Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-( n + 1 ) Functional Response

This paper discusses a predator-prey system with Holling-(n + 1) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition.The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution (?̃?, Ṽ) is asymptotically stable when the parameter k satisfies some conditions.


Introduction
In this paper, we are interested in the positive steady states of the strongly coupled predator-prey system with Holling-( + 1) functional response.The specific system is as follows: where 1 ≤  < +∞; Ω is a bounded domain in   with smooth boundary Ω; /] is the outward directional derivative normal to Ω;  and V stand for the densities of the prey and predator; the given coefficients   ( = 1, 2), , , , and  are positive constants.The term   /( +   ) is named Holling-( + 1) functional response [1,2].In the second equation, the fractional type nonlinear diffusion term Δ 2 ( 3 V/(1 + )) models a situation in which the population pressure of the predator species weakens in high-density areas of the prey species.For more precise details, we can refer to [3,4].Paper [3] discusses a strongly coupled predatorprey system with nonmonotonic functional response, the existence and nonexistence results concerning nonconstant positive steady states of the system were proved by degree theory.Paper [4] considers the positive steady states for a prey-predator model with some nonlinear diffusion terms, and the sufficient conditions for the existence of positive steady state solutions were obtained by bifurcation theory.In recent years, there has been considerable interest in the dynamics of strongly coupled reaction-diffusion systems with cross-diffusion.We point out that most efforts have concentrated on the Lotka-Volterra competition system which was proposed first by Shigesada et al. [5].Since their pioneering work, many authors have studied population models with cross-diffusion terms from various mathematical viewpoints, for example, the global existence of time-depending solutions [6][7][8][9][10][11], the stability analysis for steady states [12][13][14], and the steady state problems [15][16][17][18][19][20][21].In this paper, we mainly consider the existence of solutions of (1).The research method refers to [3,22,23].
For convenience of the research, we write (1) as the following form: where The main work of this paper is to study the effects of the fractional type nonlinear diffusion pressures on the existence of nonconstant positive steady states of (1).Here, a positive solution means a smooth solution (, V) with both  and V being positive.We will demonstrate that the cross-diffusion pressure  3 may help forming more patterns.Obviously, for system (1), one notes that when  ≤ , there holds sup ≥0 {−+   /( +   )} ≤ 0, so that the only nonnegative solutions to (1) are  = (0, 0) and  = (, 0).Consequently, (1) does not have any positive solution.On the other hand, when  > , the unique positive constant solution to ( 1) is (ũ, Ṽ); that is, ( The organization of our paper is as follows.In Section 2, we establish a priori upper and lower bounds for positive solutions of (1).In Section 3, we use a degree theory to develop a general result to enable one to conclude the existence or nonexistence of nonconstant steady-state solutions or patterns as the index of positive constant steady states changes.In Section 4, we establish the existence of nonconstant positive solutions to (1) for a large range of diffusion and cross-diffusion coefficients.Meanwhile, we prove that the positive constant solution (ũ, Ṽ) is asymptotically stable for different ranges of parameters.

Upper and Lower Bounds for Positive Solutions
The main purpose of this section is to give a priori upper and lower positive bounds for positive solutions of (1).Firstly, we cite two known results.
and noting that min Ω   ≥  3 , V  → 0, and   → Ĉ, we also have  = .By a similar argument as that in (24), for the second equation of (1), we can prove that there exists a subsequence in  2, (Ω), such that Dividing the second equation of (1) by max Ω V  (), and integrating over Ω, we have Let  → ∞, and note that V  /max Ω V  () → V 0 and   → ; then,   /( +   ) = .This contradiction to the assumption completes the proof.

A Result on Degree Theory
In this section, we obtain nonexistence of nonconstant positive solutions to (1) as  3 = 0.Meanwhile, by degree theory, a general result to establish the existence of nonconstant positive solutions to (1) in the next section is proved.Denote d = ( 1 ,  2 ,  3 ) and  = (, , , ).We will fix  ∈ (0, ∞) 4 and take d ∈ (0, ∞) 2 × [0, ∞) as bifurcation parameters, the dependence of  will often be suppressed.Define Since and det Φ  () is positive for all nonnegative , Φ −1  exists.Hence,  is a positive solution to (1) if and only if where ( − Δ) −1 is the inverse of  − Δ in .As (d; ⋅) is a compact perturbation of the identity operator , the Leray-Schauder deg((d; ⋅), 0, ) is well defined if (d; ) ̸ = 0 for all  ∈ .
For the case  > /  + , by degree invariance, we need only consider a special d; say d = (, , 0) with large .For this we can use the following nonexistence result.To compare the existence regions of (1) with and without cross-diffusion, we give a nonexistence result stronger than what is needed here.
In the following, we only calculate deg(, 0, ) when all solutions to  = 0 are positive constant solutions in ().

Existence of Nonconstant Positive Solutions
In this section, we establish the existence of positive nonconstant solutions for (1).In particular, we show that for certain ranges of parameters where (1) does not have any positive nonconstant steady state, our model can still produce patterns.The idea is as follows.First we calculate the index of (d; ⋅) at positive constant steady states.Suppose that the sum of all these indices is not equal to the degree stated in Theorem 5.Then, (d; ⋅) = 0 in () for  =  0 (d, ) must have a nonconstant positive solution, which also solves (1).
Hence, we will discuss separately the following cases:  In this subsection, we consider local stability of the constant steady state  ≡ Ũ for evolution dynamics where Ũ = (ũ, Ṽ) is a positive constant solution to (⋅) = 0 by (5).Proof.The linearization of (47) at Ũ takes the form Denote that the corresponding linear operator  := Φ  ( Ũ)Δ+   ( Ũ).By fixing the diffusion coefficients  1 (for prey) and using the diffusion coefficients  2 and  3 (for predator) as bifurcation parameters, we will show that (1) can create nonconstant positive solutions.We want to emphasize that it is caused by the presence of cross-diffusion which has a more complex role than that of the diffusion coefficients  1 and  2 .
Consequently, (d; ) = 0 has at least one nonconstant positive solution that is different from the constant function  = Ũ.Otherwise, the degree of  = 0 in () would be −1 for all large enough , which would contradict Theorem 5.This proves, the first assertion of the theorem, and the second assertion is similarly proved.
When this inequality holds, Λ 2 ( 1 ,  3 ; Ũ) is also positive provided that  3 is large, and we then can adjust to  1 and make the assumptions in (i) or (ii) of theorem hold.
Since for each integer  ≥ 1,   is invariant under   ((d;  * )), and  is an eigenvalue of    on   if and only if  is an eigenvalue of following matrix: Proof.If    is invertible, then the index of  at  * is defined as index((d; ⋅),  * ) = (−1)  , where  is the number of eigenvalues of    with negative real parts.The deg((d; ⋅), 0, ) is then equal to summation of the indexes over all solutions to  = 0 in , provided that  ̸ = 0 on .