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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response

Xiaozhou Feng / Yi Song / Xiaomin An
Published Online: 2018-06-14 | DOI: https://doi.org/10.1515/math-2018-0060

Abstract

This paper investigates the dynamic behavior analysis on the prey-predator model with ratio-dependent Monod-Haldane response function under the homogeneous Dirichlet boundary conditions, which is used to simulate a class of biological system. Firstly, the sufficient and necessary conditions on existence and non-existence of coexistence states of this model are discussed by comparison principle and fixed point index theory. Secondly, taking a as a main bifurcation parameter, the structure of global bifurcation curve on positive solutions is established by using global bifurcation theorem and properties of principal eigenvalue. Finally, the stability of coexistence states is obtained by the eigenvalue perturbation theory; the multiplicity of coexistence states is investigated when a satisfies some condition by the fixed point index theory.

Keywords: Prey-predator model; Coexistence state; Bifurcation, Stability; Multiplicity

MSC 2010: 35J25; 35B40; 35J60

1 Introduction

Population ecology is an important branch of ecology. Due to the complexity of ecological relations, mathematical methods and results have been increasingly used in ecology and population ecology. In recent years, due to the widespread application of biological models, such as predator-prey model in population ecology, this field of research has gained increasing interest. However, the predator-prey model is an important branch of reaction-diffusion equations. The dynamic relationship between predator and their prey is one of the dominant themes in ecology and mathematical ecology. During the last thirty years, the investigation on the prey-predator models has been developed, and more realistic models have been derived in view of laboratory experiments. Moreover, the research on the prey-predator models [1, 2, 3, 4] has been studied from various views and interesting results have been obtained (see [5, 6, 7, 8] and the references therein). For every specific prey-predator model, we know that the functional response of the predator to the prey density is very important, which represents the specific transformation rule of the two organisms. For example, the Holling I type functional response in a predator-prey model was considered by Cheng et al. [9] and Zhang et al. [10, 11], the Holling II type functional response was investigated by Wang and Wu [12], Zhu et al. [13] and Cui et al. [14], the Holling Tanner type functional response was studied by Casal et al. [15] and Du et al. [16], the Beddington-DeAngelis functional response was proposed by [17] and Guo and Wu [18]. In this paper, we are concerned with the prey-predator model with ratio-dependent Monod-Haldane response function [19, 20] under the homogeneous Dirichlet boundary conditions as follows:

{Δu=u(aubv1+mu+lu2+kv)xΩ,Δv=v(cv+du1+mu+lu2+kv)xΩ,u=v=0xΩ,(1)

where Ω is a bounded domain in RN (N ≥ 1) with smooth boundary Ω. u and v stand for the densities of prey and predators, respectively. The parameters a, b, c, d are assumed to be only positive constants. a and c denote the intrinsic growth rate of prey u and predator v, respectively. b and d stand for capturing rate to predator and conversion rate of prey captured by predator, respectively. Where f(u,v)=uv1+mu+lu2+kv stands for the Monod-Haldane response function with ratio-dependent, which not only has the characteristics of Monod-Haldane reaction, but also has the characteristics of Beddington-DeAngelis functional response. Thus, this response function can better simulate the transformation law of two species. In this paper, we are concerned with the coexistence states on the prey-predator model with the response in this case m > 0, k > 0 and l = 1 > 0; the specific model is as follows:

{Δu=u(aubv1+mu+u2+kv),xΩ,Δv=v(cv+du1+mu+u2+kv),xΩ,u=v=0,xΩ,(2)

where the coexistence state is a solution (u, v) of (2) satisfying u(x) > 0 and v(x) > 0 for all xΩ.

The rest of this paper is arranged as follows. In Section 2, the sufficient and necessary conditions on existence and non-existence of coexistence states of (2) are discussed by the fixed point index theory. In Section 3, taking a as a main bifurcation parameter, the structure of global bifurcation curve on coexistence states is established by using global bifurcation theorem. In Section 4, the stability of coexistence states is obtained by the eigenvalue perturbation theory; the multiplicity of coexistence stats of (2) is investigated when a satisfies some condition by the fixed point index theory.

2 Coexistence

The goal of this section is to discuss the condition on existence and non-existence of coexistence states by the fixed point index theory and properties of principal eigenvalue. In order to present the main results, we introduce some basic conclusion and notations which refer to [21, 22, 23].

Suppose λ1(q) < λ2(q) ≤ λ3(q) ≤ ⋯ are all eigenvalues of the equation

Δϕ+q(x)ϕ=λϕ,ϕ|Ω=0,

and q(x) ∈ C(Ω). And λ1(q) is simple and λ1(q) is strictly increasing which implies that q1q2 and q1q2 can deduce λ1(q1) < λ1(q2). Denote λi(0) by λi, and the eigenfunction corresponding to λ1 by ϕ1 with normalization ∥ϕ1 = 1 and ϕ1 > 0 in Ω.

Consider the following problem

Δu=ρuu2,u|Ω=0,

let θρ ( θρ < ρ ) be unique positive solution as ρ > λ1. It is easy to see that the mapping ρθρ is strictly increasing, continuously differentiable on (λ1, ∞) → C2(Ω) ∩ C0(Ω) such that θρ → 0 uniformly on Ω as ρ → λ1.

Hence, in the system (2) there exist two semi-trivial solutions (θa,0) and (0, θc) when a, c > λ1. Next, we present some a priori estimate by maximum principle (see [12, 24, 25, 26, 27, 28]). It’s proof will be omitted.

Lemma 2.1

Any coexistence state (u, v) of (2) has a priori boundary, i.e.,

ua,vB:=c+ad1+am.

Next, we give the fixed point index theory (see [29]). Let E be a Banach space. WE is named a wedge if W is the closed convex set and β WW for all β ≥ 0. For yW, let Wy = {xE : ∃ r = r(x) > 0, s.t. y + rxW}, Sy = {xWy : −xWy}, and E = WW. Let T : WyWy be a compact linear operator on E. We claim that T has property α on Wy if there exists t ∈ (0, 1) and ωWySy such that ωtTωSy. Define Bδ+(y) = Bδ(y) ∩ W, ∀δ > 0, yW. Let F : Bδ+(y) → W be a compact operator and y be an isolated fixed point of F. Denote F′ as the Fréchet differentiable at y, then F′(y) : WyWy. The fixed point index of F at y relative to W is denoted by indexW(F, y) throughout this paper. Now we give a general result on the fixed point index theory with respect to the positive cone W (see [22, 23]).

Lemma 2.2

([29, 30]). Suppose that IL is invertible on Wy. Then the following conclusions hold.

  1. If L has property α on Wy, then indexW(F, y) = 0.

  2. If L does not have property α on Wy, then indexW(F, y) = (−1)σ, where σ is the sum of algebra multiplicities of the eigenvalue of L which are greater than 1.

Let X = C01(Ω) ⨁ C01(Ω), where C01(Ω) = {ωC1(Ω) : ω|∂Ω = 0}; W = KK, where K = {φC(Ω) : φ(x) ≥ 0}; D : = {(u, v) ∈ X : ua + 1, vB + 1}; D′: = (intD) ⋂ W.

Define Ft: D′ → W as the following form:

Ft(u,v)=(Δ+P)1(tu(aubv1+mu+u2+kv)+Putv(cv+du1+mu+u2+kv)+Pv),

where t ∈ [0, 1], P > max{a+bB,c+da1+ma}. By Maximum Principle, we obtain that (−Δ + P)−1 is a compact positive operator, Ft is complete continuous and Fréchet differentiable. Let F1 = F, then (2) has a coexistence state in W if and only if Ft = F1 has a positive fixed point in D′.

If a > λ1 and c > λ1, (0, 0), (θa, 0) and (0, θc) are the non-negative fixed points of F. Hence, indexW(F, (0, 0)), indexW(F, (θa, 0)) and indexW(F, (0, θc)) are well defined. By calculating, we can get the Fréchet operator of F as follows:

(Δ+P)1(a2ubv(1u2+kv)(1+mu+u2+kv)2+Pbu(1+mu+u2)(1+mu+u2+kv)2dv(1u2+kv)(1+mu+u2+kv)2c2v+du(1+mu+u2)(1+mu+u2+kv)2+P).

Applying similar methods as in the proof of Lemma 2.1 and Lemma 2.2 in [12], we can easily establish the following Lemmas by the fixed point index theory (see [24, 25, 26, 27, 28]). We omit the proof procedure.

Lemma 2.3

Assume a > λ1.

  1. degW(IF, D′) = 1, here degW(IF, D′) is the degree of TF in Don W.

  2. If c ≠ λ1, then index W(F, (0, 0)) = 0.

  3. If c>λ1(dθa1+mθa+θa2), then index W(F, (θa, 0)) = 0.

  4. If c<λ1(dθa1+mθa+θa2), then index W(F, (θa, 0)) = 1.

Lemma 2.4

Assume c > λ1.

  1. If a>λ1(bθc1+kθc), then indexW(F, (0, θc)) = 0.

  2. If a<λ1(bθc1+kθc), then index W(F, (0, θc)) = 1.

In the following, some conditions on existence and non-existence of coexistence states of (2) are established by comparison principle.

Theorem 2.5

  1. If a ≤ λ1, then (2) does’t have any coexistence state; if a ≤ λ1 and c ≤ λ1, then (2) does’t exist non-negative non-zero solution.

  2. If c ≤ λ1 and for (2) there exists a coexistence state, then a>λ1,c+da1+ma>λ1.

  3. If c > λ1 and for (2) there exists a coexistence state, then a>λ1(bθc1+mθa+θa2+kθc).

Proof

  1. Suppose (u, v) is a coexistence state to (2), then (u, v) satisfies

    Δu=u(aubv1+mu+u2+kv)xΩ,u=0xΩ,

    and a=λ1(u+bv1+mu+u2+kv). With the aid of the comparison principle, we get a > λ1, a contradiction. We suppose that (u, v) is a non-negative non-zero solution of (2). Obviously, u ≢ 0 and v ≡ 0, then a > λ1 by the above proof. Similarly, we can deduce c > λ1 as u ≡ 0 and v ≢ 0, which derives a contradiction again.

  2. Let (u, v) be a coexistence state of (2). By (i), we know that a > λ1, and (2) has the positive semi-trivial solution θa. Due to

    Δu=u(aubv1+mu+u2+kv)u(au)xΩ,u=0xΩ,

    u is a lower solution to (2). Owing to the uniqueness of θa, uθa. It follows that v meets

    Δv=v(cv+du1+mu+u2+kv)xΩ,v=0xΩ,

    then 0=λ1(c+vdu1+mu+u2+kv)>λ1(cda1+ma), which gives the result.

  3. Suppose (u, v) is a coexistence state of (2), then (2) has the unique positive solution θa with uθa. Similarly, c > λ1 implies the existence of positive solution θc of (2) with θcv. According to the same method of (i), we directly obtain

    a=λ1(u+bv1+mu+u2+kv)>λ1(bθc(1+mθa+θa2+kθc)).

    Since the function bv1+mu+u2+kv has a minimum at u = θa and v = θc (for uθa and vθc). So the result holds.

 □

Theorem 2.6

  1. If c > λ1 and a>λ1(bθc1+kθc). Then (2) has at least a coexistence state.

  2. Suppose c < λ1. Then (2) has a coexistence state if and only if a > λ1 and c>λ1(dθa1+mθa+θa2).

Proof

  1. By Lemma 2.3-2.4 and the fixed point index theory, we have

    degW(IF,D)indexW(f,(0,0))indexW(f,(θa,0))indexW(f,(0,θc))=1.

    Then (2) has at least a coexistence state.

  2. Firstly, we demonstrate the sufficiency. Due to c < λ1, (2) does’t have the solution of the form (0, v) with v > 0. If a > λ1 and c>λ1(dθa1+mθa+θa2), since c < λ1, by Lemma 2.4 and the fixed point index theory, we get

    degW(IF,D)indexW(f,(0,0))indexW(f,(θa,0))=1.

    Hence (2) has at least a coexistence state.

    Conversely, we suppose that (u, v) is a coexistence state of (2). Then a > λ1, and u < θa. Thanks to (u, v) satisfies

    Δv=v(cv+du1+mu+u2+kv)xΩ,v=0xΩ.

    Hence, 0=λ1(c+vdu1+mu+u2+kv)>λ1(cda1+ma).

 □

Theorem 2.7

If one of the following two conditions holds, then (2) does’t have any non-negative non-zero solution:

  1. b ≥ 1 + ma + a2 + kB and ac;

  2. b < 1 + ma + a2 + kB and ca(1b(1+ma+a2+kB))B.

Proof

  1. Assume that (u, v) is a coexistence state of (2) as b ≥ 1 + ma + a2 + kB and ac. Note that u(x) ≤ a by Lemma 2.1 and 1 + mu(x) + u2(x) + kv(x) > 0 in Ω̄. we get

    0=λ1(a+u+bv1+mu+u2+kv)λ1(c+vdu1+mu+u2+kvv+bv1+mu+u2+kv+u+du1+mu+u2+kv)>λ1(c+vdu1+mu+u2+kv(1b1+mu+u2+kv)v)>λ1(c+vdu1+mu+u2+kv(1b1+ma+a2+kB)v)λ1(c+vdu1+mu+u2+kv).

    Which deduces a contradiction.

  2. Based on the proof of (i), thanks to the fact b < 1 + ma + a2 + kB and ca(1b(1+ma+a2+kB))B, we can obtain the following inequality:

    0=λ1(a+u+bv1+mu+u2+kv)>λ1(c+vdu1+mu+u2+kva+c(1b1+ma+a2+kB)B)λ1(c+vdu1+mu+u2+kv),

    which is a contradiction. The proof is completed.

 □

3 Global bifurcation

The purpose of this section is to investigate a coexistence state bifurcates from the semi-trivial non-negative branch {(0, θc, a)} depending on the change of the parameter a and c > λ1. Particularly, the structure of global bifurcation curve on coexistence states is discussed by using global bifurcation theorem and property of principal eigenvalue.

Throughout this paper, the principal eigenvalue of the following problem is denoted by ã,

Δϕ+bθc1+kθcϕ=aϕ,ϕ|Ω=0,(3)

and the corresponding eigenfunction is denoted by Φ with ∥Φ = 1.

For convenience of the calculation, we do the following variable substitution, suppose ω = u, χ = vθc, then 0 ≤ ωθa, χ ≥ 0, and ω, χ meet

{Δω=(abθc1+kθc)ω+F1(ω,χ),xΩ,Δχ=(c2θc)χ+dθc1+kθcω+F2(ω,χ),xΩ,ω=χ=0,xΩ,(4)

here

F1(ω,χ)=bωθc1+kθcbω(χ+θc)(1+mω+ω2+k(χ+θc))ω2,F2(ω,χ)=dω(χ+θc)(1+mω+ω2+k(χ+θc))dωθc1+kθcχ2.

It is easy to see that F = (F1, F2) is continuous, F(0, 0) = 0, and the Fréchet derivative D(ω, χ)F|(0,0) = 0. The inverse of −Δ is denoted by K. Then

{ω=aKωbK(ωθc1+kθc)+KF1(ω,χ),xΩ,χ=cKχ2K(χθc)+dK(ωθc1+kθc)+KF2(ω,χ),xΩ,ω=χ=0,xΩ.(5)

Now we introduce the operator T : R+ × XX as follows:

T(a;ω,χ)=(aKωbK(ωθc1+kθc)+KF1(ω,χ)cKχ2K(χθc)+dK(ωθc1+kθc)+KF2(ω,χ)).

Obviously, T(a; ω, χ) is a compact operator on X. Define G(a; ω, χ) = (ω, χ)TT(a; ω, χ), then G is continuous, and G(a; 0, 0) = 0. G(a; ω, χ) = 0 (0 ≤ ωθa, χ ≥ 0) if and only if (ω, χ+θc, a) is a positive solution of (2).

Theorem 3.1

Suppose c > λ1. Then there exists a coexistence state of (4) (or (2)) bifurcate from the point (ã; 0, θc), and a~=λ1(bθc1+kθc).

Remark 3.2

The proof of Theorem 3.1 refers to Theorem 2 in [19], more precisely, the bifurcation branch near a~=λ1(bθc1+kθc). is determined by C1 continuous curve (a(s); ϕ(s), φ(s)) : (− δ, δ) → R × Z, for some δ > 0 such that

a(0)=a~,ϕ(0)=0,φ(0)=0,and(a(s);ω(s),χ(s))=(a(s);(Φ+ϕ(s)),s(Ψ+φ(s)))

meets G(a(s); ω(s), χ(s)) = 0, here X = Zspan{(Φ, Ψ)}. Thus (a(s); U(s), V(s)) (|s| < δ) is a bifurcation solution of (4) (or (2)), here U(s) = s(Φ + ϕ(s)), V(s) = θc + s(Ψ + φ(s)), Ψ=(Δc+2θc)1(dθc1+kθcΦ).

If 0 < s < δ, we can deduce that the positive solution of (2) nearby (ã; 0, θc) lies either on the branch {(a; 0, θc) : aR+} or on the branch {(a(s); U(s), V(s)) : 0 < s < δ}.

Define T : X × RX as the compact continuously differentiable operator, and T(0, a) = 0. Let T be T(u, a) = K(a)u + R(u, a), here K(a) is a linear compact operator and the Fréchet derivative Ru(0, 0) = 0. Suppose x0 is an isolated fixed point of T, we denote the index of T at x0 by index(T, x0) = deg(IT, Uδ(x0), x0), here Uδ(x0) is a ball with center at x0, and x0 is the unique fixed point of T in Uδ(x0). If IT′(x0) is invertible, then x0 is an isolated fixed point of T, meanwhile, index (T, x0) = deg(IT, Uδ(x0), x0) = deg(IT′(x0), Uδ(x0), 0).

If x0 = 0, then deg(IK(a), Uδ (x0), 0) = (−1)σ, where σ is equal to the algebraic multiplicities of the eigenvalue of K which is greater than one.

Next, we study the structure of global bifurcation curve on coexistence states, we will extend the local bifurcation solution {(a(s); U(s), V(s)) : 0 < s < δ} established by Theorem 3.1 to the global bifurcation branch. For this purpose, we define the following notation: P1 = {uC01(Ω̄) : u(x) > 0, xΩ, un < 0, xΩ}, P = {(u, v, a) ∈ X × R+ : u, vP1}.

Theorem 3.3

If c > λ1, then the local bifurcation solution {(a(s); U(s), V(s)) : 0 < s < δ} can be extended to the global solution which is denoted by C and unbounded by going to infinity in P.

Proof

Define

T(a)(ω,χ)=D(ω,χ)T(a;0,0)(ω,χ)=(aKωbK(ωθc1+kθc),cKχ2K(χθc)+dKωθc1+kθc).

Let μ ≥ 1 be an eigenvalue of T′(a). Then

{μΔω=(abθc1+kθc)ω,xΩ,μΔχ=(c2θc)χ+dθc1+kθcω,xΩ,ω=χ=0,xΩ.

Obviously, ω ≢ 0, otherwise, ω ≡ 0, due to the fact that all eigenvalues of (−μΔc + 2θc) are greater than 0, so χ ≡ 0, a contradiction. Thus, a = ai(μ) is the eigenvalue of the following problem

μΔω+bθc1+kθcω=aω,ω|Ω=0.

So ai(μ) is increasing along with μ on [1,+∞) and can be ordered by

0<a1(μ)<a2(μ)a3(μ),a1(1)=a~.

Conversely, if μ ≥ 1, we demonstrate that all eigenvalues of (−μΔc + 2θc) are greater than 0, then χ = (−μΔc + 2θc)−1 (dθc1+kθc). Hence, μ ≥ 1 is an eigenvalue of T′(a) if and only if a = ai(μ) for some i = 1, 2, ….

If a < ã, then a < a1(1) ≤ ai(μ) for any μ ≥ 1, i ≥ 1. It follows that T′(a) has no eigenvalue greater than 1, meanwhile, index(T(a; ⋅), 0) = 1 for a < ã.

If ã < a < a2(1), then a < ai(μ). Since a1(1) = ã for any μ ≥ 1, i ≥ 2, limμ a1(μ) = +∞, and a1(μ) is increasing along with μ. Hence, there exists a unique μ1 > 1 such that a = a1(μ1). Thus N(μ1IT′(a)) = span{(ω, χ)}, dimN(μ1IT′(a)) = 1, here ω > 0 is the principal eigenvalue of the following equation

μ1Δω¯+(abθc1+kθc)ω¯=0,ω¯|Ω=0,(6)

here χ = (−μ1Δc + 2θc)−1 (dθc1+kθcω¯).

Next, we shall demonstrate that R(μ1IT′(a)) ∩ N(μ1IT′(a)) = 0. As a matter of fact, suppose the assertion is false, we may assume that (ω, χ) ∈ R(μ1IT′(a)). Then there exists (ω, χ) ∈ X such that (μ1IT′(a))(ω, χ) = (ω, χ), i.e.,

μ1Δω+(abθc1+kθc)ω=Δω¯,ω|Ω=0.

Multiplying the above equation by ω, and integrating over Ω, by Green’s formula, we have

Ωω¯Δω¯=Ω(μ1Δω+aωbωθc1+kθc)ω¯=Ω(μ1Δω¯+aω¯bω¯θc1+kθc)ω=0.

Thus Ω1μ1(abθc1+kθc)ω¯2=0, a contradiction, which has proved the claim. Then the multiplicity of μ1 is one and index(T(a; ⋅), 0) = −1 for ã < a < a2(1). With the aid of global bifurcation theory [23], we deduce that there exists a continuum C0 of zero points of G(a; ω, χ) = 0 in R+ × X bifurcating from (ã; 0, 0) and all zero points of G(a; ω, χ) nearby (ã; 0, 0) lie on the curve for which the existence was established by Theorem 3.1.

Define the maximal continuum C1 as C1 = C0 − {(a(s); s(Φ + ϕ(s)), s(Ψ + φ(s))) : − δ < s < 0}, then C1 consists of the curve {(a(s); s(Φ + ϕ(s)), s(Ψ + φ(s))):− δ < s < 0} in the neighborhood of (ã; 0, 0). Suppose = {(a; u, v) : U = ω, V = θc + χ, (ω, χ) ∈ C1}. It follows that is the solution branch of (2) which bifurcates from (ã; 0, θc) and keeps the positive near (ã; 0, θc) and P. Thus, the continuum − {(ã; 0, θc)} must satisfy one of the three alternatives:

  1. joining up with a bifurcation point of the form (ā; 0, θc), which IT′(ā) is not invertible, ãā.

  2. joining up from (ã; 0, θc) to ∞ in R × X.

  3. containing points of the form (a; u, θc + v) and (a; −u, θcv), here (u, v) ≠ (0, 0).

Now we claim that −{(ã; 0, θc)} ⊂ P. Suppose −{(ã; 0, θc)} ⊈ P, it follows that there exists (â; û, ) ∈ (−{(ã; 0, θc)}) ∩ P and the sequence {(an; un, vn)} ⊂ P, un > 0, vn > 0 such that (an; un, vn) → (â; û, ) as n → ∞. We can obtain that û P1 or P1. If û P1, then û ≥ 0, xΩ. Thus, we find either x0Ω such that û(x0) = 0 or x0Ω such that u^n|x0=0. Since û satisfies

Δu^=(a^u^bv^(1+mu^+u^2+kv^))u^,u^|Ω=0.

By maximum principle, we can get û ≡ 0. Similarly, we can deduce that ≡ 0 for P1.

Thus, we will investigate the following three cases:

(i)(u^,v^)(θa^,0);(ii)(u^,v^)(0,θc);(iii)(u^,v^)(0,0).

  1. If (û, ) ≡ (θâ, 0), then (an; un, vn) → (â; θâ, 0) as n → ∞. Set Vn=vn||vn||, then Vn meets

    ΔVn=(cvn+dun(1+mun+un2+kvn))Vn,Vn|Ω=0.(7)

    With the aid of LP estimates and Sobolev embedding theorem, we deduce that there exists a convergent sub-sequence of Vn, which be still denoted by Vn, it follows that VnV in C01(Ω) as n → ∞, and V ≥ 0, ≢ 0, xΩ. Taking the limit in (7) as n → ∞, we get

    ΔV=(c+dθa^1+mθa^+θa^2)V,V|Ω=0.

    By the maximum principle, we get V > 0, xΩ, which implies c=λ1(dθa^1+mθa^+θa^2). A contradiction to c > λ1.

  2. If (û, ) ≡ (0, θc). Then (an; un, vn) → (â; 0, θc) as n → ∞. Set Un=un||un||, hence, Un meets

    ΔUn=(anunbvn(1+mun+un2+kvn))Un,Un|Ω=0.(8)

    Similarly, using LP estimates and Sobolev embedding theorem, we can get a convergent sub-sequence of Un, which be still denoted by Un, then UnU in C01(Ω) as n → ∞, and U ≥ 0, ≢ 0, xΩ. Taking limit in (8) by n → ∞, we get

    ΔU=(a^bθc1+kθc)U,U|Ω=0.

    Due to maximum principle, we get U > 0, xΩ. Thus a^=λ1(bθc1+kθc). A contradiction to âã.

  3. If (û, ) ≡ (0, 0), then similarly as in the method from the case above, we also arrive at a contradiction. Hence, −{(ã; 0, θc)} ⊂ P. It follows from Lemma 2.1 that 0 ≤ U ≤ a, θcVc+da1+αa. With the aid of LP estimates and Sobolev embedding theorem, we prove that there exists a constant M > 0 such that ||U||C1,||V||C1M. Then, the global bifurcation solution branch of positive solutions of (2) bifurcating at (ã; 0, θc) contains points with a arbitrarily large in P.

 □

4 Stability and multiplicity

The purpose of this section is to investigate the stability and multiplicity of coexistence states of (2) by means of eigenvalue perturbation theory and the fixed point index theory.

Set X1 = [C2,α(Ω̄) × C2,α(Ω̄) ∩ X], Y = [Cα(Ω̄) × Cα(Ω̄), here 0 < α < 1. i : X1Y is the inclusion mapping. Since L1 is the linearized operator at (ã; 0, θc) of (2), according to the proof of Theorem 3.1, we get N(L1) = span{(Φ, Ψ)}, Codim R(L1) = 1, and R(L0) = {(u, v) ∈ X : ∫ΩuΦ dx = 0}. Due to i(Φ, Ψ) ∈ R(L1), it follows from [31] that 0 is an i−simple eigenvalue of L1.

Lemma 4.1

0 is the eigenvalue of L1 with the largest real part, and all the other eigenvalues of L1 lie in the left half complex plane.

Proof

The proof of Lemma 4.1 can be found in [9, 18, 24, 26], here we omit it. □

Let L(u(s), v(s), a(s)) and L(a; 0, θc) be the linearized operators of (2) at (u(s), v(s), a(s)), (a; 0, θc), respectively. Applying the linearized stability theory (see [31, 32]), we can obtain the following result.

Lemma 4.2

There exists C1function: a → (M(a), γ(a)) and s → (N(s), π(s)), which are defined by the mapping from the neighborhood of ã and 0 into X1 × R, respectively, satisfying the forms γ(ã) = π(0) = 0, M(ã) = N(0) = (Φ, Ψ) and

L(a;0,θc)M(a)=γ(a)M(a),for|aa~|1,

L(u(s),v(s),a(s))N(s)=π(s)N(s),for|s|1,

where M(a) = (ϕ1(a), ϕ2(a)), N(s) = (φ1(a)φ2(a)). Meanwhile, γ′(ã) ≠ 0, furthermore, for |s| ≪ 1, π(s) ≠ 0, π(s) and −sa′(s)γ′(ã) have the same sign, where γ′(ã) is the derivative of γ(a) at a = ã, and a′(s) is the derivative of a(s) on the point s.

Lemma 4.3

γ′(ã) > 0.

Proof

By calculating L(a; 0, θc)M(a) = γ(a)M(a), for |aã| ≪ 1, we have

{Δϕ1+(abθc1+kθc)ϕ1=γ(a)ϕ1,xΩ,Δϕ2+(c2θc)ϕ2+dθc1+kθcϕ1=γ(a)ϕ2,xΩ,ϕ1=ϕ2=0,xΩ.

Note that |aã| ≪ 1, then | γ(a)| ≪ 1. Obviously, ϕ1 ≢ 0, otherwise, ϕ1 ≡ 0, then ϕ2 ≡ 0, a contradiction. Thus γ(a) is an eigenvalue of (Δ+(abθc1+kθc)I). So Φ > 0, implies ϕ1 = ϕ1(a) > 0 as |aã| ≪ 1. It follows that γ(a) is the principal eigenvalue of (Δ+(abθc1+kθc)I), and γ(a) is increasing along with a for |aã| ≪ 1. Meanwhile, γ′(ã) ≠ 0. Thus γ′(ã) > 0. □

Lemma 4.4

a′(0) satisfies

a(0)ΩΦ2dx=Ω(1bmθc(1+kθc)2)Φ3dx+ΩbΨ(1+kθc)2Φ2dx.

Proof

By substituting (u(s), v(s), a(s)) into (2), differentiating on s, and let s = 0, we get

Δϕ(0)=(a~bθc1+kθc)ϕ(0)+[a(0)ΦbΨmθcΦ(1+kθc)2]Φ,

where ϕ′(0) is the derivative of ϕ on the point s = 0.

Multiplying the above equation by Φ, and applying Green’s formula and the definition of Φ, we obtain

a(0)ΩΦ2dx=Ω(1bmθc(1+kθc)2)Φ3dx+ΩbΨ(1+kθc)2Φ2dx. □

Taking the advantage of Lemma 4.1-Lemma 4.4, we directly derive the following theorem.

Theorem 4.5

Let σ=Ω(1bmθc(1+kθc)2)Φ3dx+ΩbΨ(1+kθc)2Φ2dx. If σ > 0, then local bifurcation solution (u(s), v(s)) is stable; if σ < 0, then local bifurcation solution (u(s), v(s)) is unstable.

Remark 4.6

In section 2, by the sufficient condition and necessary condition on coexistence states of (2) established in Theorem 2.5-2.6, we find that there exists a gap between a>λ1(bθc1+kθc)anda>λ1(bθc(1+mθa+θa2+kθc)) as c > λ1.

In the following, we shall investigate the multiplicity of coexistence states in the gap.

Theorem 4.7

Suppose that c>λ1andΩ(1bmθc(1+kθc)2)Φ3dx<0. If there exists a sufficiently small ε > 0 and d ≪1, then coexistence state (u(s), v(s)) is non-degenerate and unstable for a ∈ (ãε, ã), moreover, (2) has at least two coexistence states.

Proof

Firstly, we will demonstrate the coexistence state (u(s), v(s)) is non-degenerate and unstable. To this end, we only need to prove that there exists a sufficiently small ε > 0 such that any coexistence state (u(s), v(s)) of (2) is non-degenerate for a ∈ (ãε, ã), and the linearized eigenvalue problem

{Δξ[a(s)2u(s)bv(s)(1u(s)2+kv(s))(1+mu(s)+u(s)2+kv(s))2]ξ+bu(s)(1+mu(s)+u(s)2)(1+mu(s)+u(s)2+kv(s))2η=μξ,xΩ,Δη[c2v(s)+du(s)(1+mu(s)+u(s)2)(1+mu(s)+u(s)2+kv(s))2]ηdv(s)(1u(s)2+kv(s))(1+mu(s)+u(s)2+kv(s))2=μη,xΩ,ξ=η=0,xΩ(9)

has a unique eigenvalue μ* and Re(μ*) < 0 with algebra multiplicity one.

Let {εi > 0} and {di > 0} be sequences which converge to 0 by i → ∞. Owing to a = ã + a′(0)s + O(s2), we can get the sequence {εi > 0} and {ai} yield ai ∈ (ãεi, ã) and si → 0 as i → ∞. It is easy to see that (ui, vi) is one solution of (2). Hence, the corresponding linearized problem (9) can be written in the following form:

Li(ξiηi)=μi(ξiηi)andLi=(Mi11Mi12Mi21Mi22),

where (ξi, ηi) ≢ (0, 0) and

Mi11=Δ[ai2uibvi(1ui2+kvi)(1+mui+ui2+kvi)2],Mi12=bui(1+mui+ui2)(1+mui+ui2+kvi)2,Mi21=dvi(1ui2+kvi)(1+mui+ui2+kvi)2,Mi22=Δ[c2vi+dui(1+mui+ui2)(1+mui+ui2+kvi)2].

Observe that, as i → ∞,Li(ξiηi) converges to

L0(ξη)=(Δξ(a~bθc1+kθc)ξ00Δη(c2θc)η).

It follows that 0 is a simple eigenvalue of L0 with the corresponding eigenfunction (ξ, η)T = (Φ, 0)T. Meanwhile, all the other eigenvalues of L0 are positive and keep away from 0. Furthermore, using perturbation theory [31, 33], for large i, the operator Li has a unique eigenvalue μi which is close to zero. Moreover, all other eigenvalues of Li have positive real parts and keep away from 0. Because μi is simple real eigenvalue which tends to zero, we denote the corresponding eigenfunction by (ξi, ηi)T which satisfies (ξi, ηi) → (Φ, 0) as i → ∞.

Now we claim that Reμi < 0 for large i. Multiplying Φ on the first equation of Li(ξi,ηi)T = μi(ξi,ηi)T and integrating over Ω, we obtain

ΩΦΔξiΩ(ai2uibvi(1ui2+kvi)(1+mui+ui2+kvi)2)Φξi+Ωbui(1+mui+ui2)Φηi(1+mui+ui2+kvi)2=ΩμiΦξi.(10)

Multiplying two sides of the first equation of (2) with (a, u, v) = (ai, ui, vi) by ξi and integrating, we obtain

ΩξiΔuiΩ(aiuibvi1+mui+ui2+kvi)uiξi=0.

Taking ui = siΦ + O(si2) into the above equation we have

ΩΦΔξiΩξiΦ(aiuibvi1+mui+ui2+kvi)+O(si2)=0.(11)

By combining (10) and (11), we obtain

Ω[1bvi(m+2ui)(1+mui+ui2+kvi)2]uiΦξi+Ωbui(1+mui+ui2)Φηi(1+mui+ui2+kvi)2=ΩμiΦξi.

Notice that (ui, vi) = (siΦ+O(si2),θc+siΨdi+O(si2)), where Ψdi stands for Ψ defined in Remark 3.2. Thus, dividing the above equation by si and taking the limit, we get

limiμisi=Ω(1bmθc(1+kθc)2)Φ3/ΩΦ2<0,

which results in Reμi < 0 for large i. Hence, our claim is illustrated.

Finally, using similar method as in [25], we will demonstrate the remaining part of Theorem 4.7. In order to apply reduction to absurdity, we suppose that there exists a unique coexistence state (ũ,) for (2), then (ũ, ) must be bifurcated from (0, θc), since there exists a coexistence state near ã. Thus, (ũ, ) is non-degenerate, and the corresponding linearized problem has a unique eigenvalue μ̃ with algebra multiplicity one which satisfies Reμ̃ < 0. The above fact implies that IF′ (ũ, ) is invertible and does not have property α on W(ũ,), hence index(F, (ũ, )) = (−1)1 = −1 by Lemma 2.2(ii). Thus, according to Lemma 2.3-2.4 and the fixed point index theory, we get

1=indexW(F,D))=indexW(F,(0,0))+indexW(F,(θa,0))+indexW(F,(0,θc))+indexW(F,(u~,v~))=0+0+11=0,

which leads to a contradiction. Thus, the proof is completed. □

Remark 4.8

The proof of Theorem 4.7 implies that the multiplicity of (2) can be obtained easily as m ≥ 0. Due to a′(0) < 0 for a sufficiently small d, and Ω(1bmθc(1+kθc)2)Φ3<0, it is easy to see that a = a(s) ∈ (λ1, ã). From Theorem 2.5(i), we know that there is no coexistence state of (2) if aλ1(bθc(1+mθa+θa2+kθc)) as m ≥ 0, c > λ1. As a result, we have demonstrated that there exist at least two coexistence states for a ∈ (a*, ã) and some a(λ1(bθc(1+mθa+θa2+kθc)),λ1(bθc1+kθc)).

Finally, we study the stability of any positive solutions (if exists) as the parameter a, b, c belong to some domain.

Theorem 4.9

If a > λ1, c > λ1, then there exists some sufficiently small > 0 such that any coexistence state of (2) (if exists) is non-degenerate and stable for b.

Proof

Let be a positive solution of the following equation

Δv=v(bvdθa1+mθa+lθa2+kv)xΩ,v|Ω=0.

Hence, we need only demonstrate that the linearized problem of (2) has no the real part of eigenvalue. By reduction to absurdity, we suppose that the positive solution (ui, vi) of (2) is either non-degenerate or unstable. Suppose the sequence bi meets bi → 0 with i ≥ 1, then there exists μi such that Re μi ≤ 0, and the corresponding eigenfunction (ξi, ηi) ≠ (0, 0) with ∥ξi2 + ∥ηi2 = 1 satisfying the following linearized problem

{Δξi[a2uibvi(1ui2+kvi)(1+mui+ui2+kvi)2]ξi+bui(1+mui+ui2)(1+mui+ui2+kvi)2ηi=μξi,xΩ,Δηi[c2vi+dui(1+mui+ui2)(1+mui+ui2+kvi)2]ηidvi(1ui2+kvi)(1+mui+ui2+kvi)2=μηi,xΩ,ξi=ηi=0,xΩ(12)

Multiplying two sides of (12) with ξi,ηi), respectively, and integrating on Ω, it follows from Divergence theorem that

μi=Ω|ξi|2dxΩ[a2uibvi(1ui2+kvi)(1+mui+ui2+kvi)2]|ξi|2dx+Ωbui(1+mui+ui2)(1+mui+ui2+kvi)2ηiξi¯dx+Ω|ηi|2dxΩ[c2vi+dui(1+mui+ui2)(1+mui+ui2+kvi)2]|ηi|2dxΩdvi(1ui2+kvi)(1+mui+ui2+kvi)2ξiηi¯dx,

where ξ̄i and η̄i are the complex conjugates of ξi and ηi. According to the above equation, we can deduce that Re(μi) and Im(μi) are bounded, so we suppose that μiμ with Re(μ) ≤ 0. Meanwhile, (ξi, ηi) → (ξ, η), bi → 0 as i → ∞. Then (12) converges to

{Δξ(a2θa)ξ=μξ,xΩ,Δη[c2v~+dθa(1+mθa+θa2)(1+mθa+θa2+kv~)2]ηidv~(1θa2+kv~)(1+mθa+θa2+kv~)2=μη,xΩ.ξ=η=0,xΩ(13)

It follows that μ must be real number with μ ≤ 0. Suppose ξ ≢ 0, one can deduce that μ is an eigenvalue of the equation

Δϕ+(a+2θa)ϕ=μϕ,xΩ,ϕ|Ω=0,

Then μ ≥ λ1(−a + 2θa ), since λ1(−a + 2θa ) > λ1(−a + θa ) = 0, hence μ ≤ 0, a contradiction.

Suppose ξ ≡ 0, then η ≢ 0. By the second equation of (13), we get

Δη[c2v~+dθa(1+mθa+θa2)(1+mθa+θa2+kv~)2]ηi=μη,xΩ.η|Ω=0.

Observe that η ≢ 0, we obtain μ=λ1[c+2v~dθa(1+mθa+θa2)(1+mθa+θa2+kv~)2].Setg(x,v)=cv+dθa(1+mθa+θa2+kv~), then gv < 0. Hence,

λ1(g(x,v~))<λ1(g(x,v~)λ1(v~gv(x,v~)))=λ1[c+2v~dθa(1+mθa+θa2)(1+mθa+θa2+kv~)2]=μ.

Notice that λ1(−g(x, )) = 0, so μ > 0, a contradiction. □

5 Conclusion

This paper considers dynamic behavior of the prey-predator model with ratio-dependent Monod-Haldane response function under homogeneous Dirichlet boundary conditions. We come to the following conclusion. Firstly, the sufficient and necessary conditions on existence and non-existence of coexistence states of (2) are proved by the fixed point index theory, see Theorems 2.5, 2.6 and 2.7, which determine the conditions for the coexistence of two species of organisms. Secondly, taking a as a main bifurcation parameter, the structure of global bifurcation curve on coexistence states is established by global bifurcation theorem and property of principal eigenvalue, see Theorems 3.1 and 3.3, which show the global coexistence state of two species of organisms by controlling the change of the parameter a. Finally, the stability of coexistence states is obtained by the eigenvalue perturbation theory, see Theorems 4.5 and 4.9; the multiplicity of coexistence states to (2) is obtained when a ∈ (a*, ã) by the fixed point index theory, see Theorem 4.7, which show that two species of organisms have two coexistence states when the parameter satisfying some condition.

Acknowledgement

The work was partially supported by the National Natural Science Youth Fund of China (61102144); The Natural Science Foundation of Shaanxi Province (2016JM6041); The Shaanxi Province Department of Education Fund (14JK1353); The Shandong Provincial Natural Science Foundation of China (No.ZR2015AQ001); Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST (2014TDJH102); The principal fund of Xi’an Technological University (XAGDXJJ17028).

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About the article

Received: 2017-10-31

Accepted: 2018-04-25

Published Online: 2018-06-14


Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 623–635, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0060.

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© 2018 Feng et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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