Dynamics and Patterns of a Diffusive Prey-Predator System with a Group Defense for Prey

The predator-prey system first proposed by [1, 2] is one of the fundamental ecological systems in both ecology and mathematical ecology. Based on different settings, various types of predator-prey models described by differential systems have been proposed and the dynamics of these systems are studied [3–6]. The basic form of these models is as follows: dx dt = rx (1 − x K) − P (x, y) , dy dt = −sy + cP (x, y) , (1)


Introduction
The predator-prey system first proposed by [1,2] is one of the fundamental ecological systems in both ecology and mathematical ecology.Based on different settings, various types of predator-prey models described by differential systems have been proposed and the dynamics of these systems are studied [3][4][5][6].The basic form of these models is as follows: where  is the intrinsic growth rate and  is the environmental carrying capacity of prey population, and the function () is the functional response; the constant (>0) is the ratio of biomass conversion and  is the natural death rate of predator species.The simplest functional response is Lotka-Volterra function which is described as which is also called Holling type I function.However, the curve defined by the Lotka-Volterra response function is a straight line through the origin and is unbounded.Thus, more reasonable response functions should be nonlinear and bounded.In 1913, Michaelis and Menten proposed the response function where  > 0 denotes the maximal growth rate of the species and  > 0 is the half-saturation constant.It is now referred to as a Michaelis-Menten function or a Holling type II function.
Another class of response function is which is called a sigmoidal response function, while the simplification is known as a Holling type III function.Some authors [7,8] considered system (1) with following response function: which is called Holling type IV function.Besides, Beddington-DeAngelis type (, ) = /( +  + ) and more complicated functional response (, ) =  2 /( 2 +  2 ) are also considered by some researchers [9,10].
Recently, some works consider the case when animals join together in herds in order to provide a self-defense from predators.In [11], the authors argued that it is more appropriate to model the response functions of prey that exhibit herd behavior in terms of the square root of the prey population.Inspired by this thought, the authors in [12] choose response function () = √ to reflect this fact.When motion is allowed, [13] considered the spatiotemporal behavior of a prey-predator system with a group defense for prey by means of extensive computer simulations.The proposed model is as follows: where  and V denote, respectively, the densities of prey and predator species. is the growth rate of prey species,  is its carrying capacity,  2 is the mortality rate of predator species,  is the search efficiency of predator for prey,  is the biomass conversion coefficient, and  ∈ (0, 1) represents a kind of aggregation efficiency.The local dynamics for nonspatial model was studied, such as Hopf bifurcation and existence of extinction domain.For model (7), the authors only give some numerical simulations to find some spatiotemporal features.
Reference [14] considers the direction and the stability of the bifurcating periodic solutions for model (7) with  = 1/2 under Neumann boundary conditions.Reference [15] investigated the global dynamics of nonspatial model including the nonexistence of periodic orbits and the existence and uniqueness of limit cycles.We refer readers to [16][17][18][19][20][21] as some other related works on predator-prey model with herd behavior.
It is noted that up to now no one has studied the existence and nonexistence of positive steady state solutions of (7).Therefore, the main aim of this article is to study the existence and nonexistence of nonconstant positive solutions of the following elliptic system: where ] is the outward unit normal vector on Ω, and we impose a homogeneous Neumann type boundary condition, which implies that ( 8) is a closed system and has no flux across the boundary Ω.The structure of this paper is arranged as follows.In Section 2, we estimate the a priori bounds of positive solutions of (7).In Section 3, the local and global stabilities of nonnegative constant steady states of (7) are discussed.In Section 4, we give a priori estimate for the positive solutions of (8) by using maximum principle and Harnack inequality.In Section 5, we give a nonexistence result of nonconstant solutions of (8).In Section 6, we consider the existence of nonconstant positive solutions of (8).Finally, to support our theoretical predictions, some numerical simulations are given.

Basic Dynamics and a Priori Bound
Theorem 1.For system (7), one has the following.
For the estimate of Multiplying ( 14) by / and adding it to (15), we have Integration of the inequality leads to

Stability of the Nonnegative Constant Steady States of (7)
In this section, we will analyze the stability of nonnegative constant steady states of (7).By the direct computation, we see that the possible nonnegative constant steady states of ( 7) are where (iii) X fl {u = (, V) ∈ [ 1 (Ω)] 2 : /n = V/n = 0}, and so X = ⊕ ∞ =1 X  , where Let Ẽ be a nonnegative constant steady state of ( 7); then the linearization of ( 7) at a constant solution Ẽ can be expressed by where  = diag( 1 ,  2 ), u = ((, ), V(, ))  , and In view of Notation 1, we can induce the eigenvalues of system (19) confined on the subspace X  .If  is an eigenvalue of (19) on X  , it must be an eigenvalue of the matrix −   +  for each  ≥ 0. It is easy to see that  satisfies the characteristic equation Proof.(i) For  0 = (0, 0), the eigenvalues are Obviously,  0 is unstable.

The Prior Estimate
In this section, we will give some a priori estimates of positive solutions to (8).Firstly, we give two known lemmas.
In the following, we estimate the positive lower bound of positive solution of (8).Theorem 6.Let Ω be a bounded smooth domain in   .There exist two positive constants  <  depending possibly on  1 ,  2 , , , , , and Ω, such that such that any positive solution ((), V()) of system (8) satisfies Proof.From Lemma 5, we obtain where  depends on  1 ,  2 , , , , and .
From Lemma 3, we obtain the fact that there exists a positive constant  2 such that sup On the contrary, suppose the result is false.Then there exists a sequence {(  , V  )} of positive solutions to system (8) such that sup By the regularity theory for elliptic equations, there exists a subsequence of {(  , V  )}, will be denoted again by Observe that  0 ≤  and, from (41), either  0 ≡ 0 or V 0 ≡ 0. Therefore, we have the following two cases: (8), one can obtain the following integral equation by integrating (8) for   and V  over Ω: (i) In this case,  0 ≡ 0; then uniformly as  → ∞ and V  > 0; then for sufficiently large , we have which is a contradiction.

Nonexistence of Nonconstant Positive Steady States
In this section, we can show the nonexistence of nonconstant positive solutions to system (8) when the diffusion coefficients  1 and  2 are large.
Furthermore, multiplying the first equation of ( 8) by /, adding it to the second equation, and integrating over Ω, we get and then the Neumann boundary conditions lead to Thus Multiplying the first equation in ( 8) by  − , we have Multiplying the second equation in (8) by V − V, we have From ( 54) and ( 55) and the Poincaré inequality, we obtain where Hence, if then and (, V) must be a constant solution.

Existence of Nonconstant Positive Steady States
In this subsection, we discuss the existence of nonconstant positive solutions to system (8) when the diffusion coefficients  1 and  2 vary while the parameters  1 , , , , , and  2 are fixed by using the Leray-Schauder degree theory.Throughout this section, we assume that the positive constant steady state  * = ( * , V * ) exists.
For simplicity, denote u = (, V) and Thus, (8) can be written as and, obviously, u is a positive solution of (61) if and only if where ( − Δ) −1 is the inverse of  − Δ with the homogeneous Neumann boundary condition.As F(⋅) is a compact perturbation of the identity operator, the Leray-Schauder degree deg(F(⋅), Λ, 0) is well-defined from Theorem 6.By direct computation, we have where  is the number of negative eigenvalues of F u ( * ).Note that  is an eigenvalue of F u ( * ) on   if and only if it is an eigenvalue of the matrix Thus F u ( * ) is invertible if and only if, for all  ≥ 0, the matrix   is nonsingular.Writing we have that if ( 1 ,  2 ; ) ̸ = 0, then ( 1 ,  2 ; ) < 0 if and only if the number of negative eigenvalues of F u ( * ) in   is odd.The following lemma gives the explicit formula of calculating the index.
where (  ) is the algebraic multiplicity of   .
To facilitate our computation of deg(F(⋅),  * ), we only need consider the sign of det Obviously, nonnegative roots of (68) exist if and only if  2 1  2 − 4 1  2  3 > 0 and  1 > 0. Assume that  + and  − are the two roots of (68), we have the following conclusion.
The positive solutions of the problem are contained in Λ.Note that u is a positive solution of system (8) if and only if it is a positive solution of (71) with  = 1.u * is the unique positive constant solution of (71) for any  ∈ [0, 1].
According to the choice of  * in Theorem 7, we have  * which is the only fixed point of A 0 . deg Since F =  − (⋅, 1) and if (8) has no other solutions except the constant one  * , then we have On the other hand, by the homotopy invariance of the topological degree, which is a contradiction.Therefore, there exists at least one nonconstant solution of (8).

Conclusions
In this paper, we have investigated the existence/nonexistence of nonconstant positive steady states for a diffusive predatorprey system with a group defense for prey under Neumann boundary conditions.The existence results provide a theoretical support for pattern formation caused by diffusion.We also study the stability of nonnegative equilibria and obtain the fact that  1 is globally asymptotically stable when   <  2 .In fact, the positive steady state does not exist at this time.

Figure 1 :
Figure 1: The steady state  1 is globally asymptotically stable.

Figure 2 :
Figure 2: The positive steady state  * is locally asymptotically stable.
be the eigenvalues of −Δ on Ω under homogeneous Neumann boundary condition.We define the following space decomposition:(i) (  ) is the space of eigenfunctions corresponding to   for  = 0, 1, 2, . ...