NON-CONSTANT POSITIVE STEADY STATE OF A DIFFUSIVE LESLIE-GOWER TYPE FOOD WEB SYSTEM

A three species food web comprising of two preys and one predator in an isolated homogeneous habitat is considered. The preys are assumed to grow logistically. The predator follows modified Leslie-Gower dynamics and feeds upon the prey species according to Holling Type II functional response. The local stability of the constant positive steady state of the corresponding temporal system and the spatio-temporal system are discussed. The existence and non-existence of nonconstant positive steady states are investigated.


Introduction
In the last decades the temporal modeling of species interactions have been extensively investigated for two or more species in food chains / food webs [1,5,6,7,8,9,14,17] assuming uniform distribution of resources in the habitat under consideration. The nonlinear system may admit several steady states and the long term behavior is studied through local and global stability of these steady states. However, in reality resources are not uniformly distributed in the habitat and hence their densities are spatially inhomogeneous. In such cases, spatio-temporal dynamics of an ecological system is represented by a system of coupled nonlinear reaction diffusion equations.
The spatio-temporal dynamics of two species with variety of functional responses have been extensively investigated by many authors [2,3,13,15,16,20,21,22,24,26,27,29,30]. Although the spatio-temporal system and its corresponding temporal system admits same uniform equilibrium states, the dynamic behavior may be quite different. The presence of diffusion may destabilize the otherwise stable uniform equilibrium state of the temporal system. The diffusive instabilities may lead to the emergence of Turing patterns in the ecological system [3,24]. The existence of non-constant positive steady state is possible and its stability is of interest [2,3,13,21,22,26,27]. The complex stable pattern formations in such systems have been reported in many ecological situations.
In [13,22], the authors have considered a Leslie-Gower predator-prey system with Beddington-DeAngelis functional response and general functional response, respectively. The long time behavior of solutions of the corresponding systems have be investigated. Moreover, the existence of non-constant positive steady states of the systems have been discussed. W. Ko and K. Ryu [13] pointed out that a Leslie-Gower predator-prey system with general functional response may admit at least one non-constant positive steady state. However, the system with linear predator dependent functional response has no non-constant positive steady state.
In nature, the ecological communities exhibit a very complex dynamical behavior. Two species autonomous models are insufficient to produce realistic dynamics. The complex dynamics is possible for multi-species systems. Complex temporal dynamics in food chain/food web models are reported by many investigators [1,6,7,8,9,14]. However few attempts have been made to investigate the spatio-temporal dynamics in food chain/food web models [4,10,12,19,23,25,28]. The spatio-temporal dynamics of three species diffusive systems with different types of functional responses under Dirichlet and zero flux boundary condition have been explored. The local as well as global stability of the constant positive steady state, the existence and non-existence of the non-constant steady state of such systems have been discussed.
In [10], the authors have considered a diffusive predator-prey system with two predators (consumers) competing for one prey(resource) with Beddington-DeAngelis and Holling type II functional response and studied the non-existence of nonconstant positive steady state, the existence and bifurcation of non-constant positive steady state. Zheng [28] studied the existence and stability of semi-trivial steady states of a reaction diffusion system comprising of two competing preys and a modified Leslie-Gower type predator dynamic under both Dirichlet and homogeneous Neumann boundary conditions. However, W. Chen and M. Wang [4] studied the existence of positive solution of the corresponding elliptic system of the reaction diffusion system considered by Zheng [28] under Dirichlet boundary conditions only.
The investigation of the existence and non-existence of non-constant positive steady state for a three species food web system comprising of two preys and a predator with the modified Leslie-Gower type dynamics seems to be rare. So, in this paper, we are interested in a diffusive food web system comprising of two apparently competing logistic preys and a predator, where the predator follows modified Leslie -Gower type dynamics and feeds upon the prey species' according to Holling Type II functional response. The main objective of this paper is to study the existence of non-constant positive steady state solution by using the Leray-Schauder degree theory.
This paper is organized as follows: the model is described and preliminary analysis is carried out in section 2. Section 3 is devoted to the local stability of the positive steady state of the temporal, non-spatial, system. In section 4, the uniform asymptotical stability of the constant positive steady state of the spatio-temporal system is discussed. In section 5, a prior estimate for the positive steady state of the system is mentioned. In sections 6 and 7, the non-existence and existence of the non-constant positive steady state are investigated respectively.

Model
Consider a food web in a homogeneous bounded habitat Ω ∈ R n (n > 1) consisting of two prey species, U(X, T ),V(X, T ), and a predator species,W(X, T ). In the absence of predation, the two prey species grow logistically with carrying capacities K 1 and K 2 and intrinsic growth rates r 1 and r 2 respectively. The prey predator interaction is assumed to be Holling type II. The predator grows logistically with intrinsic growth rate r 3 and carrying capacity proportional to the renewable resources, population size of prey. The modified Leslie-Gower type dynamics for the predator species is considered. The three species are assumed to diffuse at rates D i (i = 1, 2, 3). Thus, the spatio-temporal dynamics of the three species is given by: All the parameters K j ,h j (j = 1, 2),r i ,A i ,S i ,D i (i = 1, 2, 3) in system (2.1) are assumed to be positive constants and have usual meaning. The initial data U 0 (X), V 0 (X) and W 0 (X) are non-negative continuous functions of position X in Ω. The vector n is an outward unit normal vector to the smooth boundary ∂Ω of the habitat Ω. The homogeneous Neumann boundary condition signifies that the system is self contained and there is no population flux across the boundary ∂Ω. Using the following scaling: and the parameters the system (2.1) takes the form It is necessary to investigate the temporal dynamics of the system before studying the spatio-temporal system (2.2). In the absence of population gradient the spatiotemporal system (2.2) is reduced to the system: Denoting u = (u, v, w) T and G(u) = (G 1 (u), G 2 (u), G 3 (u)), the system (2.3) can be written as:

Analysis of The Temporal Model
It can be easily proved that the system (2.4) has positive solutions.
The comparison argument applied to the first two equations of system Again, applying a comparison argument gives 0 ≤ w(t) ≤ δ 3 c 3 (1 + s 1 + s 2 ). This proves that the system (2.4) has bounded solutions.
The system (2.4) has the following equilibrium points.
(ii) Axial equilibrium points: u 1 = (1, 0, 0),u 2 = (0, 1, 0), u 3 = (0, 0, δ3 c3 ). (iii) Boundary equilibrium points: Similarly, w " = δ3 c3 (1 + s 2 v " ) and v " is the unique positive root of the quadratic polynomial The following two cases may arise. Case I: If δ 2 c 1 > c 2 then u * is the positive root of the quadratic equation Case II: If δ 2 c 1 < c 2 then v * , which is the positive root of the quadratic equation This in turn implies the unique existence and positivity of the equilibrium densities u * ,given as u * = δ2c1v * −δ2c1+c2 c2 > 0, and w * . In this case v * < u * . Thus, the positive equilibrium point u * exists uniquely if δ 2 c 1 < c 2 and c 3 δ 2 > c 2 (1 + s 1 )δ 3 . From the above two cases it can be observed that the unique existence of the constant positive steady state u * does not depend on the sign of the expression (δ 2 c 1 − c 2 ). Thus, we can have the following sufficient condition which ensures the unique existence of u * .
Remark 3.1. It may be observed from the above existence and uniqueness conditions of the three equilibrium points; the existence of the positive equilibrium point u * guarantees the existence of boundary points u ′ and u " . However, the boundary points may exist even though the coexistence equilibrium point does not exist.
In the next two sections, the local stability of the coexistence steady state u * = (u * , v * , w * ) with respect to the temporal system (2.4) and the spatio-temporal system (2.2) are discussed under condition (3.1).
The matrix coefficients are The characteristics polynomial of G u (u * ) is given by According to Routh Hurwitz criterion, the necessary and sufficient conditions for stability are Algebraic manipulations of the expressions (a 11 a 22 − a 12 a 21 ), (a 23 a 33 + a 13 a 21 ) and (a 12 a 23 + a 13 a 11 ) gives: The following are observed under condition (3.2):   It is easy to check that both the existence condition (3.1) and the local stability condition (3.2) hold. However, for the following choice of parameters, the existence condition (3.1) is satisfied but the stability condition (3.2) fails.

2)
In this section, the local stability of the constant positive steady state u * of the spatio-temporal system (2.2) will be investigated. Let 0 = µ 0 < µ 1 < µ 2 < µ 3 < ... be the eigenvalues of the operator −∆ on Ω with the homogeneous Neumann boundary condition. Let X i be the eigenspace corresponding to the eigenvalue µ i , then the solution space X = {u ∈ [C 1 Ω ] 3 | ∂u ∂n = 0 on ∂Ω} can be represented as Theorem 4.1. Assume that the parameters in (2.4) satisfy (3.2). Then the constant positive steady state u * of the spatio-temporal system (2.2) is uniformly asymptotically stable.
Proof. The linearization of the system (2.2) about the constant positive steady state u * is: 1, d 3 ).
The eigenspace X i , i ≥ 0, is invariant under the operator L. λ is an eigenvalue of L on X i if and only if it is an eigenvalue of the matrix −µ a 33 a 22 − a 23 a 32 ) + (a 11 a 33 − a 13 a 31 ) + d 3 (a 11 a 22 − a 12 a 21 )) + P 0 .

From the assumption, it follows that
Algebraic manipulation of the expression E i = P 1i P 2i − P 0i yields Clearly E i > 0; i ≥ 0, as P 1 P 2 −P 0 > 0 under the given conditions. From the Routh-Hurwitz criterion it follows that, for each i ≥ 0, all the three roots λ i,1 , λ i,2 , λ i,3 of σ i (λ) = 0 have negative real parts. Thus there exist some positive numbers κ i such that Let κ = min{κ i }. Then, κ > 0 and Re{λ i,1 }, Re{λ i,2 }, Re{λ i,3 } ≤ − κ ∀i. Consequently, the spectrum of L, which consists of eigenvalues, lies in {Reλ ≤ − κ}. Thus, theorem 5.1.1 of Dan Henry [11] concludes the uniform asymptotical stability of u * . and Turing instability cannot occur in the vicinity of u * . Hence, system (2.2) will not have a non constant positive steady state in some neighborhood of u * under condition (3.2). However,if condition (3.2) fails then there is a possibility for the occurrence of Turing instability. For example, for the data set (3.5), the condition (3.2) fails but condition (3.3) still holds and u * is temporally stable. Further, P 0i changes its sign with wave number k 2 as shown in fig. 2. In this case, Turing instability is possible. Accordingly, the system may eventually go to non-constant positive steady state. Thus, existence of non-constant positive steady state may be possible when condition (3.2) fails.

A priori estimates of non-constant positive steady state
The steady state problem corresponding to (2.2) is The classical solutions of (5.1) are assumed to be in C 2 (Ω) C 1 (Ω). The following two results are needed for the estimates.

Existence of non-constant positive steady state
The main aim of this section is to discuss the existence of non-constant positive solutions to (5.1) by using Leray-Schauder Theorem. Theorem 6.1 implies that when the assumptions of the theorem holds then (5.1) will not have non-constant positive solution. In addition to this, as a consequence of theorem 4.1 the system (2.2) will not have a non-constant positive steady state in some neighborhood of u * if the stability condition (3.2) holds. However, by properly choosing the parameters and contradicting the condition (3.2), it is possible to obtain a non-constant positive solution for (5.1) and hence a non-constant positive steady state of (2.2). This ensures that stationary patterns and more interesting Turing patterns can arise as a result of diffusion.
The linearization of (5.1) at u * is presented as follows. Let X be as in section 4 and define X + = {u ∈ X|u > 0, v > 0, w > 0} on Ω, B(C) = {u ∈ X|C −1 < u, v, w < C} on Ω where C is a positive constant in which its existence is ensured by theorems 5.1 and 5.2. Thus, (5.1) is equivalent to Also, u is a positive solution of (7.1) if and only if where, I is the identity map from C 1 (Ω) to itself and (I−∆) −1 is the inverse of I−∆ in X subject to Neumann boundary condition. It can be noticed that the Leray -Schauder degree deg(ϕ(.), 0, B) is well defined if ϕ(u) = 0 for any u ∈ ∂B(C). It is straight forward to see that and recall that if D u ϕ(u * ) is invertible then the index of ϕ at u * is defined as index(ϕ(.), u * ) = (−1) ρ , where ρ is the total number of eigenvalues with negative real parts(counting multiplicities) of D u ϕ(u * ) [18]. As in the proof of theorem (3.1), note that, λ is an eigenvalue of the matrix D u ϕ(u * ) on X i if and only if it is an eigenvalue of the matrix 1 1+µi is non-singular for any i ≥ 1. For the sake of convenience, denote 2) The number of negative eigenvalues µ of D u ϕ(u * ) on X i is odd if and only if l 3 = (a 22 a 33 − a 23 a 32 )d 1 + (a 11 a 33 − a 13 a 31 ) + (a 11 a 22 − a 12 a 21 )d 3 .
Since det (G u (u * )) < 0, i.e µ 1 µ 2 µ 3 < 0, and l 1 > 0, one of µ 1 , µ 2 , µ 3 is real and negative, and the product of the other two is positive. For a sufficiently large d 3 , i.e d 3 → ∞, we have where The following theorem proves the existence of non-constant positive solutions of (5.1) for some fixed positive constants Λ and d 1 , for sufficiently large diffusion coefficient d 3 .
Theorem 7.1. Assume that the parameters Λ and d 1 are fixed, and satisfy (3.1), a 11 > 0 and τ 2 > 0. Let µ 2 be given by the limit (7.4). If µ 2 ∈ (µ p , µ p+1 ) for some p ≥ 1 and the sum ρ = Proof. From proposition 7.2, it follows that there exists a positive constant D 3 such that,when d 3 ≥ D 3 , (7.5) holds and 0 = µ 0 < µ 3 < µ 1 , µ 2 ∈ (µ p , µ p+1 ) . (7.6) Now we prove that, for any d 3 ≥ D 3 , (5.1) admits at least one non-constant positive solution. Assume that the assertion is not true for some d 3 = d 3 ≥ D 3 . In this proof, the homotopy invariance of the topological degree is used to derive a contradiction. Thus, u is a non-constant positive solution of (5.1) if and only if it is a positive solution of (7.7) for t = 1. Clearly, u * is the unique constant positive solution of (7.7) for any 0 ≤ t ≤ 1. For any 0 ≤ t ≤ 1, u is a positive solution of (5.1) if and only if it is a solution of the following problem It is clear that ϕ(1; u) = ϕ(0; u). From theorem 6.1 it follows that ϕ(0; u) = 0 has only the positive solution u * in X + . It is easy to see that In view of theorems 5.1 and 5.2, there exists a positive constant C such that, for all 0 ≤ t ≤ 1, the positive solutions of (7.7) satisfy C −1 < u, v, w < C and hence ϕ(t; u) = 0 on ∂B(C). By the homotopy invariance of the topological degree, we have deg(ϕ(1; .), 0, B(C)) = deg(ϕ(0; .), 0, B(C)). This contradicts ( 7.11). Hence the proof is complete.