# Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

Authors: Xiaozhou Feng; Kenan Shi; Jianhui Tian; Tongqian Zhang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied whenb and d are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is sufficiently large.


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## Body

## 1. Introduction

The dynamic relationship between predator and their prey is one of dominant themes in ecology and mathematical ecology. Some models have been studied from various views and obtained many good results (see [1–16] and the references therein). In this paper, we are concerned with the predator-prey model with Dinosaur functional response under the homogeneous Dirichlet boundary conditions as follows: (1)∂u∂t-Δu=ua-u-bve-ku,x∈Ω,∂v∂t-Δv=vc-v+due-ku,x∈Ω,u=v=0,x∈∂Ωux,0=u0x,vx,0=v0x,x∈∂Ω,where Ω is a bounded domain in Rn(n≥1) with smooth boundary ∂Ω and u, v stand for the densities of prey and predators, respectively. a, b, c, d, k are positive constants. a and c denote prey intrinsic growth rate and predator intrinsic growth rate and then decrease to zero when u→∞, as follows:(2)dduue-ku>0,0≤u<1k,dduue-ku<0,u>1k.The Dinosaur reaction term is an improvement or simplification of the Ivlev-type reaction term, and the change on the species density of prey better than Ivlev-type functional response h(1-e-ku) can be explained. It is easy to see that the Dinosaur reaction term describes prey focus on the fight against predators when the species density of prey is large enough, so as to achieve better defense or disguise itself. To see more biological significance of systems with Ivlev-type functional responses, one can resort to [17–28] and their contents. The research on existence and uniqueness of the limit cycle of a predator-prey model with Ivlev response can be found in [22, 23]. The permanence and existence and stability of positive periodic solutions of the model were studied in [24–26]. The spatial pattern formation of the model was investigated by using Hopf bifurcation in [27, 28]. Some dynamical behavior analysis of the Ivlev response predator-prey systems was established in [17–21]. However, the researches on system (1) are very few. Hence, this paper mainly aims at establishing the existence, bifurcation, and multiplicity of positive solutions on the corresponding elliptic equations to system (1).The research on the steady-states in reaction-diffusion model is the hot point question [1–4, 9, 10, 13, 16]. In the present paper, we study the steady-state problem corresponding to (1), with the specific form as follows: (3)-Δu=ua-u-bve-ku,x∈Ω,-Δv=vc-v+due-ku,x∈Ω,u=v=0,x∈∂Ω.Motivated by the papers [1–4], in the present paper, we mainly consider the positive solution of (3). In Section 2, we give some basic results and calculate the fixed point index by the fixed point index theory. In Section 3, we apply the results obtained in Section 2 to study the existence of positive solutions. In Section 4, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of (3) and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when b and d are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is sufficiently large.


## 2. Preliminaries

In this section, we mainly calculate the fixed point index by the fixed point index theory, in the following, and set up some definitions which are used in last paper.LetE be a Banach space. W⊂E is called a wedge if W is a closed convex set αW⊂W for all α≥0. For any y∈W, define (4)Wy=x∈E:∃r=rx>0,s.t.y+rs∈W,Sy=x∈W¯y:-x∈W¯y,and we always assume that E=W-W¯. Let T:Wy→Wy be a compact linear operator on E. We say that T has property α on W¯y if there exist t∈(0,1) and ω∈W¯y∖Sy such that ω-tTω∈Sy. Suppose that F:W→W is a compact operator for the fixed point y0∈W, and F is Fréchet differentiable at y0. Set L=F′(y0) as the Fréchet differentiable of F at y; hence L:W¯→W¯.Lemma 1 (see [29, 30]).
Assuming thatI-L is invertible on E, then one has(i)
L has property α on W¯, and then indexw(F,y0)=0.(ii)
L does not have property α on W¯, and then indexw(F,y0)=(-1)σ, where σ is the sum of algebra multiplicities of the eigenvalues of L which are greater than one.Letq(x)∈L∞, k>0 denote the principal eigenvalue of the following problem: (5)-Δϕ+qxϕ=λϕ,x∈Ω,ϕ=0,x∈∂Ω,and, by λ1(q(x)), it follows from [3, 4] that λ1(q(x)) is strictly increasing in the case that q1(x)≤q2(x) and q1(x)≠q2(x) implies λ1(q1(x))<λ1(q2(x)). For convenience we denote λ1(0) by λ1. Let r(T) be the spectral radius of linear operator T on Banach space.Lemma 2 (see [24, 30]).
Suppose thatq(x)∈C1, and P is large enough such that ∀x∈Ω, -q(x)+P>0. Then the following results hold:(i)
λ1(q1(x))<0⇒r[(-Δ+P)-1(-q(x)+P)]>1.(ii)
λ1(q1(x))>0⇒r[(-Δ+P)-1(-q(x)+P)]<1.(iii)
λ1(q1(x))=0⇒r[(-Δ+P)-1(-q(x)+P)]=1.Next, we present some a priori estimate by maximum principle which is similar to [3, 4]; its proof will be omitted.Theorem 3.
Any positive solution(u,v) of (3) has an a priori boundary; that is, (6)ux≤a,vx≤c+dke.Consider the following equation:(7)-Δθ=θρ-θ,x∈Ω,θ=0,x∈∂Ω;it is easy to see that (7) has unique positive solution denoted by θρ when ρ>λ1,k. Hence, if a>λ1, then (3) has a semitrivial solution (θa,0); if c>λ1, (3) has a semitrivial solution (0,θc).Now, some notations are introduced as follows:(1)
E:c(Ω¯)⊕c(Ω¯), here c(Ω¯)≔{ϕ∈c(Ω¯),x∈∂Ω,v=0}.(2)
W:K⊕K, here K≔{ϕ∈CΩ¯:0≤ϕx,x∈Ω}.(3)
D:{(u,v)∈E:u≤a+1,v≤v+d/ke+1}.(4)
D′≔intD∩W.Define a compact operatorF:D′→W by the following form: (8)Fu,v=-Δ+P-1ua-u-bve-ku+Puvc-v+due-ku+Pu,where P is sufficiently large constant such that P>max⁡{a+b(c+d/ke),c+d/ke}. It follows that (3) has a positive solution in W if and only if F has a positive fixed point in D′. Hence, (0,0),(θa,0),(0,θc) are the only isolated fixed points of F if a,c>λ1. Therefore, indexw(F,(0,0)),indexw(F,(θa,0)),indexw(F,(0,θc)) are well-defined, and the indices can be calculated, similar to Lemmas  2.3-2.4 in [3]; we can directly give Lemmas 4 and 5 as follows.Lemma 4.
Supposea>λ1.(i)
indexw(F,D′)=1.(ii)
indexw(F,(0,0))=0.(iii)
Ifc>λ1(-dθae-kθa), then indexw(F,(θa,0))=0.(iv)
Ifc<λ1(-dθae-kθa), then indexw(F,(θa,0))=1.Lemma 5.
Supposec>λ1. (i) If a>λ1(bθc), then indexw(F,(0,θc))=0; (ii) if a<λ1(bθc), then indexw(F,(0,θc))=1.


## 3. Existence of Positive Solutions

In the section, we apply the result of Lemmas4 and 5 obtained in Section 2 to study the existence and nonexistence of positive solutions of (3).Theorem 6.
(i) Ifc>λ1, a>λ1(bθc), then (3) has at least one positive solution.
(ii) Ifc<λ1, then (3) has a positive solution if and only if a>λ1 and c>λ1(-dθae-kθa).Proof.
(i) According to Lemmas4 and 5 and the additivity property of the index, we have (9)indexwF,D′-indexwF,0,0-indexwF,θa,0-indexwF,0,θc=1.Hence, (3) has at least one positive solution.
(ii) Ifc<λ1, then (3) has no solution taking the form (0,v)(v>0). Supposing a>λ1 and c>λ1(-dθae-kθa), note that c<λ1, and by Lemma 5, we can get the following: (10)indexwF,D′-indexwF,0,0-indexwF,θa,0=1.Hence, (3) has at least one positive solution. So the sufficiency is proved. Conversely, suppose that (u,v) is a positive solution of (3); then a<λ1, u<θa. This is because (u,v) satisfies (11)-Δv=vc-v+due-ku,x∈Ω,u=0,x∈∂Ω.It follows that 0=λ1(-c+v-due-ku)>λ1(-c-dθae-kθa). This completes the proof of Theorem 6.Theorem 7.
(i) Ifa≤λ1, then (3) has no positive solutions; if a≤λ1 and c≤λ1, then (3) has no nonnegative nonzero solution.
(ii) Ifc≤λ1, then the conditions a>λ1, c+d/ke>λ1 are necessary for the existence of positive solution of (3). In addition, as a<1/k, the condition c>λ1(-dθae-kθa) can take place of the necessary condition c+d/ke>λ1.
(iii) Ifc>λ1, then the conditions a>max⁡{λ1(bθc),λ1(bθc)e-kθa} are necessary for the existence of positive solutions of (3).Proof.
(i) Suppose that(u,v) is a positive solution of (3); then (u,v) satisfies (12)-Δu=ua-u-bve-ku≤aa-u,x∈Ω,u=0,x∈∂Ω,so we have a=λ1(u+bve-ku). According to the comparison principle of eigenvalues, we have a>λ1, a contradiction. Next, we suppose that (u,v) is a nonnegative nonzero solution of (3). If u≢0, v≡0, then a>λ1, a contradiction to the previous proof. Similarly, it is easy to see that c>λ1, when u≡0, v≢0, which derives a contradiction.
(ii) Suppose that(u,v) is a positive solution of (3); it follows that a>λ1, and then the positive semitrivial solution θa exists. Thanks to (13)-Δu=ua-u-bve-ku≤aa-u,x∈Ω,u=0,x∈∂Ω,u is a lower solution of (3), and due to the uniqueness of θa, u<θa. Thus, since v satisfies the problem (14)-Δv=vc-v+due-ku,x∈Ω,u=0,x∈∂Ωit follows from Lemma 2(iii) that (15)0=λ1-c+v-due-ku>λ1-c-dθae-kθa.Moreover, as a<1/k, due to u<θa≤a<1/k, by the second equation of (3), we get c>λ1(-dθae-kθa).
(iii) Supposing that(u,v) is a positive solution of (3), by the proof of (ii), θa exists and u≤θa. Similarly, the given condition c>λ1 can deduce the existence of the positive θc of (3), θc≤v. Similar to the proof of (i), we also obtain the fact that a=λ1(u+bve-ku) has the minimum at u=0, v=θc or u=θa, v=θc, which derives the desired result.
This completes the proof of Theorem7.Theorem 8.
If one of the following conditions holds, then (3) has no positive solutions:(i)
b≥eka and a≤c.(ii)
b<eka and a+d/ke≤beka(c+d/ke).Proof.
(i) Assume that there is a positive solution(u,v) of (3). Recalling the results of Theorem 3, as b≥eka and a≤c, we get(16)0=λ1-a+u+bve-ku≥λ1-c+v-due-ku-v+bve-ku+u+due-ku≥λ1-c+v-due-ku-1-be-kuv≥λ1-c+v-due-ku,which is a contradiction to Lemma 2(iii).
(ii) Assume that there is a positive solution(u,v) of (3). By Theorem 3, if b<eka and a+d/ke≤beka(c+d/ke), we have (17)0=λ1-a+u-bve-ku≥λ1-c+v-due-ku-a+c-v+u+bve-ku+due-ku≥λ1-c+v-due-ku-a+c-c+dke1-be-ka≥λ1-c+v-due-ku,which is a contradiction to Lemma 2(iii).
This completes the proof of Theorem8.


## 4. Bifurcation and Multiplicity

In this section, takinga as a main bifurcation parameter, we shall prove that (3) has at least two positive solutions when the parameters involved in (3) satisfy some ranges. In particular, the uniqueness of positive solutions is established when b and d are small enough. By the local bifurcation theory [6, 31] or Section  13 in [30], the branch of positive solutions of (3) bifurcates from (0,θc,a) (or (θa,0,c)) when c>λ1 (or a>λ1) is established. Now we give the following Lemma where the proof is similar to Theorem  3.8 in [3].Lemma 9.
(i) Ifc>λ1 and a≥λ1(bθc), then (3) has a positive solution which bifurcates from (0,θc,a) if and only if a=a~, and here a~≔λ1(bθc).
(ii) Ifa>λ1 and λ1(-dθae-kθa)≤c≤λ1, then (3) has a positive solution which bifurcates from (θa,0,c) if and only if c=c~, and here c~≔λ1(-dθae-kθa).More precisely, all positive solutions neara~≔λ1(bθc) are defined by (18)us,vs,as=sΦ+Os2,θc+sΨ+Os2,a~+a1s+Os2:0<s<σ,where Φ is the principal eigenfunction of a~, and it follows from the following problem: (19)-Δξ+a~-bθcξ=0,x∈Ω,-Δη+c-2θcη=-dθcξ,x∈Ω,ξ=η=0,x∈∂Ω,where Ψ=(-Δ+c-2θc)-1(dθcΦ).Remark 10.
In Section3, by Theorems 6 and 7, we can know the sufficient condition and necessary condition on the existence of positive solutions and find that there exists a gap between a>λ1(bθc) and a>λ1(bθc)e-kθa.Next, using the constanta as a main bifurcation parameter, we get the following theorem which establishes the multiplicity and stability results of positive solution for (3) in the gap.Theorem 11.
Supposec>λ1 and ∫Ω‍Φ3(1-bkθc)<0. There exists small enough ε>0 such that the local bifurcation of the positive solution (u(s),v(s)) bifurcates from (0,θc,a~) which is nondegenerate and unstable for a∈(a~-ε,a~) and 0<d≪1. In addition, (3) has at least two positive solutions.Proof.
Firstly, we prove that any positive solutions bifurcated from(0,θc) are nondegenerate and unstable. To complete this, we need only show that there exists a sufficiently small ε>0 such that, for a∈(a~-ε,a~), any positive solution (u(s),v(s)) of (3) is nondegenerate and the linearized eigenvalue problem (20)-Δξ-ξas-2us+bvse-kus+busvske-kus+buse-kusη=μξ,x∈Ω,-Δη-ξdvse-kus-dusvske-kus-c-2vs+duse-kusη=μη,x∈Ωhas a unique eigenvalue μ^ such that Re(μ^)<0 with multiplicity one. Suppose that {εn>0}, {dn}, {sn>0} are the sequences which converge to 0 as n→∞. Thanks to a=a~+a1s+O(s2), there exists the sequence {an} such that an∈(a~-εn,a~) as n→∞. Set (un,vn) as a solution of (3). So linearized problem (20) can be denoted by (21)-Δξn-ξnan-un-bvne-kun+bunvnke-kun+bune-kunηn=μnξn,x∈Ω,-Δηn-ξndnvne-kun-dnunvnke-kun-c-2vn+dnunke-kunηn=μnηn,x∈Ω,ξn=ηn=0,x∈∂Ω.Notice that, as n→∞, problem (21) converges to the following problem: (22)Δξ-a~-bθcξ=μξ,x∈Ω,-Δη-c-2θcη=μη,x∈Ω,ξ=η=0,x∈∂Ω,and since c=λ1(θc)<λ1(2θc), (22) has 0 as a simple eigenvalue with corresponding eigenfunction (Φ,0). Moreover, all the other eigenvalues are positive and stand apart from 0. Thus, it follows from perturbation theory [32] that problem (21) has a unique eigenvalue μn which is near to zero for large n. In particular, all the other eigenvalues of problem (21) have positive real parts and stand apart from 0. Note that μn is simple real eigenvalue which converges to zero and we can denote the corresponding eigenfunction (ξn,ηn) such that (23)limn→∞ξn,ηn⟶Φ,0.Now we prove that μn<0 for large n in following. Multiplying the first equation of (21) with Φ and integrating on Ω, we get (24)-∫ΩΔξnΦ-∫ΩΦξnan-2un-bvne-kun+bunvnme-kun+∫Ωbune-kunηnΦ=∫ΩμnξnΦ.Meanwhile, multiplying the first equation of (3) with (u,v)=(un,vn) by ξn and integrating, we obtain (25)-∫ΩΔunξn=∫Ωunξnan-un-bvne-kun,Thanks to un=snΦ+O(s2), adopting variational principle, by the above equation, we have (26)-∫ΩΔξnΦ=∫ΩΦξnan-un-bve-kun+Osn2.According to (24) and (26), we have (27)-∫ΩξnΦun1-bvnme-kun-∫Ωbune-kunηnΦ=∫ΩΦμξn+Osn2.Taking (u(s),v(s))=(sΦ+O(s2),θc+sΨ+O(s2)) into (27) and so dividing by sn and taking the limit, we obtain (28)limn→∞μnsn=∫ΩΦ31-bkθc∫ΩΦ2;recalling that ∫Ω‍Φ3(1-bmθc)<0, it is easy to see that μ<0 for large n. This proves our claim.
Next, in order to prove the existence of at least two positive solutions, we may use reduction to absurdity and suppose that (3) has a unique coexistence state (u^,v^); it follows from local bifurcation theory that solutions must be positive solutions bifurcated from (0,θc) near a~; moreover, (u^,v^) is nondegenerate and the corresponding linearized eigenvalue problem has a unique eigenvalue μ^ such that Re(μ^)<0 with multiplicity one. Thanking to the above facts, it is easy to see that I-F′(u^,v^) is invertible and does not have property α on W-, and it follows from Lemma 1(ii) that indexw(F,(u^,v^))=(-1)1=-1. Thus, applying Lemmas 4-5 and the additivity property of the index, we obtain (29)1=indexwF,D′=indexw0,0+indexwF,θa,0+indexwF,u^,v^=0+0+1-1=0,which derives a contradiction. The proof is completed.Remark 12.
Theorem6 gives the multiplicity of positive solutions when k≥0. Note that since ∫Ω‍Φ3(1-bmθc)<0 with a1<0 for d≪1, there exists a=a(s)∈(λ1,a~). Remark 10 shows that there is no positive solution of (1) if a>λ1(bθce-kθc). Hence, by establishing a bifurcation sketch map as in Figure 1, we can easily get the fact that there must be at least two coexistence states for a∈(a^,a~) and some a^∈(λ1(bθce-kθc),λ1(bθc)).Figure 1
Bifurcation sketch map.Theorem 13.
Suppose thatc>λ1, a>λ1+b(c+d/(ke)), and b(c+d/(ke))k<1. If (30)Γb2+Γak+12d2+2akbd+4bkc+dke≤4,where (31)Γ≔maxsupΩ¯θaθc,supΩ¯θc+d/keθa-bc+d/ke,then (3) has a unique positive solution.Proof.
We suppose that(u1,v1) and (u2,v2) are two positive solutions of (3). Due to c>λ1 and a>λ1+b(c+d/(ke)), it follows from the comparison argument for elliptic problem that we have the following results: (32)θa-bc+d/ke≤ui≤θa;θc≤vi≤θc+d/ke,i=1,2.
SettingA≔u1-u2; B≔v1-v2, according to system (3) and the differential mean value theorem, there exists ξ1,ξ2∈[u1,u2] such that A and B satisfy (33)-ΔA-Aa-u1-bv1e-ku1+u2A-u2bv2ke-kξ1A-e-ku1B=0,x∈Ω,-ΔB-Bc-v1+du1e-ku1+v2B-e-ku2-u1kekξ2v2dA=0,x∈Ω,A=B=0,x∈∂Ω.In view of the fact that (u1,v1) is a solution of (3), 0 is the principal eigenvalue for the following two eigenvalue problems: (34)-Δχ-χa-u1-bv1e-ku1=0,x∈Ω,χ=0,x∈∂Ω,-Δω-ωc-v1+du1e-ku1=0,x∈Ω,ω=0,x∈∂Ω.Hence, applying Rayleigh’s formula for the principal eigenvalue, it follows that (35)∫Ω∇A2-A2a-u1-bv1e-ku1≥0,∫Ω∇B2-B2c-v1+du1e-ku1≥0.Multiplying two equations of (33) by A and B, respectively, and integrating on Ω, we get the fact that (36)∫Ω1-v2bkekξ1u2A2+u2be-ku1+u1v2kde-kξ2-v2de-ku2AB+v2B2≤0.Thanks to 1-v2bke-kξ1≥1-b(c+d/(ke))k>0, thus (37)I≔u2be-ku1+u1v2kde-kξ2-v2de-ku22-41-v2bke-kξ1u2v2=u2be-ku12+u1ke-kξ2-e-ku22v22d2+2u2be-ku1u1ke-kξ2-e-ku2v2d-41-v2bke-kξ1u2v2≤u2be-ku12+u1ke-kξ2-e-ku22v22d2+2u1u2kbe-ku1+kξ2v2d+4u2v22bke-kξ1-4u2v2=u2v2u2v2be-ku12+u1ke-kξ2+e-ku2v2u2d2+2u1kbde-ku1+kξ2+4v2kbe-kξ1-4≤0.Hence, the integral in (36) has a nonnegative value, and then A,B≡0 which show the desired results.


## 5. Stability and Multiplicity of Positive Solutions

In this section, the multiplicity, stability, and some uniqueness of positive solutions of (3) are considered by using k as a parameter. Next, some lemmas to obtain the main results of this section are given. These lemmas will show that there exist some upper and lower solutions which do not depend on k and indicate the nondegeneracy at any solution of (3) with certain hypothesis. In the following, an asymptotic result is given firstly.Lemma 14.
Suppose thata>λ1, c>λ1. For any given small ε>0, there exists K(ε) such that for k>K(ε), (3) has at least one positive solution (u,v) which satisfies (38)θa-ε≤u≤θa,θc≤v≤θc+ε.Proof.
SetU_=(u_,v_)=(θa-ε,θc), U¯=(u¯,v¯)=(θa,θa+ε). It is easy to see that functions u(a-u-bve-ku) and v(c-v+due-ku) satisfy the Lipschitz continuous in {U_,U¯}. If we can show that U_ and U¯ are the upper and lower solutions of (3), respectively, then (3) has at least one positive solution (u,v)∈{U_,U¯} with k>K. Thanks to Chapter  10 Comparison Theorems and Monotonicity Methods in [30] for nonquasimonotone functions, it suffices to require that the following hold: (39)Δu¯+u¯a-u¯-bve-ku¯≤0,x∈Ω,Δv¯+v¯c-u¯-dee-ku≤0,x∈Ω,Δu_+u_a-u-bve-ku_≥0,x∈Ω,Δv_+v_c-v_-due-ku≥0,x∈Ω,ux,0=u0x,vx,0=v0x,x∈Ω.
The first and fourth inequalities in (39) are obvious. Next we check the second and third of (21), by the direct calculation as follows: (40)Δv¯+v¯c-u¯-due-ku=Δv¯+v¯c+ε+v¯+v¯-ε+bue-ku=v¯-ε+bue-ku,Δu_+u_a-u_-bve-ku_=Δu_+u_a-ε-u_+u_ε-bve-ku=u_ε-bve-ku.Therefore, for k≫1, the inequalities hold when (41)Δv¯+v¯c-v¯+due-ku≤0,Δu_+u_a-u_-bve-ku_≥0.Lemma 15.
Suppose thata>λ1, c>λ1.(i)
The positive solution of (3) meets (38) and (42)u,v⟶θa,θcask⟶∞.(ii)
There exists sufficiently largeK(ε) such that, for k>K(ε), the positive solution of (3) which meets (38) is nondegenerate and linearly stable.Proof.
(i) Ask→∞, it is clear to see that the operator F(u,v) defined in Section 2 converges to the following operator: (43)A^=-Δ+p-1ua-u+Puvc-v+Pv;therefore, (u,v)→(θa,θc).
(ii) Suppose that the result does not hold. Then we can obtaink→∞; there exists μn with Re(μn)<0 and (ξn,ηn)≠0 with ξnL22+ηnL22=1 such that(44)-Δξn-ξna-un-bvne-knun+bunvnkne-knun+bune-knunηn=μnξn,x∈Ω,-Δηn-ξndvne-knun-dunvnkne-knun-c-2vn+dune-knunηn=μnηn,x∈Ω,ξn=ηn=0,x∈∂Ω,where (un,vn) is a positive solution of (1) as k=kn which meets (38). By multiplying ξ¯ and η¯ to two equations of (3), respectively, and integrating on Ω, we get (45)μn=∫Ω∇ξn2-∫Ωξn2a-un-bvne-knun+bunvnkne-knun+∫Ωbune-knunηnξn¯+∫Ω∇ηn2-ξnη¯ndvne-knun-dunvnkne-knun-∫Ωc-2vn+dune-knunηn2,where ξn¯ and η¯n are the complex conjugates of ξn and ηn, respectively. By the above equation, it is easy to see that Im(μn) and Re(μn) are bounded. Then we can suppose that μn→μ and Re(μn)≤0. Since ξn and ηn are also bounded, so let ηn→η, ξn→ξ. Taking the limit on (44), we get (46)-Δξ-ξa-2θa=μξ,x∈Ω,-Δη-ηc-2θc=μη,x∈Ω,ξ=η=0,x∈∂Ω.It follows from (i) that μ must be real number with μ≤0. If ξ≠0 then μ is an eigenvalue of the following problem: (47)-Δξ-ξa-2θa=μξ,x∈Ω,ξ=0,x∈∂Ω.Due to λ1(-a+2θa)>λ1(-a+θa)=0, 0≥μ≥λ1(-a+2θa), which is a contradiction, so we get ξ≡0. Similarly, we also can prove η≡0, a contradiction to (ξn,ηn)≠0; the proof is complete.Next, by the conditiona>λ1(bθc), one can easily get the fact that a+λ1(bθc)/2>λ1(bθc); then there exists a sufficiently small σ>0 such that a+λ1(bθc)/2>λ1(bθc+σ), and then we can know that the problem (48)-Δω=ωa+λ1bθc2-bθc+σ-ω,x∈Ω,ω=0,x∈∂Ωhas a unique positive solution ω~ by the result of (7).Lemma 16.
Suppose thatc>λ1 and a>λ1(bθc). Then, for some σ, there exists a sufficiently large K(σ) such that the following result holds when K≥K(σ).(i)
u≥ω~; that is, ω~ is a lower solution for u.(ii)
A positive solution of (3) converges to (θa,θc) as k→∞. Meanwhile, the positive solution is nondegenerate and linearly stable.Proof.
(i) According to Theorem3 and the comparison argument, we show v≤θc+d/(ke); then there exists a sufficiently large K(σ) such that k≥K(σ) with d/(ke)<σ; hence (49)-Δu=ua-u-bve-ku≥ua+λ1bθc2-bθc+d/ke-u≥ua+λ1bθc2-bθc+σ-u.So the proof is complete.
(ii) Ask→∞, ω~ is a lower solution for u which does not depend on k; the proof of (ii) is similar to Lemma 15, so we omit it.Lemma 17.
Suppose thatc∈(λ1-d/(ke),λ1) and a>λ1+τ; here τ≔b(c+d/(ke)). Then there exists a sufficiently large K such that k>K; the following claims hold:(i)
u≥θa-τ.(ii)
The positive solution of (3) is nondegenerate and linearly stable.Proof.
(i) Applying the comparison argument for elliptic problem, we obtain0≤v≤θc+d/(ke)≤c+d/(ke), and so as k is large enough we get (50)-Δu=ua-u-bve-ku≥ua-bc+dke-u,which proves claim (i).
(ii) Letc=cn, where cn∈(λ1-d/(ke),λ1). Thanks to un≥θa-τn, where τn≔b(cn+d/(kne)), then knun∞→∞. Due to cn∈(λ1-d/(kne),λ1), hence (un,vn)→(θa,0). Since λ1-d/(kne)→λ1, so cn→λ1.
Setv^n=vn/vn∞; then we get (51)Δv^n+v^ncn-vn+due-knun=0,x∈Ω,v^n=0,x∈∂Ω.According to standard elliptic regularity theory and Sobolev embedding theorems, we may suppose that v^n→v^>0. Taking the limit, the above equation satisfies (52)-Δv^=λ1v^,x∈Ω;v^=0,x∈∂Ω,v^∞=0.Hence, v^ must be the principal eigenfunction corresponding to λ1 denoted by Φ1 and Φ1∞=0.
In the following, we prove that linearized eigenvalue problem (44) has no eigenvalue Re(μn)≤0 as n≫1. Taking n→∞, problem (44) converges to the following: (53)-Δξ-a-2θaξ,x∈Ω,-Δη-λ1η=μη,x∈Ωξ=η=0,x∈∂Ω.Obviously, for (53), 0 is a simple eigenvalue with its corresponding eigenfunction (ξ,η)=(0,Φ1). Applying the method of the proof in Theorem 11, it suffices to prove μn>0 for n≫1. Take an eigenfunction (ξn,ηn) such that (ξn,ηn)→(0,Φ1).
Applying the integral equation onΩ which can be established by multiplying vn to the second equation of (44) and integrating on Ω, we get (54)-∫ΩΔηnvn-∫Ωξnvndvne-knun-dunvnkme-knun-∫Ωc-2vn+dune-knunηnvn=∫Ωμnηnvn.Multiplying ηn to the second equation of (3) and using variational principle on Ω, we can deduce the following integral equation: (55)-∫ΩΔηnvn=∫Ωvnηncn-vn+dune-knun.By substituting vn=v^nvn∞ into the above two equations and taking the limit, we can get (56)Limn→∞μnvn∞=∫ΩΦ13∫ΩΦ12>0,which get the desired result.Theorem 18.
(i) Ifa∈(λ1,λ1(bθc)), c>λ1, then there exists a sufficiently large K=K(ε) such that (3) has at least two positive solutions for k>K(ε).
(ii) Ifa>λ1(bθc), c>λ1, then there exists a sufficiently large K=K(σ) such that (3) has a unique positive solution for k>K(ε).
(iii) Ifλ1(-dθae-mθa)<c<λ1 and a>λ1+τ, where τ≔b(c+d/(ke)), then there exists a sufficiently large K=K(σ) such that (3) has a unique positive solution for k>K(ε).Proof.
(i) Suppose the result is false; we can assume that (3) has a unique positive solution (u^,v^); then (u^,v^) must satisfy (38). As k is large enough, by Lemma 14, it is easy to see that I-F′(u^,v^) is invertible and has no property α on ω¯(u^,v^). Furthermore, F′ has no real eigenvalue larger than or equal to one. It follows from Lemma 1(ii) that indexw(F,(u^,v^))=(-1)0=1. Applying Lemmas 4-5 and the additivity property of the index, we get (57)1=indexwF,D′=indexwF,0,0+indexwF,θa,0+indexwF,0,θc+indexwF,u^,v^=0+0+1+1.Which derive a contradiction, so the result of (i) is established.
(ii) Suppose that (3) has many positive solutions with the hypothesis of (ii), thanks to the compactness of the operator F which has at most finitely many positive fixed points in the given region D′ which are denoted by (58)Ui=ui,vi,i=1,2,…,k.Similar to the proof of (i), it is easy to see that indexw(F,Ui)=(-1)0=1, i=1,2,…,k. Finally, applying Lemmas 16(ii) and 1(ii), along with the additivity property of the index, we get (59)k=∑i=1kindexwF,Ui=indexwF,D′+indexwF,0,0+indexwF,θa,0+indexwF,0,θc=1-0-0-0=1.Thus, the uniqueness is proved.
(iii) By-dθae-kθa≤d/(ke), we can get c>λ1-d/(ke). The proof of (iii) is similar to (ii); just the semitrivial solution (0,θc) does not exist in this case. So we omit it.Theorem 19.
If the assumption of Theorem18(i) holds, then as n→∞, either (un,vn)→(θa,θc) or (knun,vn)→(U,θc), where U is a positive solution of (60)-ΔU=Ua-bθce-U,x∈Ω,U=0,x∈∂Ω.Proof.
In the event thatknun∞→∞, it is easy to see that (un,vn)→(θa,θc) holds. Hence, we only investigate the case of knun∞<∞. Let knun=Un, it follows that (61)-ΔUn=Una-Unkn-bvne-Un,x∈Ω,-Δvn=vnc-vn+dUnkne-Un,x∈ΩUn=vn=0,x∈∂Ω.Taking n→∞, the solution (Un,vn) of the above problem converges to a nonzero solution of the following problem: (62)-ΔU=Ua-bve-U,x∈Ω,-Δv=vc-v,x∈ΩU=v=0,x∈∂Ω.Obviously, the result (knun,vn)→(U,θc) holds.Remark 20.
By Theorem19, we can understand that the exact number of positive solutions is dependent on the multiplicity of positive solutions with the same hypothesis in Theorem 18(i).


## 6. Conclusion

The positive solution of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions is investigated. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions on the predator-prey model with Dinosaur functional response are established. See Theorems6–8. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions to (3) and obtain the stability and multiplicity of the positive solution under certain conditions. See Theorem 11. Furthermore, the local uniqueness result is studied when b and d are small. See Theorem 13. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is large. See Lemmas 15–17 and Theorems 18-19.



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