# Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Authors: Xiaozhou Feng; Changtong Li; Hao Sun; Yuzhen Wang
Journal: Complexity
 (2021)

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## Abstract

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey (λ1Ω<a<λ1Ω0), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey (a>λ1Ω0), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.


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## Body

## 1. Introduction

The population dynamics system is the basic model to study the spatial and the temporal structures of the biological population. It is used to describe the dynamic distribution of population density produced by the interaction of the species in the ecosystem or the surrounding environment. In particular, the interaction of the predator-prey system is one of the basic structures in complex ecosystems, and such models have been widely studied (see [1–17]) under the uniform condition of the space, in other words, all coefficients of the model are positive. Because the natural environment of most species is spatially heterogeneous, biologists and mathematicians believe that the inhomogeneity of the spatial environment has a significant impact on the dynamic behavior of the biological population system, which is confirmed by the biologist C. B. Huffaker’s biological experiment. To investigate the effect of spatial heterogeneity on the dynamic behavior of the biological population system, a natural way is to replace the constant in the model with a function containing spatial variables. Therefore, we will study a predator-prey system with spatial degeneracy and B-D-type response function [1, 2].(1)−Δu=au−bxu2−cuvmu+kv+1,x∈Ω,t>0,−Δv=dv−v2+euvmu+kv+1,x∈Ω,t>0,u=v=0,x∈∂Ω,where Ω is the outward unit normal vector of the boundary ∂Ω in ℜNN>1. u and v are the densities of prey and predator, respectively. a and d are the intrinsic growth rate of prey u and predator v, respectively. c and e are capturing rate to predator and conversion rate of prey captured by a predator, respectively. m denotes the saturation coefficient, and kv is the inhibition of functional response function on behalf of predators. The parameters a,c,m,k are assumed to be only positive constants, while d can be negative. Let Ω0 be a subset of Ω. Assume that bx≡0 in Ω, and bx>0 in Ω\Ω0, where bx is a function dependent on space variables x, which means that two species live in a heterogeneous environment.When the space environment is homogeneous, that is,bx is a positive number in Ω, system (1) is reduced to the constant-coefficient system, and the predator-prey system with Beddington–DeAngelis-type functional response has been studied by many scholars from different perspectives [3–5].In this paper, the different behaviors of system (1) will be mainly discussed under the condition of spatial heterogeneity. If bx is a positive function in Ω, then there is no essential difference between the model and previous research. If bx degenerates partially into zero within Ω and m=k=0, the research results of system (1) show that the behavior of solutions has changed substantially. More concretely, when m=k=0, system (1) exhibits some fixed constant α∗ such that a<α∗; then the behavior of the system (1) is similar to the case of bx as the normal number. For a≥α∗, the behavior of system (1) has undergone an essential change, which means that even if the mortality of predators is very large, the system (1) still has a stable positive solution. Moreover, the asymptotic behavior of the coexistence solution of the equilibrium state on the system (1) was discussed in detail in references [10, 18] as d⟶∞.For the case wherem>0,k>0, there is no relevant research result. Therefore, the main purpose of this paper is to investigate the change rule of solutions if bx partially degenerates to zero in Ω and m>0,k>0. Is there a fundamental change in the behavior of system (1)? Are these changes the same as the case of m=k=0? Answers to these questions are detailed in the “Conclusion” section.This paper is organized as follows. In Section2, some notations and important lemmas used in previous papers is given, including the generalized comparison principle and the convergence property of the semitrivial solution ua. In Section 3.1, the system (1) with the weak growth rate on the prey (λ1Ω<a<λ1Ω0) is studied; a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter d>0 are obtained by local bifurcation theory, and the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. In Section 3.2, the system (1) with the strong growth rate on the prey (a>λ1Ω0) is investigated; a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity; and then we prove that the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In Section 4, some discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are listed.


## 2. Preliminaries

The main purpose of this section is to give some important lemmas and notations used in later papers.In this paper, we always suppose thatbx is a nonnegative function in Ω; moreover, there exists a subset Ω0⊂Ω such that bx≡0,x∈Ω¯0 and bx>0,x∈Ω¯∖Ω¯0. The research of this paper is dependent on the above assumption and is not suitable for the case ∂Ω0∩∂Ω≠0.Obviously,v=0 satisfies the second equation of system (1). For this case, u satisfies the following logistic equation:(2)−Δu=au−bxu2,x∈Ω,u=0,x∈∂Ω.According to the result of [19], we know that Equation (2) has a unique zero solution if a∈λ1Ω,λ1Ω0; Equation (2) exhibits the unique zero solution denoted by ua if a∉λ1Ω,λ1Ω0. Moreover, the mapping a⟶uax is strictly monotone increasing with a∈λ1Ω,λ1Ω0.Next, letλ1O be the principal eigenvalues of the operator −Δ under the Dirichlet boundary condition on region O. For convenience, we introduce the notation λ1Oϕ, which stands for the principal eigenvalue of the following eigenvalue problem:(3)−Δu+ϕu=λu,x∈O,u=0,x∈∂O.Under the sense of these signs,λ1O=λ1O0, By the conclusion of [19], in the sense of the norm L∞Ω, we obtain that if a⟶λ1Ω, then ua⟶0. If a⟶λ1Ω0,(4)ua⟶∞,x∈Ω¯0,ua⟶Uλ1Ω0,x∈Ω¯∖Ω¯0,where ua represents the minimal positive solution of the following boundary value blow-up problem:(5)−ΔU=aU−bxU2,x∈Ω∖Ω¯0,U|∂Ω=0,U|∂Ω0=∞,where U|∂Ω0=∞ implies limdx,∂Ω0⟶0ux=∞.In a word, ifa∈λ1Ω,λ1Ω0, then system (1) exhibits a unique semitrivial solution ua,0. For a∉λ1Ω,λ1Ω0, system (1) does not exhibit such a semitrivial solution.Ifu=0, then v satisfies the following logistic equation:(6)−Δv=dv−v2,x∈Ω,v=0,x∈∂Ω.By [20], if d≤λ1Ω, this equation does not have a positive solution. If d>λ1Ω, this equation has a unique positive solution denoted by θd. Hence, if d>λ1Ω, then system (1) has a unique semitrivial solution 0,θd.To end this paper, we need to introduce some results. First, we introduce the following generalized comparison principle.Lemma 1.
(see [19]). If u1,u2 are two order continuous derivable positive functions in Ω∖Ω¯0, and satisfy(7)Δu1+au1−bxu1p≤0≤Δu2+au2−bxu2p,x∈Ω∖Ω¯0,Bu1≥Bu2,x∈∂Ω,limdx,∂Ω0⟶0u2−u1≤0,then u1,u2 satisfy u1≥u2 in Ω¯∖Ω¯0. Next, let’s introduce a very useful lemma, which plays a key role in proving a priori estimates of positive solutions.Lemma 2.
(see [10]). Suppose un∈C2Ω satisfies(8)−Δun≤aun,x∈Ω,un|∂Ω=0,un≥0,un∞=1,wherea is a positive constant. Then there exists a function u∞∈L∞Ω∩H01Ω such that un weakly converges to u∞ with the sense of H01Ω norm; un strongly converges to u∞ with the sense of norm LpΩ,u∞∞=1.In a paper [21], López-Gómez and Sabina de Lis have proved that dua/da converges uniformly to ∞ with a⟶λ1Ω0 on any compact subset of Ω0. Next, we demonstrate the following result for the case of a⟶λ1Ω.Lemma 3.
LetΦ1x be the corresponding eigenfunction to λ1Ω and maxΩ¯Φ1x=1. If a⟶λ1Ω in Ω¯, then(9)uaxa−λ1Ω⟶∫ΩΦ12dx∫ΩbxΦ13dxΦ1x,x∈Ω¯.Proof.Letu^a=Ω¯a/ua2, then u^a2=1 and u^a satisfies(10)−Δu^a=au^a−bxuau^a,x∈Ω,u^a=0,x∈∂Ω.By multiplying two sides of Equation (10) by u^a and integrating over Ω by parts, we get(11)∫Ω∇u^2dx=a∫Ωu^a2dx−∫Ωbxuau^a2dx≤a∫Ωu^a2dx.Hence, asa⟶λ1Ω, u^a is uniformly bounded in H01Ω; it follows that u^a weakly converges to u^ depending on the norm of H01Ω, and u^a strongly converges to u^ depending on the norm of L2Ω. Since u^a2=1, we know that u^2=1, then u^≥0≠0. We chose any function ϕ∈H01Ω, and by multiplying two sides of Equation (10) by ϕ and integrating over Ω by parts, we obtain(12)∫Ω∇u^a∇ϕdx=a∫Ωu^aϕdx−∫Ωbxuau^aϕdx.Leta⟶λ1Ω, then ∫Ω∇u^∇ϕdx=λ1Ω∫Ωu^ϕdx, which implies that u^ is a weak solution of the following equation:(13)−Δu^a=λ1Ωu^,x∈Ω,u^=0,x∈∂Ω.According to Harnack’s inequality,u^>0,x∈Ω, then u^=Φ1. Thanks to the regularization theory, u^a⟶Φ1 with a⟶λ1Ω under the norm of C1Ω¯. Equation (10) can be rewritten as follows:(14)−Δu^a=au^a−bxua2u^a2,x∈Ω,u^a=0,x∈∂Ω.By multiplying two sides of the above equation byΦ1 and integrating over Ω by Green’s formula, we obtain(15)∫Ωa−λ1Ωu^Φ1dx=∫Ωbxua2u^a2Φ1dx.Leta⟶λ1Ω, then ua2/a−λ1Ω=∫ΩΦ12dx/∫ΩbxΦ13dx. Thus, as a⟶λ1Ω, we have(16)uaxa−λ1Ω=uaxua2ua2a−λ1Ω⟶∫ΩΦ12dx∫ΩbxΦ13dxΦ1x,x∈Ω¯.This completes the proof.


## 3. Global Bifurcation Structure

In this section, we will discuss the global bifurcation structure of system (1) and give sufficient conditions for the coexistence of the two species. Let a be fixed, we will investigate the global bifurcation of system (1) by dividing two different cases as follows:(17)iλ1Ω<a<λ1Ω0,iia>λ1Ω0.At the same time, unless specifically explained, parametersc,e,m,k>0 are fixed. The parameter d will be considered as the bifurcation parameter. Next, we will discuss two cases depending on the value range of a by two subsections.


### 3.1. Weak Growth Rate on the Prey (λ1Ω<a<λ1Ω0)

For anyd>0, system (1) has two semitrivial solutions ua,0 and 0,θd. Hence, system (1) has two semitrivial positive curves as follows: (18)Γu=d,ua,0:−∞<d<∞,Γv=d,ua,θd:λ1Ω<d<∞.To get our conclusion, we first have to give a priori estimates on any positive steady-state solutions by the comparison principle as follows.Lemma 4.
Suppose thatλ1Ω<a<λ1Ω0. Any positive solution u,v of system (1) satisfies(19)0<u<ua,θd<v<θd+e/m<d+em.whereua is the unique solution of Equation (2). Moreover, if system (1) has a positive solution, then the parameter d satisfies(20)d>λ1Ω−em,a>λ1Ωcθdmuax∞+kθd+1.Proof.
Notice that the first equation of system (1), it is easy to get(21)−Δu=au−bxu2−cuvmu+kv+1<au−bxu2.
It follows from the comparison principle that0<u<ua, where ua is the unique solution of (2). Similarly, according to the second equation of (1), we directly get(22)−Δv=dv−v2+euvmu+kv+1<d+emv−v2,(23)−Δv=dv−v2+euvmu+kv+1>dv−v2.
Combining with the comparison principle, we obtain(24)θd<v<θd+e/m<d+em.
If system (1) has a positive solution, we easily get(25)d=λ1Ωv−eumu+kv+1>λ1Ω−eumu+kv+1>λ1Ω−em,(26)a=λ1Ωbxu+cvmu+kv+1>λ1Ωcvmu+kv+1>λ1Ωcθdm∥uax∥∞+kθd+1.
Next, two local bifurcation solution branches depending on the bifurcation parameterd>0 are obtained by local bifurcation theory as follows. Let w=ua−u,X=u∈W2,pΩ:u=0,x∈∂Ω,Y=LpΩ. The operator F:ℜ×X×X⟶Y×Y can be defined as follows:(27)Fd,w,v=Δw+aw−2bxuaw+bxw2+cua−wvmua−w+kv+1Δv+dv−v2+eua−wvmua−w+kv+1.
Next, we prove thatd,w,v=d^,0,0 is a local bifurcation point of system (1), where d^=λ1Ω−eua/mua+1. For simplicity, let A=mua−w+kv+1,B=vkv+1,C=ua−wmua−w+1,D=mua−w2kv+1+kv+1. By direct calculation, we obtain(28)Fw,vd,w,v=Δ+a−2bxua+2bxw−cBA2cCA2−eBA2Δ+d−2v+eCA2,Fw,vd,w,vϕφ=Δϕ+ϕa−2bxua+2bxw+cCφ−BϕA2Δφ+φd−2v+eCφ−BϕA2,Fdd,w,v=0v,Fdw,vd,w,vϕφ=0φ,Fw,vw,vd,w,vϕφ2=2bxϕ2−2cDϕφ−mBϕ2+kCφ2A3−2φ2−2eDϕφ−mBϕ2+cCφ2A3.
Ford,w,v=d^,0,0, it is easy to get NFw,vd^,0,0=spanϕ1,φ1, where ϕ1,φ1 satisfies(29)−Δϕ+aϕ−2bxuaϕ−cuaϕmua+1=0,x∈Ω,−Δφ+d^φ+v2+euaφmua+1=0,x∈Ω,ϕ=φ=0,x∈∂Ω.
Sinced^=λ1Ω−eua/mua+1, we can choose φ1>0. On the other hand, ua is a positive solution of Equation (2), which demonstrates that −Δ−a+2bxua is a positive operator, and ϕ1=−Δ−a+2bxua−1−euaφ1/mua+1>0. The range can be represented as RFw,vu1,0,0=f,g∈Y2:∫Ωgxφ1dx=0, since ∫Ωφ12dx>0, we have Fdw,vu1,0,0ϕ1,ψ1=0,ψ1∉RFw,vd,0,0. Hence, by the local bifurcation theory [20], we obtain the following result on the local bifurcation solution.Theorem 1.
Suppose thatλ1Ω<a<λ1Ω0, the local positive bifurcation solution set that bifurcates from the point d^,ua,0 of system (1) forms a smooth curve(30)Γ1=d^s,ua−u1s,v1s:s∈0,δ,whered^0=λ1Ω−eua/mua+1,u1s=sϕ1x+os,v1s=sφ1x+os. By calculating, we get(31)d^′0=−Fw,vw,vd^,0,0ϕ1,φ12,l12Fdw,vd^,0,0ϕ1,φ1,l1=∫Ωφ13dx+e∫Ωϕ1φ12+cuaφ13/mua+12dx∫Ωφ12dx>0,wherel1 is a linear functional in space Y2, defined as follows.Similarly, we can prove that(32)f,g,l1=∫Ωgxφ1xdx,is also ad^,0,θd^ local bifurcation point of system (1), where d˜ is determined uniquely by a=λ1Ωcθd^/kθd^+1.Setχ=v−θd, define the operator G:ℜ×X×y⟶Y×Y by(33)Gd,u,χ=Δu+au−bxu2−cuχ+θdmu+kχ+θd+1Δχ+dχ−χ2−2χθd+euχ+θdmu+kχ+θd+1.LetA1=mu+kχ+θd+1,B1=umu+1,C1=χ+θdkχ+θd+1,D1=kχ+θd2mu+1+mu+1. By directly calculating the Fréchet derivative of the operator, we get(34)Gu,wμ,u,w=Δ+a−2bxu−cC1A12−cB1A12eC1A12Δ+d−2χ−2θd+eB1A12.It follows that(35)Gu,χd,u,χϕφ=Δϕ+aϕ−2bxuϕ−cC1ϕ+B1φA12Δφ+dφ−2χφ−2θdφ+eC1ϕ+B1φA12,Gdd,u,χ=−cθd′B1A12χ−2χθd′+eθd′B1A12,Gdu,χd,u,χϕφ=cθd′−D1ϕ+2kB1φA13φ−2θd′φ+eθd′−D1ϕ+2kB1φA13,Gu,χ2d,u,χϕφ2=−2bxϕ2+2cmϕ2C1−ϕφD1+kφ2B1A132eϕφD1−mϕ2C1−kB1φ2A13.Ford,u,χ=d˜,0,0, it is easy to get NGu,χd˜,0,0=spanϕ2,φ2, where ϕ2,φ2 satisfies(36)Δϕ+aϕ−cθd˜kθd˜+1=0,x∈Ω,Δφ+d˜φ−2θd˜φ+eθd˜kθd˜+1=0,x∈Ω,ϕ=φ=0,x∈∂Ω.Sincea=λ1Ωcθd˜/kθd˜+1, we can choose φ2>0. On the other hand, −Δ−d˜+2θd˜ is a positive operator, and φ2=−Δ−u2+2θd˜−1cθd˜ϕ2/kθd˜+1>0. The range can be represented as RGu,χd˜,0,0=f,g∈Y2:∫Ωfxϕ2xdx=0, since ∫Ω−cθd˜ϕ22/kθd˜+1dx<0, it follows that(37)Gdu,χd˜,0,0ϕ2,φ2=−cθd˜′ϕ2kθd˜+12,φ2−2θd˜′φ2−eθd˜′φ2kθd˜+12∉RGu,χu2,0,0.Thus, according to the local bifurcation theory [22], we obtain similar result as follows.Theorem 2.
Suppose thatλ1Ω<a<λ1Ω0, the local bifurcation branch near d˜,0,θd˜ of system (1) forms a smooth curve(38)Γ2=d˜s,u2s,θd+v2s:s∈0,δ,whered˜0=d˜,u2s=sϕ2x+os,v2s=sφ2x+os. By calculating, we get(39)d˜0=−Gu,χu,χd˜,0,0ϕ2,φ22,l22Gdu,χd˜,0,0ϕ2,φ2,l2=∫Ωbxϕ2+cϕ2/kθd+12ϕ22dx−∫Ωcmθdϕ23/kθd+12dx−∫Ωcmθd′ϕ22/kθd+12dx,wherel2 is a linear functional in space Y×Y, defined as follows: (40)f,g,l2=∫Ωfxϕ2xdx.Next, we use the modified global bifurcation theorem [23] to prove the global bifurcation structure of the system (1) under the condition of the weak growth rate on the prey.Theorem 3.
Suppose thatλ1Ω<a<λ1Ω0, then the positive solution set of system (1) forms a bounded smooth curve Γ that connects Γ1 and Γ2 and satisfies(41)projuΓ=d∗,d∗ord∗,d∗,whered∗=d^,d˜≤d∗>∞. Moreover, the bifurcation direction of Γ1 at point λ1Ω−eua/mua+1,ua,0 is supercritical (d^′0>0). If 0≤m<m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is supercritical (d^′0>0); if m>m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is subcritical (d˜′0<0), where m0 is determined by(42)m0=∫Ωbxϕ2+cϕ2/kθd+12ϕ22dx∫Ωcθdϕ23/kθd+12dx.Proof.
The proof of this theorem is similar to Theorem4 below; the detailed proofs are omitted here.


### 3.2. Strong Growth Rate on the Prey (a>λ1Ω0)

Comparing with the weak growth rate on the prey, there exists only one semitrivial solution curveΓ2 for the system (1) under the strong growth rate on the prey. In this case, d,u,v=d˜,0,θd˜ is still a local bifurcation point. The positive solution near d˜,0,θd˜ of system (1) forms a smooth curve Γ2. By the method of Lemma 4, we can prove that d>λ1Ω−e/m if system (1) has positive solutions. Next, we establish a boundary result of any positive solutions of system (1) when d is bounded.Lemma 5.
Ifa>λ1Ω0,dn≤M, then there exists a positive constant C that does not depend on n such that any positive solution of system (1) satisfies(43)un∞+vn∞≤C.Proof.
Sincedn≤M, it follows that(44)−Δvn≤M+emvn−vn2.
Hence,un∞≤θM+e/m≤M+e/m. Suppose the conclusion of the lemma does not hold. Then for d=dn, there exists some positive solution sequence un,vn of system (1) such that un∞⟶∞ with n⟶∞.
Setu^n=un/un∞. The first equation of un system (1) implies −Δu^n≤λu^n. Following Lemma 2, u^n weakly converges to u^ in H01Ω and strongly converges to u^ in LpΩ, and v^n∞=1. Similarly, let v^n=vn/vn∞. The first equation of system (1) implies(45)−Δv^n≤M+emv^n.
Hence,v^n weakly converges to v^ in H01Ω and strongly converges to v^ in LpΩ, and v^n∞=1.
Next, it turns out thatu^ is almost zero in Ω∖Ω0. According to Lemma 1, it is easy to see that un≤Ua in Ω∖Ω0. Hence, u^n is uniformly bounded on an arbitrary subset of Ω¯∖Ω¯0. Thanks to un∞⟶∞, so u^n is uniformly convergent to 0 on any subset of Ω¯∖Ω¯0. Thus, u^ is almost zero in the Ω¯∖Ω¯0. And because ∂Ω0 is smooth enough, then u^∈H01Ω.
According to the first equationun of system (1), we have(46)−Δu^n=au^n−bxun∞u^n2−cu^nmu^n∞+kvn+1,x∈Ω,u^n=0,x∈∂Ω.
By multiplying two sides of the above equation byϕ and integrating over Ω, where the support set of the function ϕ is Ω0, we obtain(47)∫Ω0∇u^n∇ϕdx=a∫Ω0u^nϕdx−∫Ω0cu^nvnϕmu^n∞vn+kvn+1dx.
Lettingn⟶∞, we get(48)∫Ω0∇u^∇ϕdx=λ∫Ω0u^ϕdx.
But asn⟶∞,(49)∫Ωu^nvnϕmun∞u^n+kvn+1dx≤vn∞mun∞ϕLΩ01⟶0.
Hence,u^≥0 is a weak solution to the following problem:(50)−Δu^=λu^,x∈Ω0,u^=0,x∈∂Ω0.
Thus, by using Harnack’s inequality, we know thatu^>0 or u^≡0 in Ω0. If u^>0, then a=λ1Ω0; this is in contradiction with a>λ1Ω0. If u^≡0, then u^≡0 in Ω; this is in contradiction with u^n∞=1. Therefore, it follows that vn∞ is uniformly bounded by the antievidence method.Theorem 4.
Suppose thata>λ1Ω0, then the positive solution set of system (1) forms an unbounded smooth curve Γ that extends Γ2 to ∞ by d. If 0≤m<m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is supercritical (d^′0>0); if m>m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is subcritical (d^′0<0), where m0 is defined in Theorem 3.Proof.
In order to apply the global bifurcation theorem of [23] better, we define the following mapping H:ℜ×R×R⟶Y×Y by(51)Hd,u,v=uv−−Δ−1au−bxu2−cuvmu+kv+1dv−v2+euvmu+kv+1.
According to the regularization theory of elliptic equations and the Sobolev embedding theorem, the second term of the mappingH is a compact mapping. Moreover, the nonnegative solution of the system (1) is equivalent to the zero point of the mapping Hd,u,v.
The Fréchet derivative of the mappingHd,u,v at u,v=0,θd is given as follows:(52)Hu,vd,0,θd=I−−Δ−1a−cθdkθd+10eθdkθd+1d−2θd.
The corresponding adjoint operator is recorded asHu,v∗d,0,θd, that is,(53)Hu,v∗d,0,θd=I−a−cθdkθd+1−Δ−1eθdkθd+1−Δ−10d−2θd−Δ−1.
By standard calculation, we obtain(54)NHu,vd˜,0,θd˜=spanϕ2,φ2,NHu,v∗d˜,0,θd˜=span−Δϕ2,0.
Thanks to the global bifurcation theorem, the local bifurcation curveΓ2 can be extended to a smooth global curve Γ, and(55)Γ2⊂Γ⊂d,u,v∈ℜ×X×X∖d˜,0,θd˜:Hd,u,v=0.
Moreover, according to Theorem 6.4.3 in document [22], the global curve Γ must satisfy one of the following three alternatives:aΓ is unbounded in ℜ×X×Xb There exists a constant d≠d˜ such that dd,0,θd∈Γc There exists a function d,ϕ,φ∈ℜ×Z∖0,θd such that d,ϕ,φ∈Γwhere Z is the complementary space of NHu,vd˜,0,θd˜ and satisfies(56)Z=f,g∈X×X:∫Ωfϕ2dx=0.
The positive one can be defined as follows:(57)P=w∈W2,PΩ:w>0,x∈Ω,∂w/∂n<0,x∈∂Ω.
Next, we proveΓ∖d˜,0,θd˜⊂ℜ×P×P. Applying reduction to absurdity, we assume that Γ∖d˜,0,θd˜⊈ℜ×P×P. There exists a sequence(58)dj,uj,vjj=1∞⊂Γ∖d˜,0,θd˜∩ℜ×P×P,
such that(59)limj⟶∞dj,uj,vj=d∞,u∞,v∞,
inℜ×X×X, where d∞,u∞,v∞∈Γ∖d˜,0,θd˜∩ℜ×∂P×P, and u∞,v∞ is a nonnegative solution of system (1) corresponding to d=d∞. According to the strong maximum principle [22, 24], we know that u∞,v∞ satisfies one of the three following alternatives:(i)
u∞≡0,x∈Ω,v∞≡0,x∈Ω(ii)
u∞>0,x∈Ω,v∞≡0,x∈Ω(iii)
u∞≡0,x∈Ω,v∞>0,x∈Ω
Similar to the method to Theorem1 in [24], we can find a contradiction to each of these cases. Therefore, we have(60)Γ∖d˜,0,θd˜⊂ℜ×P×P.
Since equation (60) holds, it is obviously impossible for case (b) to occur. Because ϕ2>0 in Ω¯; obviously, case (c) does not impossible hold. Therefore, the global curve Γ just belongs to case (a), combined with Lemma 3.2; the projection of the global curve Γ on the d axis contains 0,∞.
The specific analysis and calculation of the direction of local bifurcation branches have been given in the previous section, so we have completed the proof of this theorem.


## 3.1. Weak Growth Rate on the Prey (λ1Ω<a<λ1Ω0)

For anyd>0, system (1) has two semitrivial solutions ua,0 and 0,θd. Hence, system (1) has two semitrivial positive curves as follows: (18)Γu=d,ua,0:−∞<d<∞,Γv=d,ua,θd:λ1Ω<d<∞.To get our conclusion, we first have to give a priori estimates on any positive steady-state solutions by the comparison principle as follows.Lemma 4.
Suppose thatλ1Ω<a<λ1Ω0. Any positive solution u,v of system (1) satisfies(19)0<u<ua,θd<v<θd+e/m<d+em.whereua is the unique solution of Equation (2). Moreover, if system (1) has a positive solution, then the parameter d satisfies(20)d>λ1Ω−em,a>λ1Ωcθdmuax∞+kθd+1.Proof.
Notice that the first equation of system (1), it is easy to get(21)−Δu=au−bxu2−cuvmu+kv+1<au−bxu2.
It follows from the comparison principle that0<u<ua, where ua is the unique solution of (2). Similarly, according to the second equation of (1), we directly get(22)−Δv=dv−v2+euvmu+kv+1<d+emv−v2,(23)−Δv=dv−v2+euvmu+kv+1>dv−v2.
Combining with the comparison principle, we obtain(24)θd<v<θd+e/m<d+em.
If system (1) has a positive solution, we easily get(25)d=λ1Ωv−eumu+kv+1>λ1Ω−eumu+kv+1>λ1Ω−em,(26)a=λ1Ωbxu+cvmu+kv+1>λ1Ωcvmu+kv+1>λ1Ωcθdm∥uax∥∞+kθd+1.
Next, two local bifurcation solution branches depending on the bifurcation parameterd>0 are obtained by local bifurcation theory as follows. Let w=ua−u,X=u∈W2,pΩ:u=0,x∈∂Ω,Y=LpΩ. The operator F:ℜ×X×X⟶Y×Y can be defined as follows:(27)Fd,w,v=Δw+aw−2bxuaw+bxw2+cua−wvmua−w+kv+1Δv+dv−v2+eua−wvmua−w+kv+1.
Next, we prove thatd,w,v=d^,0,0 is a local bifurcation point of system (1), where d^=λ1Ω−eua/mua+1. For simplicity, let A=mua−w+kv+1,B=vkv+1,C=ua−wmua−w+1,D=mua−w2kv+1+kv+1. By direct calculation, we obtain(28)Fw,vd,w,v=Δ+a−2bxua+2bxw−cBA2cCA2−eBA2Δ+d−2v+eCA2,Fw,vd,w,vϕφ=Δϕ+ϕa−2bxua+2bxw+cCφ−BϕA2Δφ+φd−2v+eCφ−BϕA2,Fdd,w,v=0v,Fdw,vd,w,vϕφ=0φ,Fw,vw,vd,w,vϕφ2=2bxϕ2−2cDϕφ−mBϕ2+kCφ2A3−2φ2−2eDϕφ−mBϕ2+cCφ2A3.
Ford,w,v=d^,0,0, it is easy to get NFw,vd^,0,0=spanϕ1,φ1, where ϕ1,φ1 satisfies(29)−Δϕ+aϕ−2bxuaϕ−cuaϕmua+1=0,x∈Ω,−Δφ+d^φ+v2+euaφmua+1=0,x∈Ω,ϕ=φ=0,x∈∂Ω.
Sinced^=λ1Ω−eua/mua+1, we can choose φ1>0. On the other hand, ua is a positive solution of Equation (2), which demonstrates that −Δ−a+2bxua is a positive operator, and ϕ1=−Δ−a+2bxua−1−euaφ1/mua+1>0. The range can be represented as RFw,vu1,0,0=f,g∈Y2:∫Ωgxφ1dx=0, since ∫Ωφ12dx>0, we have Fdw,vu1,0,0ϕ1,ψ1=0,ψ1∉RFw,vd,0,0. Hence, by the local bifurcation theory [20], we obtain the following result on the local bifurcation solution.Theorem 1.
Suppose thatλ1Ω<a<λ1Ω0, the local positive bifurcation solution set that bifurcates from the point d^,ua,0 of system (1) forms a smooth curve(30)Γ1=d^s,ua−u1s,v1s:s∈0,δ,whered^0=λ1Ω−eua/mua+1,u1s=sϕ1x+os,v1s=sφ1x+os. By calculating, we get(31)d^′0=−Fw,vw,vd^,0,0ϕ1,φ12,l12Fdw,vd^,0,0ϕ1,φ1,l1=∫Ωφ13dx+e∫Ωϕ1φ12+cuaφ13/mua+12dx∫Ωφ12dx>0,wherel1 is a linear functional in space Y2, defined as follows.Similarly, we can prove that(32)f,g,l1=∫Ωgxφ1xdx,is also ad^,0,θd^ local bifurcation point of system (1), where d˜ is determined uniquely by a=λ1Ωcθd^/kθd^+1.Setχ=v−θd, define the operator G:ℜ×X×y⟶Y×Y by(33)Gd,u,χ=Δu+au−bxu2−cuχ+θdmu+kχ+θd+1Δχ+dχ−χ2−2χθd+euχ+θdmu+kχ+θd+1.LetA1=mu+kχ+θd+1,B1=umu+1,C1=χ+θdkχ+θd+1,D1=kχ+θd2mu+1+mu+1. By directly calculating the Fréchet derivative of the operator, we get(34)Gu,wμ,u,w=Δ+a−2bxu−cC1A12−cB1A12eC1A12Δ+d−2χ−2θd+eB1A12.It follows that(35)Gu,χd,u,χϕφ=Δϕ+aϕ−2bxuϕ−cC1ϕ+B1φA12Δφ+dφ−2χφ−2θdφ+eC1ϕ+B1φA12,Gdd,u,χ=−cθd′B1A12χ−2χθd′+eθd′B1A12,Gdu,χd,u,χϕφ=cθd′−D1ϕ+2kB1φA13φ−2θd′φ+eθd′−D1ϕ+2kB1φA13,Gu,χ2d,u,χϕφ2=−2bxϕ2+2cmϕ2C1−ϕφD1+kφ2B1A132eϕφD1−mϕ2C1−kB1φ2A13.Ford,u,χ=d˜,0,0, it is easy to get NGu,χd˜,0,0=spanϕ2,φ2, where ϕ2,φ2 satisfies(36)Δϕ+aϕ−cθd˜kθd˜+1=0,x∈Ω,Δφ+d˜φ−2θd˜φ+eθd˜kθd˜+1=0,x∈Ω,ϕ=φ=0,x∈∂Ω.Sincea=λ1Ωcθd˜/kθd˜+1, we can choose φ2>0. On the other hand, −Δ−d˜+2θd˜ is a positive operator, and φ2=−Δ−u2+2θd˜−1cθd˜ϕ2/kθd˜+1>0. The range can be represented as RGu,χd˜,0,0=f,g∈Y2:∫Ωfxϕ2xdx=0, since ∫Ω−cθd˜ϕ22/kθd˜+1dx<0, it follows that(37)Gdu,χd˜,0,0ϕ2,φ2=−cθd˜′ϕ2kθd˜+12,φ2−2θd˜′φ2−eθd˜′φ2kθd˜+12∉RGu,χu2,0,0.Thus, according to the local bifurcation theory [22], we obtain similar result as follows.Theorem 2.
Suppose thatλ1Ω<a<λ1Ω0, the local bifurcation branch near d˜,0,θd˜ of system (1) forms a smooth curve(38)Γ2=d˜s,u2s,θd+v2s:s∈0,δ,whered˜0=d˜,u2s=sϕ2x+os,v2s=sφ2x+os. By calculating, we get(39)d˜0=−Gu,χu,χd˜,0,0ϕ2,φ22,l22Gdu,χd˜,0,0ϕ2,φ2,l2=∫Ωbxϕ2+cϕ2/kθd+12ϕ22dx−∫Ωcmθdϕ23/kθd+12dx−∫Ωcmθd′ϕ22/kθd+12dx,wherel2 is a linear functional in space Y×Y, defined as follows: (40)f,g,l2=∫Ωfxϕ2xdx.Next, we use the modified global bifurcation theorem [23] to prove the global bifurcation structure of the system (1) under the condition of the weak growth rate on the prey.Theorem 3.
Suppose thatλ1Ω<a<λ1Ω0, then the positive solution set of system (1) forms a bounded smooth curve Γ that connects Γ1 and Γ2 and satisfies(41)projuΓ=d∗,d∗ord∗,d∗,whered∗=d^,d˜≤d∗>∞. Moreover, the bifurcation direction of Γ1 at point λ1Ω−eua/mua+1,ua,0 is supercritical (d^′0>0). If 0≤m<m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is supercritical (d^′0>0); if m>m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is subcritical (d˜′0<0), where m0 is determined by(42)m0=∫Ωbxϕ2+cϕ2/kθd+12ϕ22dx∫Ωcθdϕ23/kθd+12dx.Proof.
The proof of this theorem is similar to Theorem4 below; the detailed proofs are omitted here.


## 3.2. Strong Growth Rate on the Prey (a>λ1Ω0)

Comparing with the weak growth rate on the prey, there exists only one semitrivial solution curveΓ2 for the system (1) under the strong growth rate on the prey. In this case, d,u,v=d˜,0,θd˜ is still a local bifurcation point. The positive solution near d˜,0,θd˜ of system (1) forms a smooth curve Γ2. By the method of Lemma 4, we can prove that d>λ1Ω−e/m if system (1) has positive solutions. Next, we establish a boundary result of any positive solutions of system (1) when d is bounded.Lemma 5.
Ifa>λ1Ω0,dn≤M, then there exists a positive constant C that does not depend on n such that any positive solution of system (1) satisfies(43)un∞+vn∞≤C.Proof.
Sincedn≤M, it follows that(44)−Δvn≤M+emvn−vn2.
Hence,un∞≤θM+e/m≤M+e/m. Suppose the conclusion of the lemma does not hold. Then for d=dn, there exists some positive solution sequence un,vn of system (1) such that un∞⟶∞ with n⟶∞.
Setu^n=un/un∞. The first equation of un system (1) implies −Δu^n≤λu^n. Following Lemma 2, u^n weakly converges to u^ in H01Ω and strongly converges to u^ in LpΩ, and v^n∞=1. Similarly, let v^n=vn/vn∞. The first equation of system (1) implies(45)−Δv^n≤M+emv^n.
Hence,v^n weakly converges to v^ in H01Ω and strongly converges to v^ in LpΩ, and v^n∞=1.
Next, it turns out thatu^ is almost zero in Ω∖Ω0. According to Lemma 1, it is easy to see that un≤Ua in Ω∖Ω0. Hence, u^n is uniformly bounded on an arbitrary subset of Ω¯∖Ω¯0. Thanks to un∞⟶∞, so u^n is uniformly convergent to 0 on any subset of Ω¯∖Ω¯0. Thus, u^ is almost zero in the Ω¯∖Ω¯0. And because ∂Ω0 is smooth enough, then u^∈H01Ω.
According to the first equationun of system (1), we have(46)−Δu^n=au^n−bxun∞u^n2−cu^nmu^n∞+kvn+1,x∈Ω,u^n=0,x∈∂Ω.
By multiplying two sides of the above equation byϕ and integrating over Ω, where the support set of the function ϕ is Ω0, we obtain(47)∫Ω0∇u^n∇ϕdx=a∫Ω0u^nϕdx−∫Ω0cu^nvnϕmu^n∞vn+kvn+1dx.
Lettingn⟶∞, we get(48)∫Ω0∇u^∇ϕdx=λ∫Ω0u^ϕdx.
But asn⟶∞,(49)∫Ωu^nvnϕmun∞u^n+kvn+1dx≤vn∞mun∞ϕLΩ01⟶0.
Hence,u^≥0 is a weak solution to the following problem:(50)−Δu^=λu^,x∈Ω0,u^=0,x∈∂Ω0.
Thus, by using Harnack’s inequality, we know thatu^>0 or u^≡0 in Ω0. If u^>0, then a=λ1Ω0; this is in contradiction with a>λ1Ω0. If u^≡0, then u^≡0 in Ω; this is in contradiction with u^n∞=1. Therefore, it follows that vn∞ is uniformly bounded by the antievidence method.Theorem 4.
Suppose thata>λ1Ω0, then the positive solution set of system (1) forms an unbounded smooth curve Γ that extends Γ2 to ∞ by d. If 0≤m<m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is supercritical (d^′0>0); if m>m0, then the bifurcation direction of Γ2 at point d˜,0,θd˜ is subcritical (d^′0<0), where m0 is defined in Theorem 3.Proof.
In order to apply the global bifurcation theorem of [23] better, we define the following mapping H:ℜ×R×R⟶Y×Y by(51)Hd,u,v=uv−−Δ−1au−bxu2−cuvmu+kv+1dv−v2+euvmu+kv+1.
According to the regularization theory of elliptic equations and the Sobolev embedding theorem, the second term of the mappingH is a compact mapping. Moreover, the nonnegative solution of the system (1) is equivalent to the zero point of the mapping Hd,u,v.
The Fréchet derivative of the mappingHd,u,v at u,v=0,θd is given as follows:(52)Hu,vd,0,θd=I−−Δ−1a−cθdkθd+10eθdkθd+1d−2θd.
The corresponding adjoint operator is recorded asHu,v∗d,0,θd, that is,(53)Hu,v∗d,0,θd=I−a−cθdkθd+1−Δ−1eθdkθd+1−Δ−10d−2θd−Δ−1.
By standard calculation, we obtain(54)NHu,vd˜,0,θd˜=spanϕ2,φ2,NHu,v∗d˜,0,θd˜=span−Δϕ2,0.
Thanks to the global bifurcation theorem, the local bifurcation curveΓ2 can be extended to a smooth global curve Γ, and(55)Γ2⊂Γ⊂d,u,v∈ℜ×X×X∖d˜,0,θd˜:Hd,u,v=0.
Moreover, according to Theorem 6.4.3 in document [22], the global curve Γ must satisfy one of the following three alternatives:aΓ is unbounded in ℜ×X×Xb There exists a constant d≠d˜ such that dd,0,θd∈Γc There exists a function d,ϕ,φ∈ℜ×Z∖0,θd such that d,ϕ,φ∈Γwhere Z is the complementary space of NHu,vd˜,0,θd˜ and satisfies(56)Z=f,g∈X×X:∫Ωfϕ2dx=0.
The positive one can be defined as follows:(57)P=w∈W2,PΩ:w>0,x∈Ω,∂w/∂n<0,x∈∂Ω.
Next, we proveΓ∖d˜,0,θd˜⊂ℜ×P×P. Applying reduction to absurdity, we assume that Γ∖d˜,0,θd˜⊈ℜ×P×P. There exists a sequence(58)dj,uj,vjj=1∞⊂Γ∖d˜,0,θd˜∩ℜ×P×P,
such that(59)limj⟶∞dj,uj,vj=d∞,u∞,v∞,
inℜ×X×X, where d∞,u∞,v∞∈Γ∖d˜,0,θd˜∩ℜ×∂P×P, and u∞,v∞ is a nonnegative solution of system (1) corresponding to d=d∞. According to the strong maximum principle [22, 24], we know that u∞,v∞ satisfies one of the three following alternatives:(i)
u∞≡0,x∈Ω,v∞≡0,x∈Ω(ii)
u∞>0,x∈Ω,v∞≡0,x∈Ω(iii)
u∞≡0,x∈Ω,v∞>0,x∈Ω
Similar to the method to Theorem1 in [24], we can find a contradiction to each of these cases. Therefore, we have(60)Γ∖d˜,0,θd˜⊂ℜ×P×P.
Since equation (60) holds, it is obviously impossible for case (b) to occur. Because ϕ2>0 in Ω¯; obviously, case (c) does not impossible hold. Therefore, the global curve Γ just belongs to case (a), combined with Lemma 3.2; the projection of the global curve Γ on the d axis contains 0,∞.
The specific analysis and calculation of the direction of local bifurcation branches have been given in the previous section, so we have completed the proof of this theorem.


## 4. Conclusion

In this work, the effect of spatial degradation on the steady-state problem of a predator-prey system with B-D functional response has been investigated. By studying the bifurcation structure of the system, sufficient conditions for the coexistence of two species are obtained. However, due to the phenomenon of space degradation, there are some phenomena that are different from the traditional predator-prey system (i.e., all the coefficients of the system are normal or positive). Specifically, for the traditional predator-prey system, the two species do not coexist when the predator’s own growth rated is large, but for the system (1), the growth rate d of the predator is very large, and the two species can still coexist in the common habitat Ω, even if the predator growth rate d is very large. The results show that spatial degradation has a significant effect on the steady-state behavior of the system. On the other hand, our results are essentially different from the results of the Lotka–Volterra predator-prey system (m=k=0). Specifically, under the case of the strong growth rate on the prey, for any large d, the two species of the system (1) can coexist, but for the smaller d≤λ1Ω−e/m, the two species cannot coexist. However, for the Lotka–Volterra predator-prey system, no matter how small the predator’s self-growth rate d, the two species can continue to coexist, but for the larger d, the two species cannot coexist. The result reveals that the B-D reaction function also has a significant effect on the steady-state behavior of the system.



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*Source: 9970255-2021-06-21.xml*