Citation: Xiaoling Li, Guangping Hu, Xianpei Li, Zhaosheng Feng. Positive steady states of a ratio-dependent predator-prey system with cross-diffusion[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6753-6768. doi: 10.3934/mbe.2019337
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The dynamic relationship between prey and predator has been one of the most interesting topics in the qualitative theory of population dynamics [1,2]. For example, Bazykin [2] considered a prey-predator system with the bounded functional response:
{dudt=u−ku2−uv1+mu,dvdt=−av−bv2+uv1+mu, | (1.1) |
where u and v represent the densities of the prey and predator, respectively. Here, a, b, k and m are positive constants. The predator consumes the prey based on the prey-dependent functional response u/(1+mu), and the term −bv2 represents the self-limitation for the predator. The above model takes account of not only the interspecies interactions between the prey species and predators, but also the density dependence of predator species.
Recently, there is much evidence from biology and physiology to show that in many situations, especially when predators have to search, share or compete for food, the modeling and analysis of predator-prey systems for each specific case should be further developed according to the ratio-dependent theory [3,4,5], which can be roughly stated as that the per capita predator growth rate should depend on the ratio of prey to predators, but not just prey numbers, so it should be the ratio-dependent function response [6,7,8,9], i.e., a function of u/v. On the basis of this consideration, substituting u/v1+mu/v for u1+mu in the above-mentioned model (1.1), we arrive at the following system:
{dudt=u−ku2−uvmu+v,dvdt=−av−bv2+uvmu+v. | (1.2) |
If ma<1, system (1.2) has a unique positive equilibrium ˜w=(˜u,˜v)T, where
˜u=m(a+b˜v)k, ˜v=m(1−ma)k+m2b. | (1.3) |
Taking into account the inhomogeneous distribution of predator and prey populations in different spatial locations, in this study we will incorporate diffusion and cross-diffusion to system (1.2) and consider a more general system:
{ut=d1Δu+d12Δv+u−ku2−uvmu+v, x∈Ω, t>0,vt=d21Δu+d2Δv−av−bv2+uvmu+v, x∈Ω, t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω, t>0,u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0,x∈Ω, | (1.4) |
where Ω is a fixed bounded domain with the sufficient smooth boundary ∂Ω in RN, ν is the outward unit normal vector of the boundary ∂Ω, and Δ is the Laplace operator. The positive constants d1 and d2 are the random diffusion coefficients of prey and predator, respectively, which describe the natural dispersive forces of random movement of individuals. Here, d12 and d21 are the cross-diffusion coefficients that express population fluxes of preys and predators due to the presence of other species [10,11,12]. The non-flux boundary condition indicates that the system is self-contained with no external energy exchange. For more related studies on dynamics of population models with diffusion and cross-diffusion, we refer the reader to [12,13,14,15,16,17] and references therein.
One of our motivations of this study lies in a fact that, in the past few decades there have been continuous interests in the existence of positive steady states of diffusive predator-prey systems, and the majority of works have devoted to discovering the effect of diffusion on positive steady states [18,19,20,21,22,23,24]. However, little attention has been paid to the case that both diffusion and cross-diffusion are present in population systems [25,26,27,28]. Therefore, the main purpose of this paper is to make an attempt on the existence and non-existence of non-constant positive steady states of system (1.4). The corresponding steady-state equations of system (1.4) are:
{−d1Δu−d12Δv=u−ku2−uvmu+v,−d21Δu−d2Δv=−av−bv2+uvmu+v,∂u∂ν=∂v∂ν=0,x∈Ω,x∈Ω,x∈∂Ω. | (1.5) |
This paper is organized as follows. In Section 2, we discuss a priori estimate of the positive steady state for system (1.4), that is, a priori upper and lower bounds for positive solutions of the elliptic system (1.5). In Section 3, we investigate the existence and non-existence of non-constant positive solutions of system (1.5). In particular, the impact of diffusion and cross-diffusion on the existence of non-constant positive steady states is explored.
In this section, to establish a priori estimates of upper and lower bounds of positive solutions for system (1.5) in a straightforward manner, we need the following two lemmas.
Lemma 2.1. [22,Maximun Principle] Assume that g∈C(¯Ω×R1).
(i) If ω∈C2(Ω)∩C1(¯Ω) satisfies Δω(x)+g(x,ω(x))≥0, x∈Ω;∂ω∂ν≤0, x∈∂Ω; and ω(x0)=max¯Ωω, then g(x0,ω(x0))≥0.
(ii) If ω∈C2(Ω)∩C1(¯Ω) satisfies Δω(x)+g(x,ω(x))≤0, x∈Ω;∂ω∂ν≥0, x∈∂Ω; and ω(x0)=min¯Ωω, then g(x0,ω(x0))≤0.
Lemma 2.2. [27,Harnack's Inequality] Suppose that c(x)∈C(¯Ω). Let ω(x)∈C2(Ω)∩C1(¯Ω) be a positive solution to
Δω(x)+c(x)ω(x)=0,x∈Ω;∂ω∂ν=0,x∈∂Ω. |
Then there exists a positive constant C∗=C∗(‖c(x)‖∞,Ω) such that
max¯Ωω≤C∗min¯Ωω. |
In the following, the positive constants C1, C2, C∗1, C∗2, c1 and c2 will rely upon the domain Ω. However, while Ω is fixed, we will not mention this dependence explicitly. Also, for convenience, we shall write Λ instead of the set of constants (a,b,m,k) in the sequel.
Theorem 2.1. [Upper Bounds] Suppose that d12d1≤D and d21d2≤D for an arbitrary fixed number D. Then there exist positive constants Ci=Ci(D,Λ), i=1,2, such that the positive solution (u,v) of system (1.5) satisfies
max¯Ωu≤C1 and max¯Ωv≤C2. | (2.1) |
Proof. Set ϕ=d1u+d12v and ψ=d21u+d2v. Then the problem (1.5) becomes
{−Δϕ=1−ku−vmu+vd1+d12vuϕ,−Δψ=−a−bv+umu+vd21uv+d2ψ,∂ϕ∂ν=∂ψ∂ν=0,x∈Ω,x∈Ω,x∈∂Ω. | (2.2) |
Let x1∈¯Ω be a point satisfying ϕ(x1)=max¯Ωϕ. Applying Lemma 2.1 to the first equation of (2.2), we can obtain u(x1)≤1k and v(x1)≤mk(1−ku(x1))≤mk. Hence, we obtain
max¯Ωu≤1d1max¯Ωϕ=1d1(d1u(x1)+d12v(x1))≤1k+d12d1⋅mk≜C1. | (2.3) |
Analogously, let x2∈¯Ω be a point satisfying ψ(x2)=max¯Ωψ. Using Lemma 2.1 to the second equation of (2.2), we have v(x2)≤1−mabm. Then we have
max¯Ωv≤1d2max¯Ωψ=1d2(d21u(x2)+d2v(x2))≤d21C1d2+1−mabm≜C2. | (2.4) |
Before presenting the lower bound, we present the following lemma.
Lemma 2.3. Let ma<1, d1,n,d2,n,d12,n,d21,n be positive constants, n=1,2,..., and (un,vn) be the positive solution of system (1.5) with d1=d1,n,d12=d12,n,d21=d21,n and d2=d2,n. Suppose that (d1,n,d2,n,d12,n,d21,n)→(d1,d2,d12,d21), and (un,vn)→(u∗,v∗) uniformly on ¯Ω. If u∗ and v∗ are positive constants, then (u∗,v∗) satisfies
1−ku∗−v∗mu∗+v∗=0, −a−bv∗+u∗mu∗+v∗=0. |
That is, (u∗,v∗)=(˜u,˜v), is the unique positive constant solution of system (1.5).
Proof. It is clear that ∫Ωun(1−kun−vnmun+vn)dx=0 holds for all n. If 1−ku∗−v∗mu∗+v∗>0, then 1−kun−vnmun+vn>0 as n is getting large. However, un is positive, so this is impossible. Similarly, 1−ku∗−v∗mu∗+v∗<0 is impossible either. Hence, 1−ku∗−v∗mu∗+v∗=0. Using the same argument leads to −a−bv∗+u∗mu∗+v∗=0. Consequently, (u∗,v∗)=(˜u,˜v).
Theorem 2.2. (Lower Bounds). Assume that d and D are fixed positive constants. Then there exist positive constants ci=ci(d,D,Λ), i=1,2, such that when d1,d2≥d and d12d1, d21d2≤D, the positive solution (u,v) of system (1.4) satisfies
min¯Ωu≥c1 and min¯Ωv≥c2. | (2.5) |
Proof. By a straightforward calculation, we have
‖u(1−ku−vmu+v)d1u+d12v‖∞≤1d1(1+kmax¯Ωu+max¯Ω{1muv+1})=1d1(1+kmax¯Ωu+1min¯Ω(muv+1))≤1d1(2+kmax¯Ωu)≤1d1(2+kC1), |
and
‖v(−a−bv+umu+v)d21u+d2v‖∞≤1d2(a+bmax¯Ωv+max¯Ω{1m+vu})=1d2(a+bmax¯Ωv+1min¯Ω(m+vu))≤1d2(a+bmax¯Ωv+1m)≤1d2(a+1m+bC2). |
Then, Lemma 2.2 indicates that there exist positive constants C∗1=C∗1(d,Λ) and C∗2=C∗2(d,Λ) such that max¯Ωϕ≤C∗1min¯Ωϕ and max¯Ωψ≤C∗2min¯Ωψ. Thus we obtain
max¯Ωumin¯Ωu≤max¯Ωϕmin¯Ω(d1+d12vu)/min¯Ωϕmax¯Ω(d1+d12vu)=max¯Ωϕmin¯Ωϕ⋅max¯Ω(d1+d12vu)min¯Ω(d1+d12vu)≤C∗1, | (2.6) |
and
max¯Ωvmin¯Ωv≤max¯Ωψmin¯Ω(d21uv+d2)/min¯Ωψmax¯Ω(d21uv+d2)=max¯Ωψmin¯Ωψ⋅max¯Ω(d21uv+d2)min¯Ω(d21uv+d2)≤C∗2. | (2.7) |
Now, we estimate the positive lower bounds of u and v. Suppose that (2.5) is not true. Then there exists a sequence {d1,n,d2,n,d12,n,d21,n}∞n=1 with (d1,n,d2,n,d12,n,d21,n)∈[d,∞)×[d,∞)×(0,∞)×(0,∞) such that the positive solution (un,vn) of system (1.5) satisfies
{−Δ(d1,nun+d12,nvn)=un−ku2n−unvnmun+vn,−Δ(d21,nun+d2,nvn)=−avn−bv2n+unvnmun+vn,∂νun=∂νvn=0,x∈Ω,x∈Ω,x∈∂Ω, | (2.8) |
and
min¯Ωun→0 or min¯Ωvn→0, as n→∞. | (2.9) |
For (2.8), it follows from the standard regularity theorem of elliptic equations [29] that there exists a subsequence of {(un,vn)}∞n=1, still denoted by {(un,vn)}∞n=1, and two nonnegative functions u,v∈C2(¯Ω) satisfying (un,vn)→(u,v) as n→∞. Assume that (d1,n,d2,n,d12,n,d21,n)→(d1,d2,d12,d21)∈[d,∞)×[d,∞)×(0,∞)×(0,∞). In view of inequalities (2.6) and (2.7), it is easy to see that either max¯Ωun→0 or max¯Ωvn→0, as n→∞.
So, there are three possible cases.
If max¯Ωun→0 and max¯Ωvn→0, then (un,vn)→(0,0) uniformly on ¯Ω, which is a contradiction to Lemma 2.3.
If max¯Ωun→0 and max¯Ωvn→¯v, where ¯v is a positive constant. By integrating the second equation of system (2.8) in Ω, it gives
∫Ωvn(a+bvn)dx=0. |
Then we get vn→−ab, which yields a contradiction to ¯v>0. Hence, min¯Ωu>0.
Similarly, if max¯Ωun→¯u and max¯Ωvn→0, where ¯u is a positive constant. By integrating the first equation of system (2.8) in Ω, it gives
∫Ωun(1−kun)dx=0. |
Then ¯u=1k. It implies that (un,vn)→(1k,0) uniformly on ¯Ω. Apparently, this leads to a contradiction to Lemma 2.3. Consequently, min¯Ωv>0.
In this section, we study the non-existence and existence of non-constant positive steady states of system (1.5).
To consider the non-existence of non-constant solutions of system (1.5) in this subsection, for convenience, we denote
f(u,v)=u−ku2−uvmu+v, g(u,v)=−av−bv2+uvmu+v. |
Theorem 3.1. Let d∗2 be a fixed positive constant and satisfy μ1d∗2>1m, where μ1 is the least positive eigenvalue of −Δ on Ω under the Neumann boundary condition. Then there exists a positive constant D1=D1(Λ,d∗2) such that when d1≥D1 and d2≥d∗2, system (1.5) with d12=d21=0 has no non-constant positive solution.
Proof. Suppose that w=(u,v)T is a positive solution of system (1.5). Let ˜φ=1|Ω|∫Ωφdx for any φ∈L1(Ω). Multiplying the first two equations of (1.5) by u−˜u and v−˜v, respectively, and then integrating on Ω, by integration by parts we obtain
∫Ωd1|∇u|2dx+∫Ωd2|∇v|2dx=∫Ωf(u,v)(u−˜u)dx+∫Ωg(u,v)(v−˜v)dx=∫Ω[f(u,v)−f(˜u,˜v)](u−˜u)dx+∫Ω[g(u,v)−g(˜u,˜v)](v−˜v)dx=∫Ω{1−k(u+˜u)−v˜v(mu+v)(m˜u+˜v)}(u−˜u)2+∫Ω{−mu˜u(mu+v)(m˜u+˜v)+v˜v(mu+v)(m˜u+˜v)}(u−˜u)(v−˜v)dx+∫Ω{−a−b(v+˜v)+mu˜u(mu+v)(m˜u+˜v)}(v−˜v)2dx≤∫Ω{(u−˜u)2+mu˜u+v˜v(mu+v)(m˜u+˜v)|u−˜u||v−˜v|+1m(v−˜v)2}dx. | (3.1) |
In view of Theorems 2.1 and 2.2, using Young's inequality to (3.1) we find
∫Ωd1|∇u|2dx+∫Ωd2|∇v|2dx≤∫Ω{(u−˜u)2+2L|u−˜u||v−˜v|+1m(v−˜v)2}dx≤∫Ω{(1+Lε)(u−˜u)2+(1m+εL)(v−˜v)2}dx | (3.2) |
for some positive constant L, where ε is the arbitrary small positive constant arising from Young's inequality.
Using Poincaré's inequality, we see that μ1∫Ω(u−˜u)2dx≤∫Ω|∇u|2dx and μ1∫Ω(v−˜v)2dx≤∫Ω|∇v|2dx, where μ1 is the least positive eigenvalue of −Δ on Ω under the Neumann boundary condition. It follows from inequality (3.2) that
μ1∫Ω[d1(u−˜u)2+d2(v−˜v)2]dx≤∫Ω{(1+Lε)(u−˜u)2+(1m+εL)(v−˜v)2}dx. |
Choosing a sufficiently small ε0>0 such that d∗2μ1≥1m+ε0L, and taking D1≜1μ1(1+Lε0), we arrive at the desired result (u,v)=(˜u,˜v).
From the discussion in the preceding subsection, we know that when the cross-diffusion terms are absent, there might be no non-constant positive solutions for system (1.5). In this subsection, we shall discuss the existence of non-constant positive solutions of system (1.5) with respect to cross-diffusion coefficients d21 and d12 as the other parameters d1 and d2 are fixed by means of the Leray-Schauder degree theory.
To facilitate the discussion, we rewrite system (1.5) as
\begin{eqnarray} \left\{ \begin{array}{l} -\Delta \mathit{\boldsymbol{\Phi}}(\textbf{w}) = \textbf{F}(\textbf{w}), \\ \frac{\partial \textbf{w}}{\partial \nu} = \textbf{0}, \end{array}\; \; \begin{array}{l} x\in \Omega, \\[1.5mm] x\in \partial\Omega, \end{array} \right. \end{eqnarray} | (3.3) |
where \textbf{w} = (u, v)^\mathit{T} , \mathit{\boldsymbol{\Phi}}(\textbf{w}) = (\phi, \psi)^\mathit{T} , and \textbf{F}(\textbf{w}) = (f(u, v), \, g(u, v))^\mathit{T} .
Let \textbf{X} = \left\{\textbf{w}\in \left[C^1(\overline{\Omega})\right]^2 \Big|\frac{\partial\textbf{w}}{\partial\nu} = \textbf{0} \ \text{on}\ \partial\Omega \right\} , and define
\begin{eqnarray*} &&\textbf{X}^+ = \{\textbf{w}\in \textbf{X} |\textbf{w} \gt 0 \; \; on\; \; \overline{\Omega}\}, \\ &&\mathcal{B}(c) = \{\textbf{w}\in \textbf{X}|c^{-1} \lt u, v \lt c \; \; on\; \; \overline{\Omega}\}, \end{eqnarray*} |
where c is a positive constant that is guaranteed to exist by Theorems 2.1 and 2.2.
Assume that
\begin{eqnarray} d_1d_2-d_{12}d_{21}\neq0. \end{eqnarray} | (3.4) |
Since \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right) , the determinant \det \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) is nonzero for all non-negative \textbf{w} , \mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w}) exists and \det\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w}) is of the same sign as \det\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) . Thus, \textbf{w} is a positive solution of system (3.3) if and only if
\begin{eqnarray} \textbf{G}(\textbf{w})\triangleq \textbf{w}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w})[\textbf{F}(\textbf{w}) +\nabla\textbf{w}\mathit{\boldsymbol{\Phi}}_{\textbf{ww}}(\textbf{w})\nabla\textbf{w}] +\textbf{w}\big\} = 0, \; \; \textbf{w}\in\textbf{X}^+, \end{eqnarray} | (3.5) |
where (\textbf{I}-\Delta)^{-1} is the inverse of \textbf{I}-\Delta in \textbf{X} under the homogeneous Neumann boundary condition. Since \textbf{G}(\cdot) is a compact perturbation of the identity operator, for any \mathcal{B} = \mathcal{B}(c) , the Leray-Schauder degree of \deg(\textbf{G}(\cdot), \textbf{0}, \mathcal{B}) is well-defined if \textbf{G}(\textbf{w})\neq 0 on \partial\mathcal{B} .
Note that
\begin{eqnarray*} D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}, \end{eqnarray*} |
where
\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right), \; \; \ \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} -\frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2} & -\frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\\ \frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2} & -\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}-b\tilde{v} \end{matrix}\right). |
We recall that if D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) is invertible, the index of \textbf{G} at \widetilde{\textbf{w}} is defined by
{\rm index}(\textbf{G}(\cdot), \widetilde{\textbf{w}}) = (-1)^{\gamma}, |
where \gamma is the total number of eigenvalues of D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) with negative real parts (counting multiplicities), then the degree \deg(\textbf{G}(\cdot), \textbf{0}, \mathcal{B}) is equal to the sum of the indices over all isolated solutions when \textbf{G} = \textbf{0} in \mathcal{B}(c) , provided that \textbf{G}\neq \textbf{0} on \partial\mathcal{B} .
Let 0 = \mu_{0} < \mu_{1} < \mu_{2} < \cdots be the eigenvalues of the operator -\Delta on \Omega under the homogeneous Neumann boundary condition and E(\mu_{i}) be the eigenspaces with respect to \mu_{i} . Let \{\phi_{ij}; j = 1, 2, \cdots, \dim E(\mu_{i})\} be a set of orthonormal basis of E(\mu_{i}) and \textbf{X}_{ij} = \{\textbf{c}\phi_{ij}|\textbf{c}\in \mathbb{R}^2\} .
Denote
\textbf{X} = \left\{\textbf{w}\in \left[C^1(\overline{\Omega})\right]^2 \Big|\frac{\partial\textbf{w}}{\partial\nu} = \textbf{0} \ \text{on}\ \partial\Omega \right\} \ \, \text{and}\ \, \textbf{X}_{ij} = \{\textbf{c}\phi_{ij}\ |\, \textbf{c}\in \mathbb{R}^2\}. |
Then
\begin{eqnarray*} \textbf{X} = \bigoplus\limits_{i = 1}^{\infty}\textbf{X}_{i} \; \; \; and\; \; \; \textbf{X}_{i} = \bigoplus \limits_{j = 1}^{\dim E(\mu_{i})}\textbf{X}_{ij}. \end{eqnarray*} |
We refer to the decomposition above in the following discussions of the eigenvalues of D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) . We know that \textbf{X}_{ij} is invariant under D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) for each i\in \mathbb{N} and each j\in [1, \dim E(\mu_{i})]\cap\mathbb{N} , i.e., D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}})\textbf{w}\in \textbf{X}_{ij} for any \textbf{w}\in \textbf{X}_{ij} . Thus, \lambda is an eigenvalue of D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) on \textbf{X}_{ij} if and only if it is an eigenvalue of the matrix
\begin{eqnarray*} \textbf{I}-\frac{1}{1+\mu_{i}}\left[\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})+\textbf{I}\right] = \frac{1}{1+\mu_{i}}\left[\mu_{i}\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right]. \end{eqnarray*} |
Hence, D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) is invertible if and only if the matrix
\begin{eqnarray*} \textbf{I}-\frac{1}{1+\mu_{i}}\left[\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})+\textbf{I}\right] \end{eqnarray*} |
is non-singular for any i\geq 0 .
Denote
\begin{eqnarray} H(\mu) = H(\widetilde{\textbf{w}}; \mu)\triangleq \det \left\{\mu\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\}. \end{eqnarray} | (3.6) |
We observe that if H(\mu_{i})\neq 0 , then for each 1\leq j\leq \dim E(\mu_{i}) , the number of negative eigenvalues of D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) on \textbf{X}_{ij} is odd if and only if H(\mu_{i}) < 0 .
We summarize our result as follows.
Theorem 3.2. Assume that the matrix \mu_{i}\bf{I}-\mathit{\boldsymbol{\Phi}}_{\bf{w}}^{-1}(\widetilde{\bf{w}}) \bf{F}_{\bf{w}}(\widetilde{\bf{w}}) is non-singular for an arbitrary i\geq 0 . Then there holds
\begin{eqnarray*} {\rm index}(\bf{G}(\cdot), \widetilde{\bf{w}}) = (-1)^{\sigma}, \end{eqnarray*} |
where \sigma = \sum \limits_{i\geq 0, \, H(\mu_{i}) < 0}\dim E(\mu_{i}) .
According to the above Theorem, we need to consider the sign of H(\mu_{i}) in order to calculate the index of (\textbf{G}(\cdot), \widetilde{\textbf{w}}) . In addition, by (3.6) we have
H(\mu) = \det \left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\right\} \det \left\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\}. |
Hence, we need to consider the signs of \det\big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} and \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}, respectively.
By a direct calculation we get
\begin{equation} \begin{split} \det\left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\right\} & = \frac 1 A, \\ \det \left\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\} & = A\mu^2-B\mu+C\triangleq\mathcal{A}(\mu), \end{split} \end{equation} | (3.7) |
where
\begin{eqnarray*} &&A = d_{1}d_{2}-d_{12}d_{21}, \\ &&B = -\frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}d_2-\left(\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+b\tilde{v}\right)d_1 -\frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2}d_{12}+\frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}d_{21}, \\ &&C = \det\big\{\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\} = \frac{ma\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+kb\tilde{u}\tilde{v} \gt 0. \end{eqnarray*} |
Let \tilde{\mu}_{1} and \tilde{\mu}_{2} be the two roots of \mathcal{A}(\mu) = 0 with Re\{\tilde{\mu}_{1}\}\leq Re\{\tilde{\mu}_{2}\} . Then \tilde{\mu}_{1}\tilde{\mu}_{2} = \frac C A , which is of the same sign as A .
Next, we discuss the dependence of \mathcal{A}(\mu) on d_{12} and d_{21} , respectively. Due to the following limits:
\begin{eqnarray*} &&\lim\limits_{d_{12}\rightarrow\infty}\frac{A}{d_{12}} = -d_{21} \lt 0, \ \; \; \lim\limits_{d_{21}\rightarrow\infty}\frac{A}{d_{21}} = -d_{12} \lt 0, \\ &&\lim\limits_{d_{12}\rightarrow\infty}\frac{B}{d_{12}} = -\frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2} \lt 0, \; \; \ \lim\limits_{d_{21}\rightarrow\infty}\frac{B}{d_{21}} = \frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2} \gt 0, \end{eqnarray*} |
it follows from (3.7) that
\begin{eqnarray} \begin{split} & \lim\limits_{d_{12}\rightarrow\infty}\frac{\mathcal{A}(\mu)}{d_{12}} = \mu\left[-d_{21}\mu+ \frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2}\right], \\ & \lim\limits_{d_{21}\rightarrow\infty}\frac{\mathcal{A}(\mu)}{d_{21}} = \mu\left[-d_{12}\mu- \frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\right]. \end{split} \end{eqnarray} | (3.8) |
Note that the above two limits hold only when d_{12} (or d_{21} ) is chosen to be large enough. This is certainly possible if A = d_1d_2-d_{12}d_{21} < 0 for the fixed d_1 and d_2 . In this case, we have
\det\big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \frac 1 A \lt 0\ \, \text{and}\ \, \tilde{\mu}_{1}\tilde{\mu}_{2} = \frac C A \lt 0. |
Based on the above analyses, in summary we obtain the following results.
Theorem 3.3. Assume A < 0 . Then there exists a positive constant d_{12}^* such that when d_{12}\geq d_{12}^* , both of the two roots \tilde{\mu}_{1}(d_{12}) and \tilde{\mu}_{2}(d_{12}) of \mathcal{A}(\mu) = 0 are real and satisfy
\begin{equation} \begin{split} & \lim\limits_{d_{12}\rightarrow\infty}\tilde{\mu}_1(d_{12}) = 0, \\ & \lim\limits_{d_{12}\rightarrow\infty}\tilde{\mu}_2(d_{12}) = \frac{\tilde{v}^2}{d_{21}(m\tilde{u}+\tilde{v})^2}\triangleq \bar{\mu} \gt 0, \end{split} \end{equation} | (3.9) |
where
\begin{equation} \left\{ \begin{aligned} &\tilde{\mu}_{1}(d_{12}) \lt 0 \lt \tilde{\mu}_{2}(d_{12}), \\ &\mathcal{A}(\mu; d_{12}) \lt 0 \ \mathit{\text{when}}\ \mu\in(-\infty, \tilde{\mu}_{1}(d_{12}))\cup(\tilde{\mu}_{2}(d_{12}), \infty), \\ &\mathcal{A}(\mu; d_{12}) \gt 0 \ \mathit{\text{when}}\ \mu\in(\tilde{\mu}_{1}(d_{12}), \tilde{\mu}_{2}(d_{12})). \end{aligned} \right. \end{equation} | (3.10) |
Theorem 3.4. Assume A < 0 . Then there exists a positive constant d_{21}^* such that when d_{21}\geq d_{21}^* , both of the two roots \tilde{\mu}_{1}(d_{21}) and \tilde{\mu}_{2}(d_{21}) of \mathcal{A}(\mu) = 0 are real and satisfy
\begin{equation} \begin{split} & \lim\limits_{d_{21}\rightarrow\infty}\tilde{\mu}_1(d_{21}) = -\frac{m\tilde{u}^2}{d_{12}(m\tilde{u}+\tilde{v})^2}\triangleq \bar{\bar{\mu}} \lt 0, \\ &\lim\limits_{d_{21}\rightarrow\infty}\tilde{\mu}_2(d_{21}) = 0, \end{split} \end{equation} | (3.11) |
where
\begin{equation} \left\{ \begin{aligned} &\tilde{\mu}_{1}(d_{21}) \lt 0 \lt \tilde{\mu}_{2}(d_{21}), \\ &\mathcal{A}(\mu; d_{21}) \lt 0 \; \; when\; \; \mu\in(-\infty, \tilde{\mu}_{1}(d_{21}))\cup(\tilde{\mu}_{2}(d_{21}), \infty), \\ &\mathcal{A}(\mu; d_{21}) \gt 0 \; \; when\; \; \mu\in(\tilde{\mu}_{1}(d_{21}), \tilde{\mu}_{2}(d_{21})). \end{aligned} \right. \end{equation} | (3.12) |
The following theorem is regarding the existence of non-constant positive solutions to system (3.3) with respect to the cross-diffusion coefficient d_{12} , while all of other parameters are fixed.
Theorem 3.5. Suppose that A < 0 , and the parameters \Lambda , d_{1} , d_{2} and d_{21} are fixed. Let \bar{\mu} be given by (3.9). If \bar{\mu}\in (\mu_{n}, \mu_{n+1}) for some n\geq 1 , and the sum \sigma_{n} = \sum\limits_{i = 1}^{n}\dim E(\mu_{i}) is odd, then there exists a positive constant d_{12}^* such that when d_{12}\geq d_{12}^* , system (3.3) has at least one non-constant positive solution.
Proof. According to Theorem 3.3, there exists a positive constant d_{12}^* such that, if d_{12}\geq d_{12}^* , then (3.10) holds and
\begin{eqnarray} \tilde{\mu}_{1}(d_{12}) \lt 0 = \mu_{0} \lt \tilde{\mu}_{2}(d_{12}), \; \; \ \tilde{\mu}_{2}(d_{12})\in(\mu_{n}, \mu_{n+1}). \end{eqnarray} | (3.13) |
It suffices to prove that for all d_{12}\geq d_{12}^* , system (3.3) has at least one non-constant positive solution. By the way of contradiction, we assume that this is not true for some d_{12}\ (\geq d_{12}^*) . By applying the homotopy invariance of the topological degree, we can see a contradiction explicitly.
For any fixed d_{12} that satisfies d_{12}\geq d_{12}^* , we take \hat{d}_2\geq d_{2}^* and \hat{d}_1\geq D_1 , where \mu_{1}d_{2}^* > \frac{1}{m} and D_{1} = D_{1}(\Lambda, d_{2}^*) are given by Theorem 3.1. For t\in [0, 1] , we define
\mathit{\boldsymbol{\Phi}}(t; \textbf{w}) = \left(\begin{matrix} [td_{1}+(1-t)\hat{d}_1]u+td_{12}v\\ td_{21}u+[td_{2}+(1-t)\hat{d}_2]v \end{matrix}\right). |
Consider the system
\begin{eqnarray} \left\{ \begin{array}{l} -\Delta \mathit{\boldsymbol{\Phi}}(t; \textbf{w}) = \textbf{F}(\textbf{w}), \\ \frac{\partial \textbf{w}}{\partial \nu} = \textbf{0}, \end{array}\; \; \begin{array}{l} x\in \Omega, \\[1.5mm] x\in \partial\Omega. \end{array} \right. \end{eqnarray} | (3.14) |
Then \textbf{w} is a non-constant positive solution of system (3.3) if and only if it is a positive solution of system (3.14) when t = 1 . It is obvious that \widetilde{\textbf{w}} is the unique positive constant solution of system (3.14) for any t\in[0, 1] . From (3.5), we know that for any t\in[0, 1] , \textbf{w} is a positive solution of system (3.3) if and only if
\begin{eqnarray*} \textbf{G}(t; \textbf{w}) &\triangleq& \textbf{w}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t; \textbf{w})[\textbf{F}(\textbf{w}) +\nabla\textbf{w}\mathit{\boldsymbol{\Phi}}_{\textbf{ww}}(t; \textbf{w})\nabla\textbf{w}] +\textbf{w}\big\} \\ & = &0, \ \text{for}\ \textbf{w}\in\textbf{X}^+. \end{eqnarray*} |
Analogous to (3.6), we set
H(t, \mu) = \det \left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t, \widetilde{\textbf{w}})\right\} \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(t, \widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}. |
It is easy to see that \textbf{G}(1; \textbf{w}) = \textbf{G}(\textbf{w}) and
\begin{eqnarray*} D_{\textbf{w}}\textbf{G}(t; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t; \widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}. \end{eqnarray*} |
In particular, we obtain
\begin{eqnarray*} &&D_{\textbf{w}}\textbf{G}(0; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}, \\ &&D_{\textbf{w}}\textbf{G}(1; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\} = D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}), \end{eqnarray*} |
where
\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} \hat{d}_{1} & 0\\ 0 & \hat{d}_{2} \end{matrix}\right)\ \, \text{and}\ \, \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right). |
More specifically, when t = 1 , from (3.6) and (3.7) we have
\begin{eqnarray} H(1, \mu) & = & H(\mu) \\ & = & \det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}\\ & = & \det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\}\mathcal{A}(\mu), \end{eqnarray} | (3.15) |
where
\det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \left( {\det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}}\right)^{-1} = \frac 1A \lt 0, |
and \mathcal{A}(\mu) is defined by (3.7).
According to (3.10), (3.13) and (3.15), we deduce that
\begin{equation*} \left\{ \begin{aligned} &H(1, \mu_{0}) = H(0) \lt 0, \\ &H(1, \mu_{i}) \lt 0 \; \; \; \; \; \; \; \; \; \; \; \; \; \text{when}\; \; 1\leq i\leq n, \\ &H(1, \mu_{i+1}) \gt 0 \; \; \; \; \; \; \; \; \; \; \text{when}\; \; i\geq n+1. \end{aligned} \right. \end{equation*} |
Therefore, zero is not an eigenvalue of the matrix \mu_{i}\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) for any i\geq 0 , and
\sum\limits_{i\geq 0, H(1, \mu_{i}) \lt 0}\dim E(\mu_{i}) = \sum \limits_{i = 1}^{n}\dim E(\mu_{i}) = \sigma_{n} |
is odd. It follows from Theorem 3.2 that
\begin{equation} {\rm index}(\mathit{\boldsymbol{\Phi}}(1; \cdot), \widetilde{\textbf{w}}) = (-1)^{\gamma} = (-1)^{\sigma_{n}} = -1. \end{equation} | (3.16) |
When t = 0 , we have
\begin{eqnarray} \begin{split} H(0, \mu)& = \det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} \det \big\{\mu\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}\\ & = \det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\}\mathcal{\hat{A}}(\mu), \end{split} \end{eqnarray} | (3.17) |
where
\det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \left(\hat{d}_1\hat{d}_2\right)^{-1} \gt 0, |
and we use \mathcal{\hat{A}}(\mu) to represent \mathcal{A}(\mu) given in (3.7) under the restriction of d_{12} = d_{21} = 0 . In this case, by Theorem 3.1 it implies that \textbf{G}(0; \textbf{w}) = 0 only has the positive constant solution \widetilde{\textbf{w}} in \textbf{X}^+ . By a direct calculation we have
\mathcal{\hat{A}}(\mu) = \hat{d}_1\hat{d}_2\mu^2+\left[\left(\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2} +b\tilde{v}\right)\hat{d}_1+ \frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\hat{d}_2\right]\mu+\frac{ma\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+kb\tilde{u}\tilde{v}, |
and so H(0, \mu_{i}) > 0 for all i\geq 0 .
Discussing in the same manner, we can prove that
\begin{equation} {\rm index}(\mathit{\boldsymbol{\Phi}}(0; \cdot), \widetilde{\textbf{w}}) = (-1)^{0} = 1. \end{equation} | (3.18) |
According to Theorems 2.1 and 2.2, there exists a positive constant c such that the positive solution of system (3.14) satisfies c^{-1} < u and v < c for any t\in[0, 1] . Thus for any t\in [0, 1] , there holds \mathit{\boldsymbol{\Phi}}(t; \textbf{w})\neq 0 on \partial\mathcal{B}(M) . By the homotopy invariance of the topological degree, we have
\begin{equation} {\rm deg}(\mathit{\boldsymbol{\Phi}}(1; \cdot), 0, \mathcal{B}(M)) = {\rm deg}(\mathit{\boldsymbol{\Phi}}(0; \cdot), 0, \mathcal{B}(M)). \end{equation} | (3.19) |
Note that both \mathit{\boldsymbol{\Phi}}(1; \textbf{w}) = 0 and \mathit{\boldsymbol{\Phi}}(0; \textbf{w}) = 0 have the unique positive solution \widetilde{\textbf{w}} in \mathcal{B}(M) . From (3.16) and (3.18) we get
\begin{equation} \left\{ \begin{aligned} &{\rm deg}(\mathit{\boldsymbol{\Phi}}(0; \cdot), 0, \mathcal{B}(M)) = {\rm index}(\mathit{\boldsymbol{\Phi}}(0; \cdot), \widetilde{\textbf{w}}) = 1, \\ &{\rm deg}(\mathit{\boldsymbol{\Phi}}(1; \cdot), 0, \mathcal{B}(M)) = {\rm index}(\mathit{\boldsymbol{\Phi}}(1; \cdot), \widetilde{\textbf{w}}) = -1, \end{aligned} \right. \end{equation} | (3.20) |
which yields a contradiction with (3.19).
Processing in an analogous way as we just did in the proof of Theorem 3.5, we can obtain the following result regarding the existence of non-constant positive steady states for system (3.3) with respect to the cross-diffusion coefficient d_{21} . So, we omit the proof.
Theorem 3.6. Suppose that A < 0 , and the parameters \Lambda , d_{1} , d_{2} and d_{12} are fixed. Let \bar{\bar{\mu}} be given by (3.11). If \bar{\bar{\mu}}\in (\mu_{n}, \mu_{n+1}) for some n\geq 1 , and the sum \sigma_{n} = \sum\limits_{i = 1}^{n}\dim E(\mu_{i}) is odd, then there exists a positive constant d_{21}^* such that when d_{21}\geq d_{21}^* , system (3.3) has at least one non-constant positive solution.
This work is supported by NSF of China (No. 41575004) and UTRGV Faculty Research Council Award 1100127.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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