state of the corresponding system. By considering Det(λ) in three situations we can easily compute index Q(CM, (u * , v * )). We proposed some parameter conditions to guarantee the local stability of the unique interior equilibrium.
It is worth noting that our methods in this paper can be applied to investigation of a class of nonlinear prey-taxis model with more general functional responses. By utilizing the fixed point index theory to get the existence of nonconstant interior steady states it is necessary to provide an a priori estimate and regularity of steady states, which plays an important role (see [15]).