# Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

Authors: Zongmin Yue; Wenjuan Wang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We investigated the dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition. Some qualitative properties, including the dissipation, persistence, and local and global stability of positive constant solution, are discussed. Moreover, we give the refined a priori estimates of positive solutions and derive some results for the existence and nonexistence of nonconstant positive steady state.


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## Body

## 1. Introduction

In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, sparrow and sparrow hawk, and some other species [1, 2], Robert May developed a prey-predator model of Holling-type functional response [3, 4] to describe the predation rate and Leslie’s formulation [5, 6] to describe predator dynamics. This model is known as Holling-Tanner model for prey-predator interaction, which takes the form of
(1)dNdt=rN(1-NK)-mNPa+N,dPdt=sP(1-hPN),
where N(t) and P(t) stand for prey and predator population (density) at any instant of time t. r, K, m, a, s, h are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively.The dynamics of model (1) has been considered in many articles. For example, Hsu and Huang [7] obtained some results on the global stability of the positive equilibrium, more precisely, under the conditions which local stability of the positive equilibrium implies its global stability. Gasull and coworkers [8] investigated the conditions of the asymptotic stability of the positive equilibrium which does not imply global stability. Sáez and González-Olivares [9] showed the asymptotic stability of a positive equilibrium and gave a qualitative description of the bifurcation curve.Recently, there is a growing explicit biological and physiological evidence [10–12] that in many situations, especially, when the predator has to search for food (and therefore has to share or compete for food), a more suitable general predator-prey theory should be based on the so-called radio-dependent theory which can be roughly stated as that the per capital predator growth rate should be a function of the ratio of prey to predator abundance, and so would be the so-called predator functional responses [13]. This is supported by numerous fields and laboratory experiments and observations [14, 15]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of
(2)dNdt=rN(1-NK)-mNPN+aP,dPdt=sP(1-hPN).For model (2), in [13], the authors investigated the effect of time delays on the stability of the model and discussed the local asymptotic stability and the Hopf-bifurcation. Liang and Pan [16] have studied the local and global asymptotic stability of the coexisting equilibrium point and obtained the conditions for the Poincaré-Andronov-Hopf-bifurcating periodic solution. M. Banerjee and S. Banerjee [17] have studied the local asymptotic stability of the equilibrium point and obtained the conditions for the occurrence of the Turing-Hopf instability for PDE model. It is shown that prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability.On the other hand, an implicit assumption contained in the logistic equation(3)dNdt=rN(1-NK)
is that the average growth rate N′(t)/N is a linear function of the density N(t). It has been shown that this assumption is not realistic for a food-limited population under the effects of environmental toxicants. The following alternative model has been proposed by several authors [18–23] for the dynamics of a population where the growth limitations are based upon the proportion of available resources not utilized:
(4)dNdt=rNK-NK+cN,
where r/c is the replacement of mass in the population at K. Equation (4) takes into account both environmental and food chain effects of toxicant stress.Based on the above discussions, in this paper, we rigorously consider the radio-dependent Holling-Tanner model with Smith growth that takes the form of(5)dNdt=rNK-NK+cN-mNPaP+N,dPdt=sP(1-hPN).Also considering the spatial dispersal and environmental heterogeneity, in this paper, we study the following generalized reaction-diffusion system for model (5):
(6)∂Ndt=rNK-NK+cN-mNPaP+N+d1ΔN,x∈Ω,t>0,∂Pdt=sP(1-hPN)+d2ΔP,x∈Ω,t>0,∂N∂ν=∂P∂ν=0,x∈∂Ω,t>0,
where Ω⊂ℝn  (n≥1) is a bounded domain with a smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω. The nonnegative constants d1 and d2 are the diffusion coefficients of N and P, respectively. The zero-flux boundary condition indicates that predator-prey system is self-contained with zero population flux across the boundary. From the standpoint of biology, we are interested only in the dynamics of model (6) in the closed first quadrant ℝ+2={(N,P):N≥0,P≥0}. Thus, we consider only the biologically meaningful initial conditions
(7)N(x,0)=N0(x)>0,P(x,0)=P0(x)>0,x∈Ω,
which are continuous functions due to its biological sense. Straightforward computation shows that model (6) are continuous and Lipschizian in ℝ+2 if we redefine that when
(8)∂N∂t=∂P∂t=0,if(N,P)=(0,0).
Hence, the solution of model (6) with positive initial conditions exists and is unique.The stationary problem of model (6), which may display the dynamical behavior of solutions to model (6) as time goes to infinity, satisfies the following elliptic system:
(9)-d1ΔN=rNK-NK+cN-mNPN+aP,x∈Ω,t>0,-d2ΔP=sP(1-hPN),x∈Ω,t>0,∂N∂ν=∂P∂ν=0,x∈∂Ω,t>0,N(x,0)=N0(x)>0,P(x,0)=P0(x)>0,x∈Ω.Simple computation shows that ifm<r(a+h), then model (6) and (9) possess a unique positive constant solution, denoted by E*=(N*,P*), where
(10)N*=K(ar+hr-m)ar+cm+hr,P*=1hN*.
In addition, (K,0) is the second nonnegative constant steady state of model (6) and (9).The rest of the paper is organized as follows. In Section2, we investigate the lager time behavior of model (6), including the dissipation, persistence property, and local and global stability of positive constant solution E*. In Section 3, we first give a priori upper and lower bounds for positive solutions of model (9), and then we deal with existence and nonexistence of nonconstant positive solutions of model (9), which imply some certain conditions under which the pattern happens or not.


## 2. Large Time Behavior of Solution to Model (6)

In this section, the dissipation and persistence properties are studied for solution of model (6). Moreover, the local and global asymptotic stability of positive constant solution E*=(N*,P*) are investigated.


### 2.1. The Properties of Dissipation and Persistence of Solution to Model (6)

Theorem 1.
All the solutions of model (6) are nonnegative and defined for all t>0. Furthermore, the nonnegative solution (N,P) of model (6) satisfies
(11)limsupt→∞maxΩ¯N(·,t)≤K,limsupt→∞maxΩ¯P(·,t)≤Kh.Proof.
The nonnegativity of the solution of model (6) is clear since the initial value is nonnegative. We only consider the latter of the theorem.
Note thatN satisfies
(12)∂N∂t-d1ΔN≤rN(K-N)K+cN,x∈Ω,t>0,∂N∂ν=0,x∈∂Ω,t>0,N(x,0)=N0(x),x∈Ω.
Letz(t) be a solution of the ordinary differential equation:
(13)z˙(t)=rN(K-N)K+cN,x∈Ω,t>0,z(0)=maxΩ¯N(x,0)>0.
Then, limt→∞z(t)=K. From the comparison principle, one can get N(x,t)≤z(t); hence,
(14)limsupt→∞maxΩ¯N(x,t)≤K.
As a result, for anyε>0, there exists t0>0, such that N(x,t)≤K+ε for all x∈Ω¯ and t≥t0. Hence, P(x,t) is a lower solution
(15)∂z∂t-d2Δz=sz(1-hwK+ε),x∈Ω,t>t0,∂z∂ν=0,x∈∂Ω,t>t0,z(x,t0)=P(x,t0).
LetP(t) be the unique positive solution of problem
(16)w˙(t)=sw(1-hwK+ε),t>t0,w(t0)=maxΩ¯P(x,t0).
Then, P(t) is an upper solution of (15). As limt→∞P(t)=(K+ε)/h, we get from the comparison principle that
(17)limsupt→∞maxΩ¯P(x,t)≤K+εh,
which implies the second assertion by the arbitrariness of ε>0. This ends the proof.Definition 2 (see [24]).
The spatial model (6) is said to have the persistence property if for any nonnegative initial data (N0(x),P0(x)), there exists a positive constant ε=ε(N0,P0), such that the corresponding solution (N,P) of model (6) satisfies
(18)liminft→∞minΩ¯N(x,t)≥ε,liminft→∞minΩ¯P(x,t)≥ε.Theorem 3.
Ifm(1+c)<ar, then model (6) has the persistence property.Proof.
LetN(x,t) be an upper solution of the following problem:
(19)∂z∂t-d1Δz=z(r1+c-ma-rzK(1+c)),x∈Ω,t>T,∂z∂ν=0,x∈∂Ω,t>T,z(x,T)=N0(x,T)≥0,x∈Ω¯.
LetN(t) be the unique positive solution to the following problem:
(20)dwdt=w(r1+c-ma-rwK(1+c)),t>T,w(T)=maxΩ¯N0(x,T)≥0.
Due tom(1+c)<ar, we have that limt→∞w(t)=K(ar-m(1+c))/ar. By comparison, it follows that
(21)liminft→∞minΩ¯N(x,t)≥K(ar-m(1+c))ar≜η.
Hence, N(x,t)>η-ε for t>T and x∈Ω¯.
Similarly, by the second equation of model (6), we have that P(x,t) is an upper solution of problem
(22)∂z∂t-d2Δz=sz(1-hzη-ε),x∈Ω,t>T,∂z∂ν=0,x∈∂Ω,t>T,z(x,T)=P0(x,T)≥0,x∈Ω¯.
LetP(t) be the unique positive solution to the following problem:
(23)dwdt=sz(1-hzη-ε),t>T,w(T)=maxΩ¯P0(x,T)≥0.
Then, limt→∞w(t)=η/h for the arbitrariness of ε, and an application of the comparison principle gives
(24)liminft→∞minΩ¯P(x,t)≥ηh.
The proof is complete.


### 2.2. The Local Stability of the Constant Steady State

In this subsection, we shall analyze the asymptotical stability of the positive constant solutionE* for model (6). Before developing our argument, let us set up the following notations.(i)
Let0=μ0<μ1<μ2<⋯→∞ be the eigenvalues of the operator –Δ on Ω with the zero-flux boundary condition;(ii)
LetE(μ)={ϕ∣-Δϕ=μϕinΩ,  ∂νϕ=0on∂Ω} with μ∈ℝ1;(iii)
Let{ϕij∣j=1,…,dimE(μi)} be an orthonormal basis of E(μi), and Xij={cϕij∣c∈ℝ2};(iv)
Let(25)X={(N,P)∈[H2(Ω)]2∣∂νN=∂νP=0on∂Ω},then(26)X=⨁i=1∞Xi,whereXi=⨁j=1dimE(μi)Xij.Theorem 4.
Assume that(27)m(ar+cm+2hr-acr)<r(a+h)2(r+s+cs)
and the first eigenvalues μ1 of the Dirichlet operator subject to zero-flux boundary conditions satisfy
(28)μ1>max{m(ar+cm+2hr-acr)-r2(a+h)2d1r(1+c)(a+h)2-sd2,0}.
Then the positive constant solution E* of model (6) is locally asymptotically stable.Proof.
Defineℒ:X→C(Ω¯)×C(Ω¯) by
(29)ℒ=(d1Δ+J1-mh2(a+h)2-shd2Δ-s),
where J1=(m(ar+cm+2hr-acr)-r2(a+h)2)/r(1+c)(a+h)2.
For eachi=0,1,2,…,Xi is invariant under the operator ℒ, and λ is an eigenvalue of this operator on Xi if and only if it is an eigenvalue of the following matrix:
(30)Ai=(-d1μi+J1-mh2(a+h)2-sh-d2μi-s).
Moreover,
(31)det(λI-Ai)=λ2-tr(Ai)λ+det(Ai),
where
(32)det(Ai)=d1d2μi2+(d1s-d2J1)μi+s(ar+hr-m)(ar+hr+cm)r(1+c)(a+h)2,tr(Ai)=-(d1+d2)μi+J1-s.
In view of (27) and (28), we have det(Ai)>0>tr(Ai) for any i≥0. Therefore, the eigenvalues of the matrix Ai have negative real parts.
In the following, we prove that there existsδ>0 such that
(33)Re{λi1}≤-δ,Re{λi2}≤-δ.
Letλ=μiξ, then
(34)φ~i(λ)≜μi2ξ2-tr(Ai)μiξ+det(Ai).
Since μi→∞ as i→∞, it follows that
(35)limi→∞φ~i(λ)μi2=ξ2+(d1+d2)ξ+d1d2.
By the Routh-Hurwitz criterion, it follows that the two rootsξ1, ξ2 of φ~i(λ)=0 all have negative real parts. Thus, let d~=min{d1,d2}, we have that Re{ξ1},Re{ξ2}≤-d~. By continuity, we see that there exists i0 such that the two roots ξi1, ξi2 of φ~i(λ)=0 satisfy Re{ξi1}≤-d~/2,  Re{ξi2}≤-d~/2, for all i≥i0. In turn, Re{λi1},Re{λi2}≤-μid~/2≤-d~/2, for all i≥i0.
Let-δ~=max1≤i≤i0{Re{λi1},Re{λi2}}, then δ~>0 and (33) hold for δ=min{δ~,d~/2}. Consequently, the spectrum of ℒ which consists of eigenvalues, lies in {Reλ≤-δ}. In the sense of [25], we obtain that the positive constant solution E*=(N*,P*) of model (6) is uniformly asymptotically stable. This ends the proof.


### 2.3. The Global Stability of the Constant Solution

This subsection is devoted to the global stability of the constant solutionE* for model (6).Theorem 5.
Assume that the following hold:(A1)
m(1+c)<ar;(A2)
h(K+cN*)(N*+2P*)+(K+cN*)(N*+aP*)(a+h)≤2ηr(N*+aP*)(a+h);(A3)
hKN*≤η(2h-1)(N*+aP*)(a+h),where η=K(ar-m(1+c))/ar. Then the constant solution E* is globally asymptotically stable.Proof.
In order to give the proof, we need to construct a Lyapunov function. Define(36)V(N,P)=∫N*N‍ξ-N*ξdξ+∫P*P‍ζ-P*ζdζ,E(t)=∫ΩV(N(x,t),P(x,t))dx.
We note that E(t) is nonnegative, E(t)=0 if and only if (N(x,t),P(x,t))=(N*,P*). Furthermore, by simple computations, it follows that
(37)dE(t)dt=∫Ω(VN(N(x,t),P(x,t))Nt+VP(N(x,t),P(x,t))Pt)dx=∫Ω(d1(N-N*)NΔN+d2(P-P*)PΔP)dx+∫Ω((N-N*)(r(K-N)K+cN-mPN+aP)+(P-P*)(1-hPN))dx=-∫Ω(d1N*N2|∇N|2+d2P*P2|∇P|2)dx+I(t),
where
(38)I(t)=∫Ω((N-N*)2(-rK(1+c)(K+cN*)(K+cN)+P*(N*+aP*)(N+aP)))dx+∫Ω(hN(N-N*)(P-P*)×(1N-N*(N*+aP*)(N+aP))-hN(P-P*)2)dx.
Setφ=N-N*,ϕ=P-P*. We have
(39)I(t)=-∫Ω(rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N)φ2dx-∫Ω(2h-12N-N*2(N*+aP*)(N+aP))ϕ2dx-∫Ω((φ+ϕ)2N*2(N*+aP*)(N+ap)12N(φ-ϕ)2+N*2(N*+aP*)(N+aP)×(φ+ϕ)2N*2(N*+aP*)(N+ap))dx≤-∫Ω(rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N)φ2dx-∫Ω(2h-12N-N*2(N*+aP*)(N+aP))ϕ2dx.
By virtue of Theorems1 and 3 and under the assumption of Theorem, we have
(40)rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N≥rK+cN*-h(N*+2P*)2η(N*+aP*)(a+h)-12η=(2ηr(N*+aP*)(a+h)-h(K+cN*)(N*+2P*)-(K+cN*)(N*+aP*)(a+h))×(2η(K+cN*)(N*+aP*)(a+h))-1≥0,2h-12N-N*2(N*+aP*)(N+aP)≥2h-12K-hN*2η(N*+aP*)(a+h)=η(2h-1)(N*+aP*)(a+h)-hKN*2Kη(N*+aP*)(a+h)≥0.
As a result, we haveI(t)≤0. Thus dE(t)/dt≤0, which implies the desired assertion. The proof is completed.


## 2.1. The Properties of Dissipation and Persistence of Solution to Model (6)

Theorem 1.
All the solutions of model (6) are nonnegative and defined for all t>0. Furthermore, the nonnegative solution (N,P) of model (6) satisfies
(11)limsupt→∞maxΩ¯N(·,t)≤K,limsupt→∞maxΩ¯P(·,t)≤Kh.Proof.
The nonnegativity of the solution of model (6) is clear since the initial value is nonnegative. We only consider the latter of the theorem.
Note thatN satisfies
(12)∂N∂t-d1ΔN≤rN(K-N)K+cN,x∈Ω,t>0,∂N∂ν=0,x∈∂Ω,t>0,N(x,0)=N0(x),x∈Ω.
Letz(t) be a solution of the ordinary differential equation:
(13)z˙(t)=rN(K-N)K+cN,x∈Ω,t>0,z(0)=maxΩ¯N(x,0)>0.
Then, limt→∞z(t)=K. From the comparison principle, one can get N(x,t)≤z(t); hence,
(14)limsupt→∞maxΩ¯N(x,t)≤K.
As a result, for anyε>0, there exists t0>0, such that N(x,t)≤K+ε for all x∈Ω¯ and t≥t0. Hence, P(x,t) is a lower solution
(15)∂z∂t-d2Δz=sz(1-hwK+ε),x∈Ω,t>t0,∂z∂ν=0,x∈∂Ω,t>t0,z(x,t0)=P(x,t0).
LetP(t) be the unique positive solution of problem
(16)w˙(t)=sw(1-hwK+ε),t>t0,w(t0)=maxΩ¯P(x,t0).
Then, P(t) is an upper solution of (15). As limt→∞P(t)=(K+ε)/h, we get from the comparison principle that
(17)limsupt→∞maxΩ¯P(x,t)≤K+εh,
which implies the second assertion by the arbitrariness of ε>0. This ends the proof.Definition 2 (see [24]).
The spatial model (6) is said to have the persistence property if for any nonnegative initial data (N0(x),P0(x)), there exists a positive constant ε=ε(N0,P0), such that the corresponding solution (N,P) of model (6) satisfies
(18)liminft→∞minΩ¯N(x,t)≥ε,liminft→∞minΩ¯P(x,t)≥ε.Theorem 3.
Ifm(1+c)<ar, then model (6) has the persistence property.Proof.
LetN(x,t) be an upper solution of the following problem:
(19)∂z∂t-d1Δz=z(r1+c-ma-rzK(1+c)),x∈Ω,t>T,∂z∂ν=0,x∈∂Ω,t>T,z(x,T)=N0(x,T)≥0,x∈Ω¯.
LetN(t) be the unique positive solution to the following problem:
(20)dwdt=w(r1+c-ma-rwK(1+c)),t>T,w(T)=maxΩ¯N0(x,T)≥0.
Due tom(1+c)<ar, we have that limt→∞w(t)=K(ar-m(1+c))/ar. By comparison, it follows that
(21)liminft→∞minΩ¯N(x,t)≥K(ar-m(1+c))ar≜η.
Hence, N(x,t)>η-ε for t>T and x∈Ω¯.
Similarly, by the second equation of model (6), we have that P(x,t) is an upper solution of problem
(22)∂z∂t-d2Δz=sz(1-hzη-ε),x∈Ω,t>T,∂z∂ν=0,x∈∂Ω,t>T,z(x,T)=P0(x,T)≥0,x∈Ω¯.
LetP(t) be the unique positive solution to the following problem:
(23)dwdt=sz(1-hzη-ε),t>T,w(T)=maxΩ¯P0(x,T)≥0.
Then, limt→∞w(t)=η/h for the arbitrariness of ε, and an application of the comparison principle gives
(24)liminft→∞minΩ¯P(x,t)≥ηh.
The proof is complete.


## 2.2. The Local Stability of the Constant Steady State

In this subsection, we shall analyze the asymptotical stability of the positive constant solutionE* for model (6). Before developing our argument, let us set up the following notations.(i)
Let0=μ0<μ1<μ2<⋯→∞ be the eigenvalues of the operator –Δ on Ω with the zero-flux boundary condition;(ii)
LetE(μ)={ϕ∣-Δϕ=μϕinΩ,  ∂νϕ=0on∂Ω} with μ∈ℝ1;(iii)
Let{ϕij∣j=1,…,dimE(μi)} be an orthonormal basis of E(μi), and Xij={cϕij∣c∈ℝ2};(iv)
Let(25)X={(N,P)∈[H2(Ω)]2∣∂νN=∂νP=0on∂Ω},then(26)X=⨁i=1∞Xi,whereXi=⨁j=1dimE(μi)Xij.Theorem 4.
Assume that(27)m(ar+cm+2hr-acr)<r(a+h)2(r+s+cs)
and the first eigenvalues μ1 of the Dirichlet operator subject to zero-flux boundary conditions satisfy
(28)μ1>max{m(ar+cm+2hr-acr)-r2(a+h)2d1r(1+c)(a+h)2-sd2,0}.
Then the positive constant solution E* of model (6) is locally asymptotically stable.Proof.
Defineℒ:X→C(Ω¯)×C(Ω¯) by
(29)ℒ=(d1Δ+J1-mh2(a+h)2-shd2Δ-s),
where J1=(m(ar+cm+2hr-acr)-r2(a+h)2)/r(1+c)(a+h)2.
For eachi=0,1,2,…,Xi is invariant under the operator ℒ, and λ is an eigenvalue of this operator on Xi if and only if it is an eigenvalue of the following matrix:
(30)Ai=(-d1μi+J1-mh2(a+h)2-sh-d2μi-s).
Moreover,
(31)det(λI-Ai)=λ2-tr(Ai)λ+det(Ai),
where
(32)det(Ai)=d1d2μi2+(d1s-d2J1)μi+s(ar+hr-m)(ar+hr+cm)r(1+c)(a+h)2,tr(Ai)=-(d1+d2)μi+J1-s.
In view of (27) and (28), we have det(Ai)>0>tr(Ai) for any i≥0. Therefore, the eigenvalues of the matrix Ai have negative real parts.
In the following, we prove that there existsδ>0 such that
(33)Re{λi1}≤-δ,Re{λi2}≤-δ.
Letλ=μiξ, then
(34)φ~i(λ)≜μi2ξ2-tr(Ai)μiξ+det(Ai).
Since μi→∞ as i→∞, it follows that
(35)limi→∞φ~i(λ)μi2=ξ2+(d1+d2)ξ+d1d2.
By the Routh-Hurwitz criterion, it follows that the two rootsξ1, ξ2 of φ~i(λ)=0 all have negative real parts. Thus, let d~=min{d1,d2}, we have that Re{ξ1},Re{ξ2}≤-d~. By continuity, we see that there exists i0 such that the two roots ξi1, ξi2 of φ~i(λ)=0 satisfy Re{ξi1}≤-d~/2,  Re{ξi2}≤-d~/2, for all i≥i0. In turn, Re{λi1},Re{λi2}≤-μid~/2≤-d~/2, for all i≥i0.
Let-δ~=max1≤i≤i0{Re{λi1},Re{λi2}}, then δ~>0 and (33) hold for δ=min{δ~,d~/2}. Consequently, the spectrum of ℒ which consists of eigenvalues, lies in {Reλ≤-δ}. In the sense of [25], we obtain that the positive constant solution E*=(N*,P*) of model (6) is uniformly asymptotically stable. This ends the proof.


## 2.3. The Global Stability of the Constant Solution

This subsection is devoted to the global stability of the constant solutionE* for model (6).Theorem 5.
Assume that the following hold:(A1)
m(1+c)<ar;(A2)
h(K+cN*)(N*+2P*)+(K+cN*)(N*+aP*)(a+h)≤2ηr(N*+aP*)(a+h);(A3)
hKN*≤η(2h-1)(N*+aP*)(a+h),where η=K(ar-m(1+c))/ar. Then the constant solution E* is globally asymptotically stable.Proof.
In order to give the proof, we need to construct a Lyapunov function. Define(36)V(N,P)=∫N*N‍ξ-N*ξdξ+∫P*P‍ζ-P*ζdζ,E(t)=∫ΩV(N(x,t),P(x,t))dx.
We note that E(t) is nonnegative, E(t)=0 if and only if (N(x,t),P(x,t))=(N*,P*). Furthermore, by simple computations, it follows that
(37)dE(t)dt=∫Ω(VN(N(x,t),P(x,t))Nt+VP(N(x,t),P(x,t))Pt)dx=∫Ω(d1(N-N*)NΔN+d2(P-P*)PΔP)dx+∫Ω((N-N*)(r(K-N)K+cN-mPN+aP)+(P-P*)(1-hPN))dx=-∫Ω(d1N*N2|∇N|2+d2P*P2|∇P|2)dx+I(t),
where
(38)I(t)=∫Ω((N-N*)2(-rK(1+c)(K+cN*)(K+cN)+P*(N*+aP*)(N+aP)))dx+∫Ω(hN(N-N*)(P-P*)×(1N-N*(N*+aP*)(N+aP))-hN(P-P*)2)dx.
Setφ=N-N*,ϕ=P-P*. We have
(39)I(t)=-∫Ω(rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N)φ2dx-∫Ω(2h-12N-N*2(N*+aP*)(N+aP))ϕ2dx-∫Ω((φ+ϕ)2N*2(N*+aP*)(N+ap)12N(φ-ϕ)2+N*2(N*+aP*)(N+aP)×(φ+ϕ)2N*2(N*+aP*)(N+ap))dx≤-∫Ω(rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N)φ2dx-∫Ω(2h-12N-N*2(N*+aP*)(N+aP))ϕ2dx.
By virtue of Theorems1 and 3 and under the assumption of Theorem, we have
(40)rK(1+c)(K+cN*)(K+cN)-N*+2P*2(N*+aP*)(N+aP)-12N≥rK+cN*-h(N*+2P*)2η(N*+aP*)(a+h)-12η=(2ηr(N*+aP*)(a+h)-h(K+cN*)(N*+2P*)-(K+cN*)(N*+aP*)(a+h))×(2η(K+cN*)(N*+aP*)(a+h))-1≥0,2h-12N-N*2(N*+aP*)(N+aP)≥2h-12K-hN*2η(N*+aP*)(a+h)=η(2h-1)(N*+aP*)(a+h)-hKN*2Kη(N*+aP*)(a+h)≥0.
As a result, we haveI(t)≤0. Thus dE(t)/dt≤0, which implies the desired assertion. The proof is completed.


## 3. A Priori Estimates and Existence of Nonconstant Positive Solution

In this section, we will deduce a priori estimates of positive upper and lower bounds for positive solution of model (9). Then, based on a priori estimates, we discuss the existence of nonconstant positive solution of model (9) for certain parameter ranges.


### 3.1. A Priori Estimates

In order to obtain the desired bound, we recall the following two lemmas which are due to Lin et al. [26] and Lou and Ni [27], respectively.Lemma 6 (Harnack’s inequality [26]).
Assume thatc∈C(Ω¯) and let w∈C2(Ω)∩C1(Ω¯) be a positive solution to
(41)Δw(x)+c(x)w(x)=0inΩ,∂w∂ν=0on∂Ω.
Then there exists a positive constant C*=C*(∥c∥∞) such that
(42)maxΩ¯w≤C*minΩ¯w.Lemma 7 (maximum principle [27]).
LetΩ be a bounded Lipschitz domain in ℝ2 and g∈C(Ω¯×ℝ).(a)
Assume thatw∈C2(Ω)∩C1(Ω¯) and satisfies
(43)Δw(x)+g(x,w(x))≥0inΩ,∂w∂ν≤0on∂Ω.Ifw(x0)=maxΩ¯w(x), then g(x0,w(x0))≥0.(b)
Assume thatw∈C2(Ω)∩C1(Ω¯) and satisfies
(44)Δw(x)+g(x,w(x))≤0inΩ,∂w∂ν≥0on∂Ω.Ifw(x0)=minΩ¯w(x), then g(x0,w(x0))≤0.For convenience, let us denote the constantsa,  c,  h,  m,  K,  r,  s collectively by Λ. The positive constants C,  C_,  C¯, and so forth will depend only on the domain Ω and Λ. Now, we can state the main result which will play a critical role in Section 3.3.Theorem 8.
For any positive solution(N,P) of model (9),
(45)maxΩ¯N(x)≤K,maxΩ¯P(x)≤Kh.Proof.
Assume that(N,P) is a positive solution of model (9). Set
(46)N(x1)=maxΩ¯N(x).
Then, by Lemma 7, it follows from the first equation of (9) that
(47)rN(x1)(K-N(x1))K+cN(x1)≥mN(x1)P(x1)N(x1)+aP(x1)>0.
This clearly gives N(x1)<K.
Since0<P(x)≤(1/h)∥N(x)∥∞, we have P(x)≤K/h in Ω¯.Theorem 9.
Letd be a fix positive constant. Then there exists positive constant C_=C_(Λ,d) such that if d1,d2>d, any positive solution (N,P) of model (6) satisfies
(48)minΩ¯N(x)≥C_,minΩ¯P(x)≥C_.Proof.
Let(49)N(x0)=minΩ¯N(x),P(y0)=minΩ¯P(x),P(y1)=maxΩ¯P(x).
By Lemma7, it is clear that
(50)r(K-N(x0))K+cN(x0)-mP(x0)N(x0)+aP(x0)≤0,1-hP(y0)N(y0)≤0,1-hP(y1)N(y1)≥0.
So, we have
(51)1hN(x0)≤1hN(y0)≤P(y0),(52)P(y1)≤1hN(y1)≤1hmaxΩ¯N(x).
Sincem(K+cN(x0))/(N(x0)+aP(x0))≤C with C>0, then, by virtue of (52), we derive
(53)K-N(x0)≤mP(x0)(K+cN(x0))r(N(x0)+aP(x0))≤CrP(x0)≤CrP(y1)≤ChrmaxΩ¯N(x),
which implies that
(54)K≤minΩ¯N(x)+ChrmaxΩ¯N(x).
Define c(x)=d1-1(r(K-N)/(K+cN)-mP/(N+aP)), then N satisfies
(55)ΔN(x)+c(x)N(x)=0inΩ,∂N∂ν=0on∂Ω.
Therefore, we have
(56)maxΩ¯N(x)≤C*minΩ¯N(x),
herein a positive constant C*=C*(∥c∥∞). Hence, we obtain
(57)minΩ¯N(x)≥hKrhr+CC*.
It follows from (51) that
(58)minΩ¯P(x)≥Krhr+CC*.
The proof is completed.


### 3.2. Nonexistence of the Nonconstant Positive Solutions

Note thatμ1 is the smallest positive eigenvalues of the operator –Δ in Ω subject to the zero-flux boundary condition. Now, using the energy estimates, we can claim the following results.Theorem 10.
LetD>s/μ1 be a fixed positive constant. Then there exists a positive constant d*=d*(Λ,D) such that model (9) has no positive nonconstant solution provided that d1>d* and d2>D.Proof.
Let(N,P) be any positive solution of model (9) and denote g¯=|Ω|-1∫Ω‍gdx. Then
(59)∫Ω‍(N-N¯)dx=∫Ω‍(P-P¯)dx=0.
Then, multiplying the first equation of model (9) by (N-N¯), integrating over Ω, we have that
(60)d1∫Ω‍|∇(N-N¯)|2dx=∫Ω‍rN(K-N)(N-N¯)K+cNdx-∫Ω‍mNP(N-N¯)N+aPdx≤∫Ω‍rN(1-NK)(N-N¯)dx-∫Ω‍amP¯P(N-N¯)2(N+aP)(N¯+aP¯)dx-∫Ω‍mN¯N(N-N¯)(P-P¯)(N+aP)(N¯+aP¯)dx≤∫Ω‍(r(N-N¯)2+m|N-N¯||P-P¯|)dx.
In a similar manner, we multiply the second equation in model (9) by (P-P¯) to have
(61)d2∫Ω‍|∇(P-P¯)|2dx=∫Ω‍sP(1-hPN)(P-P¯)dx=∫Ω‍s(P-P¯)×(hP¯2(N-N¯)N¯NP-P¯-h(P+P¯)(P-P¯)N+hP¯2(N-N¯)N¯N)dx≤∫Ω‍s((P-P¯)2+hP¯2N¯N|N-N¯||P-P¯|)dx.
By theε-Young inequality and the Poincaré inequality, we obtain that
(62)∫Ω‍(d1|∇(N-N¯)|2+d2|∇(P-P¯)|2)dx≤∫Ω‍(r(N-N¯)2+2M(N-N¯)(P-P¯)+s(P-P¯)2)dx≤1μ1∫Ω‍(|∇(N-N¯)|2(r+Mϵ)+|∇(P-P¯)|2(s+ϵM)(r+Mε))dx
for some positive constant M and an arbitrary small positive constant ϵ.
In view ofd2>D>s/μ1, we can find a sufficiently small ϵ0>0 such that d2μ1≥s+ϵ0M. Let d*=(1/μ1)(r+M/ϵ), then
(63)∇(N-N¯)=∇(P-P¯)=0
and (N,P) must be a constant solution. This completes the proof.


### 3.3. Existence of the Nonconstant Positive Solutions

In this subsection, we shall discuss the existence of the positive nonconstant solution of model (9).Unless otherwise specified, in this subsection, we always require thatm<r(a+h) holds, which guarantees that model (9) has the unique positive constant solution E*=(N*,P*). From now on, we denote w=(N,P)T and w0=E*.LetX be the space defined in (25) and let
(64)X+={(N,P)∈X∣N,P>0onC(Ω¯)}.
We write model (9) in the following form:
(65)-Δw=𝒢(w),w∈X+,∂νw=0on∂Ω,
where
(66)𝒢(w)=(Nd1(r(K-N)K+cN-mPN+aP)sPd2(1-hPN)).Thenw is a positive solution of model (65) if and only if w satisfies
(67)ℱ(w)=w-(I-Δ)-1{𝒢(w+w)}=0,inX+,
where (I-Δ)-1 is the inverse operator of I-Δ subject to the zero-flux boundary condition. Then
(68)∇ℱ(w0)=I-(I-Δ)-1(I+𝒜),
where
(69)𝒜≜∇𝒢(w0)=(J1d1-mh2d1(a+h)2-sd2h-sd2).If∇ℱ(w0) is invertible, by Theorem 2.8.1 of [28], the index of ℱ at w0 is given by
(70)index(ℱ,w0)=(-1)γ,
where γ is the multiplicity of negative eigenvalues of ∇ℱ(w0).On the other hand, using the decomposition (26), we have that Xi is an invariant space under ∇ℱ(w0) and ξ∈ℝ is an eigenvalue of ∇ℱ(w0) in Xi, if and only if, ξ is an eigenvalue of (μi+1)-1(μiI-𝒜). Therefore, ∇ℱ(w0) is invertible, if and only if, for any i≥0 the matrix μiI-𝒜 is invertible.Letm(μi) be the multiplicity of μi. For the sake of convenience, we denote
(71)H(μ)=det(μI-𝒜).
Then, if μiI-𝒜 is invertible for any i≥0, with the same arguments as in [29], we can assert the following conclusion.Lemma 11.
Assume that, for alli≥0, the matrix μiI-𝒜 is nonsingular, then
(72)index(ℱ,w0)=(-1)γ,whereγ=∑i≥0,H(μi)<0‍m(μi).To computeindex(ℱ,w0), we have to consider the sign of H(μ). A straightforward computation yields
(73)H(μ)=μ2-θ1μ+θ2,
where θ1=(d2J1-d1s)/d1d2,  θ2=s(ar+hr-m)(ar+hr+cm)/d1d2r(1+c)(a+h)2.Ifθ12-4θ2>0, then H(μ)=0 has two positive solutions μ± given by
(74)μ±=12(θ1±θ12-4θ2).Theorem 12.
Assume thatm(ar+cm+2hr-acr)>r2(a+h)2 and θ12-4θ2>0. If μ-∈(μi,μi+1) and μ+∈(μj,μj+1) for some 0≤i<j, and ∑k=i+1j‍m(μk) is odd, then model (9) has at least one nonconstant solution.Proof.
By Theorem10, we can fix d¯1>d1 and d¯2>d2 such that model (9) with diffusion coefficients d¯1 and d¯2 has no nonconstant solutions.
By virtue of Theorems8 and 9, there exists a positive constant C_,  C¯ such that C_<N,  P<C¯.
Set(75)ℳ={(N,P)∈C(Ω¯)×C(Ω¯):C_<N,P<C¯inΩ¯},
and define
(76)Ψ:ℳ×[0,1]⟶C(Ω¯)×C(Ω¯)
by
(77)Ψ(w,t)=(I-Δ)-1{G(w,t)+w},
where
(78)G(w,t)=((td1+(1-t)d¯1)-1(rN(K-N)K+cN-mNPN+aP)(td2+(1-t)d¯2)-1(sP(1-hPN))).
It is clear that finding the positive solution of model (9) becomes equivalent to finding the positive solution of Ψ(w,1)=0 in ℳ. Further, by virtue of the definition of ℳ, we have that Ψ(w,t)=0 has no positive solution in ∂ℳ for all 0≤t≤1.
SinceΨ(w,t) is compact, the Leray-Schauder topological degree deg(I-Ψ(w,t),ℳ,0) is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce
(79)deg(I-Ψ(w,1),ℳ,0)=deg(I-Ψ(w,0),ℳ,0).
Clearly,I-Ψ(w,1)=ℱ. Thus, if model (9) has no other solutions except the constant one w0, then Lemma 11 shows that
(80)deg(I-Ψ(w,1),ℳ,0)=index(ℱ,w0)=(-1)∑k=i+1jm(μk)=-1.
On the contrary, by the choice ofd¯1 and d¯2, we have that w0 is the only solution of Ψ(w,0)=0. Furthermore, we have
(81)deg(I-Ψ(w,0),ℳ,0)=index(I-Ψ(w,0),w0)=1.
From (79)–(81), we get a contradiction, and the proof is completed.


## 3.1. A Priori Estimates

In order to obtain the desired bound, we recall the following two lemmas which are due to Lin et al. [26] and Lou and Ni [27], respectively.Lemma 6 (Harnack’s inequality [26]).
Assume thatc∈C(Ω¯) and let w∈C2(Ω)∩C1(Ω¯) be a positive solution to
(41)Δw(x)+c(x)w(x)=0inΩ,∂w∂ν=0on∂Ω.
Then there exists a positive constant C*=C*(∥c∥∞) such that
(42)maxΩ¯w≤C*minΩ¯w.Lemma 7 (maximum principle [27]).
LetΩ be a bounded Lipschitz domain in ℝ2 and g∈C(Ω¯×ℝ).(a)
Assume thatw∈C2(Ω)∩C1(Ω¯) and satisfies
(43)Δw(x)+g(x,w(x))≥0inΩ,∂w∂ν≤0on∂Ω.Ifw(x0)=maxΩ¯w(x), then g(x0,w(x0))≥0.(b)
Assume thatw∈C2(Ω)∩C1(Ω¯) and satisfies
(44)Δw(x)+g(x,w(x))≤0inΩ,∂w∂ν≥0on∂Ω.Ifw(x0)=minΩ¯w(x), then g(x0,w(x0))≤0.For convenience, let us denote the constantsa,  c,  h,  m,  K,  r,  s collectively by Λ. The positive constants C,  C_,  C¯, and so forth will depend only on the domain Ω and Λ. Now, we can state the main result which will play a critical role in Section 3.3.Theorem 8.
For any positive solution(N,P) of model (9),
(45)maxΩ¯N(x)≤K,maxΩ¯P(x)≤Kh.Proof.
Assume that(N,P) is a positive solution of model (9). Set
(46)N(x1)=maxΩ¯N(x).
Then, by Lemma 7, it follows from the first equation of (9) that
(47)rN(x1)(K-N(x1))K+cN(x1)≥mN(x1)P(x1)N(x1)+aP(x1)>0.
This clearly gives N(x1)<K.
Since0<P(x)≤(1/h)∥N(x)∥∞, we have P(x)≤K/h in Ω¯.Theorem 9.
Letd be a fix positive constant. Then there exists positive constant C_=C_(Λ,d) such that if d1,d2>d, any positive solution (N,P) of model (6) satisfies
(48)minΩ¯N(x)≥C_,minΩ¯P(x)≥C_.Proof.
Let(49)N(x0)=minΩ¯N(x),P(y0)=minΩ¯P(x),P(y1)=maxΩ¯P(x).
By Lemma7, it is clear that
(50)r(K-N(x0))K+cN(x0)-mP(x0)N(x0)+aP(x0)≤0,1-hP(y0)N(y0)≤0,1-hP(y1)N(y1)≥0.
So, we have
(51)1hN(x0)≤1hN(y0)≤P(y0),(52)P(y1)≤1hN(y1)≤1hmaxΩ¯N(x).
Sincem(K+cN(x0))/(N(x0)+aP(x0))≤C with C>0, then, by virtue of (52), we derive
(53)K-N(x0)≤mP(x0)(K+cN(x0))r(N(x0)+aP(x0))≤CrP(x0)≤CrP(y1)≤ChrmaxΩ¯N(x),
which implies that
(54)K≤minΩ¯N(x)+ChrmaxΩ¯N(x).
Define c(x)=d1-1(r(K-N)/(K+cN)-mP/(N+aP)), then N satisfies
(55)ΔN(x)+c(x)N(x)=0inΩ,∂N∂ν=0on∂Ω.
Therefore, we have
(56)maxΩ¯N(x)≤C*minΩ¯N(x),
herein a positive constant C*=C*(∥c∥∞). Hence, we obtain
(57)minΩ¯N(x)≥hKrhr+CC*.
It follows from (51) that
(58)minΩ¯P(x)≥Krhr+CC*.
The proof is completed.


## 3.2. Nonexistence of the Nonconstant Positive Solutions

Note thatμ1 is the smallest positive eigenvalues of the operator –Δ in Ω subject to the zero-flux boundary condition. Now, using the energy estimates, we can claim the following results.Theorem 10.
LetD>s/μ1 be a fixed positive constant. Then there exists a positive constant d*=d*(Λ,D) such that model (9) has no positive nonconstant solution provided that d1>d* and d2>D.Proof.
Let(N,P) be any positive solution of model (9) and denote g¯=|Ω|-1∫Ω‍gdx. Then
(59)∫Ω‍(N-N¯)dx=∫Ω‍(P-P¯)dx=0.
Then, multiplying the first equation of model (9) by (N-N¯), integrating over Ω, we have that
(60)d1∫Ω‍|∇(N-N¯)|2dx=∫Ω‍rN(K-N)(N-N¯)K+cNdx-∫Ω‍mNP(N-N¯)N+aPdx≤∫Ω‍rN(1-NK)(N-N¯)dx-∫Ω‍amP¯P(N-N¯)2(N+aP)(N¯+aP¯)dx-∫Ω‍mN¯N(N-N¯)(P-P¯)(N+aP)(N¯+aP¯)dx≤∫Ω‍(r(N-N¯)2+m|N-N¯||P-P¯|)dx.
In a similar manner, we multiply the second equation in model (9) by (P-P¯) to have
(61)d2∫Ω‍|∇(P-P¯)|2dx=∫Ω‍sP(1-hPN)(P-P¯)dx=∫Ω‍s(P-P¯)×(hP¯2(N-N¯)N¯NP-P¯-h(P+P¯)(P-P¯)N+hP¯2(N-N¯)N¯N)dx≤∫Ω‍s((P-P¯)2+hP¯2N¯N|N-N¯||P-P¯|)dx.
By theε-Young inequality and the Poincaré inequality, we obtain that
(62)∫Ω‍(d1|∇(N-N¯)|2+d2|∇(P-P¯)|2)dx≤∫Ω‍(r(N-N¯)2+2M(N-N¯)(P-P¯)+s(P-P¯)2)dx≤1μ1∫Ω‍(|∇(N-N¯)|2(r+Mϵ)+|∇(P-P¯)|2(s+ϵM)(r+Mε))dx
for some positive constant M and an arbitrary small positive constant ϵ.
In view ofd2>D>s/μ1, we can find a sufficiently small ϵ0>0 such that d2μ1≥s+ϵ0M. Let d*=(1/μ1)(r+M/ϵ), then
(63)∇(N-N¯)=∇(P-P¯)=0
and (N,P) must be a constant solution. This completes the proof.


## 3.3. Existence of the Nonconstant Positive Solutions

In this subsection, we shall discuss the existence of the positive nonconstant solution of model (9).Unless otherwise specified, in this subsection, we always require thatm<r(a+h) holds, which guarantees that model (9) has the unique positive constant solution E*=(N*,P*). From now on, we denote w=(N,P)T and w0=E*.LetX be the space defined in (25) and let
(64)X+={(N,P)∈X∣N,P>0onC(Ω¯)}.
We write model (9) in the following form:
(65)-Δw=𝒢(w),w∈X+,∂νw=0on∂Ω,
where
(66)𝒢(w)=(Nd1(r(K-N)K+cN-mPN+aP)sPd2(1-hPN)).Thenw is a positive solution of model (65) if and only if w satisfies
(67)ℱ(w)=w-(I-Δ)-1{𝒢(w+w)}=0,inX+,
where (I-Δ)-1 is the inverse operator of I-Δ subject to the zero-flux boundary condition. Then
(68)∇ℱ(w0)=I-(I-Δ)-1(I+𝒜),
where
(69)𝒜≜∇𝒢(w0)=(J1d1-mh2d1(a+h)2-sd2h-sd2).If∇ℱ(w0) is invertible, by Theorem 2.8.1 of [28], the index of ℱ at w0 is given by
(70)index(ℱ,w0)=(-1)γ,
where γ is the multiplicity of negative eigenvalues of ∇ℱ(w0).On the other hand, using the decomposition (26), we have that Xi is an invariant space under ∇ℱ(w0) and ξ∈ℝ is an eigenvalue of ∇ℱ(w0) in Xi, if and only if, ξ is an eigenvalue of (μi+1)-1(μiI-𝒜). Therefore, ∇ℱ(w0) is invertible, if and only if, for any i≥0 the matrix μiI-𝒜 is invertible.Letm(μi) be the multiplicity of μi. For the sake of convenience, we denote
(71)H(μ)=det(μI-𝒜).
Then, if μiI-𝒜 is invertible for any i≥0, with the same arguments as in [29], we can assert the following conclusion.Lemma 11.
Assume that, for alli≥0, the matrix μiI-𝒜 is nonsingular, then
(72)index(ℱ,w0)=(-1)γ,whereγ=∑i≥0,H(μi)<0‍m(μi).To computeindex(ℱ,w0), we have to consider the sign of H(μ). A straightforward computation yields
(73)H(μ)=μ2-θ1μ+θ2,
where θ1=(d2J1-d1s)/d1d2,  θ2=s(ar+hr-m)(ar+hr+cm)/d1d2r(1+c)(a+h)2.Ifθ12-4θ2>0, then H(μ)=0 has two positive solutions μ± given by
(74)μ±=12(θ1±θ12-4θ2).Theorem 12.
Assume thatm(ar+cm+2hr-acr)>r2(a+h)2 and θ12-4θ2>0. If μ-∈(μi,μi+1) and μ+∈(μj,μj+1) for some 0≤i<j, and ∑k=i+1j‍m(μk) is odd, then model (9) has at least one nonconstant solution.Proof.
By Theorem10, we can fix d¯1>d1 and d¯2>d2 such that model (9) with diffusion coefficients d¯1 and d¯2 has no nonconstant solutions.
By virtue of Theorems8 and 9, there exists a positive constant C_,  C¯ such that C_<N,  P<C¯.
Set(75)ℳ={(N,P)∈C(Ω¯)×C(Ω¯):C_<N,P<C¯inΩ¯},
and define
(76)Ψ:ℳ×[0,1]⟶C(Ω¯)×C(Ω¯)
by
(77)Ψ(w,t)=(I-Δ)-1{G(w,t)+w},
where
(78)G(w,t)=((td1+(1-t)d¯1)-1(rN(K-N)K+cN-mNPN+aP)(td2+(1-t)d¯2)-1(sP(1-hPN))).
It is clear that finding the positive solution of model (9) becomes equivalent to finding the positive solution of Ψ(w,1)=0 in ℳ. Further, by virtue of the definition of ℳ, we have that Ψ(w,t)=0 has no positive solution in ∂ℳ for all 0≤t≤1.
SinceΨ(w,t) is compact, the Leray-Schauder topological degree deg(I-Ψ(w,t),ℳ,0) is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce
(79)deg(I-Ψ(w,1),ℳ,0)=deg(I-Ψ(w,0),ℳ,0).
Clearly,I-Ψ(w,1)=ℱ. Thus, if model (9) has no other solutions except the constant one w0, then Lemma 11 shows that
(80)deg(I-Ψ(w,1),ℳ,0)=index(ℱ,w0)=(-1)∑k=i+1jm(μk)=-1.
On the contrary, by the choice ofd¯1 and d¯2, we have that w0 is the only solution of Ψ(w,0)=0. Furthermore, we have
(81)deg(I-Ψ(w,0),ℳ,0)=index(I-Ψ(w,0),w0)=1.
From (79)–(81), we get a contradiction, and the proof is completed.



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*Source: 267173-2013-03-25.xml*