Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.


Introduction
Three-species reaction-diffusion models with Holling-type II functional response have been a familiar subject for the analysis. Taking more practical factors into consideration, a model with a modified Leslie-Gower scheme is worthy to explore. Leslie-Gower's scheme indicates that the carrying capacity of the predator is proportional to the population size of the prey. The existing works [1][2][3] are all about models with this scheme. As a matter of fact, predators prefer to prey on other prey in the event of a shortage of favorite prey, so the research of the modified Leslie-Gower model springs up. Aziz-Alaoui and Okiye [4] focused on a twodimensional continuous time dynamical system modeling a predator-prey food chain and gave the main result of the boundedness of solutions, the existence of an attracting set, and the global stability of the coexisting interior equilibrium, which was based on a modified version of the Leslie-Gower scheme and Holling-type II scheme. Singh and Gakkhar [5] investigated the stabilization problem of the modified Leslie-Gower type prey-predator model with the Holling-type II functional response. The analysis of models with a modified Leslie-Gower scheme can be also found in [6][7][8][9][10].
Nonconstant positive steady states have received increasing attention in recent years, see [11][12][13][14][15][16][17][18] and references therein. Ko and Ryu [19] showed that the predator-prey model with Leslie-Gower functional response had no nonconstant positive solution in homogeneous environment, but the system with a general functional response might have at least one nonconstant positive steady state under some conditions. Zhang and Zhao [20] analyzed a diffusive predator-prey model with toxins under the homogeneous Neumann boundary condition, including the existence and nonexistence of nonconstant positive steady states of this model by considering the effect of large diffusivity. Shen and Wei [21] considered a reaction-diffusion mussel-algae model with state-dependent mussel mortality which involved a positive feedback scheme. Wang and his partners [22] considered a tumor-immune model with diffusion and nonlinear functional response and investigated the effect of diffusion on the existence of nonconstant positive steady states and the steady-state bifurcations. Hu and Li [23] were concerned about a strongly coupled diffusive predator-prey system with a modified Leslie-Gower scheme and established the existence of nonconstant positive steady states. Qiu and Guo [24] analyzed a stationary Leslie-Gower model with diffusion and advection.
Motivated by the mentioned above, we consider a threespecies reaction-diffusion model with a modified Leslie-Gower and Holling-type II scheme under the homogeneous Neumann boundary condition as follows: where u and v represent the density of two competitors, respectively, while w stands for the density of the predator who preys on u. A, B, and C are all positive as the intrinsic growth rates, A 1 and A 2 regard as influencing factors within diverse populations themselves while B 1 and B 2 are influencing factors between different populations. All of them are nonnegative. C 1 w/ð1 + D 1 uÞ and C 2 w/ð1 + D 2 uÞ are the modified Leslie-Gower scheme, and C 1 , C 2 , D 1 , and D 2 are positive. Applying the following scaling to (1), as well as assuming C 1 D 2 /D 1 C 2 = 1 for simplicity of calculation: still using u, v, w, t replace m 1 , m 2 , m 3 , s, the following ODE system can be logically obtained: It is clear that ð0, 0, 0Þ, ða, 0, 0Þ, ð0, b, 0Þ, ð0, 0, cβ 2 Þ, and ð0, b, cβ 2 Þ are nonnegative constant solutions of system (3). ðða − α 1 bÞ/ð1 − α 1 α 2 Þ, ðb − α 2 aÞ/ð1 − α 1 α 2 Þ, 0Þ is a semitrivial solution when it satisfies ða − α 1 bÞðb − α 2 aÞ > 0. When System (3) yields that If the following alternative conditions hold: ii there exists the unique positive equilibrium ðu * , v * , w * Þ as where Taking the diffusion into account, the corresponding PDE system can be written as where Ω ⊂ R N is a smooth bounded domain, n is the outward unit normal vector on ∂Ω, Δ is the Laplace operator, and diffusion coefficients are d 1 , d 2 , d 3 > 0: The rest of this paper is arranged as follows. In Section 2, the stability of constant solutions for the ODE system is discussed. In Section 3, the stability of constant solutions for the PDE system is studied. In Section 4, we focus on the priori estimates of positive steady states. In the last two sections, we have a discussion about the nonexistence and existence of nonconstant positive steady states under different conditions.

Stability of Constant Solutions for the ODE System
In this section, we discuss the stability of constant solutions with the condition of their existence for the ODE system.
The Jacobian matrix of the ODE system at ð _ u, 0, _ wÞ is The characteristic polynomial is When the eigenvalue satisfies ð17Þ (4) and (5), we know that With the existence condition aβ 1 > cβ 2 , p 1 > 0 and p 2 > 0 hold, such that equation (17) has two solutions with negative real parts.
Because of aβ 1 > cβ 2 , holds, then The Jacobian matrix of the ODE system at ð0, b, cβ 2 Þ is The characteristic polynomial is The corresponding eigenvalues are The corresponding characteristic polynomial is
The proof is complete.

Stability of Constant Solutions for the PDE System
In this section, the stability of the constant solutions with the condition of their existence for the PDE system is discussed. Let 0 = μ 0 < μ 1 < μ 2 < μ 3 < ⋯ as the eigenvalues of the operator −Δ over Ω under the homogeneous Neumann boundary condition and Eðμ i Þ be the corresponding eigenspace while fφ ij | j = 1, 2,⋯, dim Eðμ i Þg is a set of the orthogonal basis of Eðμ i Þ, Theorem 2. For the PDE system (11), let Γ = fa, b, c, α 1 , α 2 , β 1 , β 2 g and 1/ðβ 1 + u * Þ ≜ B.
The Jacobian matrix of the PDE system at ð _ u, 0, _ wÞ is The characteristic polynomial is When the eigenvalue satisfies λ 1 = b − α 2 _ u > 0, it deduces that a < b/α 2 , there exists an eigenvalue with positive real part, and ð _ u, 0, _ wÞ is unstable to PDE system (11). It is clear that eigenvalue Then, we discuss the following equation emphatically: Journal of Function Spaces Let It shows that p 3 > 0 on account of p 1 > 0. When cd 1 + ½ _ u − ðð _ uða − _ uÞÞ/ðβ 1 + _ uÞÞd 3 > 0, we know p 4 > 0 holds. So the eigenvalues all have negative real parts.
When A 1 > 0 and d 1 , have d 3 a 11 a 22 + d 2 a 11 a 33 + d 1 a 22  a 33 − d 3 a 12 a 21 − d 2 a 13 a 31 > 0 and d 1 d 2 a 33 + d 1 d 3 a 22 + d 2 d 3 a 11 < 0. As a result of A 3 > 0 and d 1 , d 2 , d 3 > 0, A 3μ i > 0 can be obtained. What is more, Thus, the eigenvalues all have negative real parts.
In the following, we shall prove that there exists a positive constant κ when the corresponding eigenvalues all have negative real parts, such that Let λ = μ i ζ, then

Journal of Function Spaces
Since μ i → ∞ as i → ∞, it follows that Applying the Hurwitz criterion, the three roots ζ 1 , ζ 2 , ζ 3 of ψðζÞ = 0 all have negative real parts. Thus, there exists a positive constant κ ′ such that Re ðζ 1 Þ, Re ðζ 1 Þ, Re ðζ 1 Þ ≤ −κ ′ . By continuity, there exists i 0 such that the three roots ζ i1 , ζ i2 , ζ i3 of ψðζÞ = 0 satisfy Therefore, the constant solutions are uniformly asymptotically stable when the corresponding eigenvalues all have negative real parts.
The proof is complete.

A Priori Estimates of Positive Steady States
The corresponding steady-state problem of system (11) is Two lemmas are listed here for the preliminary.

Theorem 6. (lower bounds).
Fix Γ and d 1 , d 2 , d 3 as positive constants. Assume that then there exists a positive constant C = CðΓ, Ω, N, d 1 , d 2 , d 3 Þ who can make every positive solution ðu, v, wÞ of system (37) satisfy Journal of Function Spaces Proof. Let In view of (41), (42), and (43), a positive constant C = C ðΩ, N, D, ΓÞ can be easily found, such that where d 1 , d 2 , d 3 > D. Thus, u, v, and w satisfy that According to the Harnack inequality in Lemma 3, there must be a positive constant C * = C * ðΩ, N, D, ΓÞ, such that Then, we apply ðu i , v i , w i Þ to the system of (37) and integrate by parts, so we obtain that There exists a subsequence of fðd 1i , d 2i , d 3i Þg ∞ i=1 according to the L p -regularity theory and Sobolev embedding theorem, but we still use fðd 1i , d 2i , d 3i Þg ∞ i=1 to represent for convenience. So there must be u * , v * , w * and ð d 1 , They can be written as follows: Let i → ∞, we get that We now discuss the following three cases.

Journal of Function Spaces
The proof is complete.

Existence of Nonconstant Positive Steady States
In this part, we discuss the existence of nonconstant positive solutions of (37) by using the degree theorem. Fix the Γ, d 1 , d 3 still as positive number and define X + = fU ∈ X | U > 0, x ∈ Ω, i = 1, 2, 3g, BðlÞ = fU ∈ X | l −1 < u, v , w < l, x ∈ Ωg, l > 0. Then, (37) can be noted as where ðI − ΔÞ −1 is the inverse of I − Δ in X under the homogeneous Neumann boundary condition. And if FðUÞ ≠ 0 on ∂B, the Leray-Schauder degree deg ðFð·Þ, 0, BÞ can be well defined. Besides, we note that The index of FðUÞ at U * can be either 1 or -1 if D U FðU * Þ is invertible, which is defined as indexðFð·Þ, U * Þ = ð−1Þ r , where r is the total number of eigenvalues with negative real parts of D U FðU * Þ.
Let λ be an eigenvalue of D U FðU * Þ on X ij for each integer i ≥ 1 and each integer 1 ≤ j ≤ dim Eðμ i Þ, if and only if it is an eigenvalue of the matrix Hence, D U FðU * Þ is invertible if and only if, for all i ≥ 1, i ∈ Z, the matrix I − ð1/ð1 + μ i ÞÞ½D −1 G U ðU * Þ + I is nonsingular. Let We can know that if Hðμ i Þ ≠ 0, the number of negative eigenvalues of D U FðU * Þ on X ij is odd if and only if Hðμ i Þ < 0 for every 1 ≤ j ≤ dim Eðμ i Þ. According to this, we can form the following result.