# Structure and Stability of Steady State Bifurcation in a Cannibalism Model with Cross-Diffusion

Authors: Meijun Chen; Shengmao Fu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper deals with spatial patterns of a predator-prey crossdiffusion model with cannibalism. By applying the asymptotic analysis and Rabinowitz bifurcation theorem, we consider the local structure of steady state to the model and determine an explicit formula of the nonconstant steady state. Furthermore, the criteria of the stability/instability for the steady state with small amplitude are established.


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## Body

## 1. Introduction

Multicomponent system is widely existed in nature and engineering apllications, for instance, in combustion, chemical reactors, tumor growth, gas mixtures, and animal crowds. On the diffusive level, these systems can be described by crossdiffusion equations taking into account multicomponent diffusion and reaction [1]. Specifically, crossdiffusion is a phenomenon in which the concentration gradient of one species induces a flux of the other species. The possibility of crossdiffusion terms in multicomponent systems was proposed by Onsager and Fuoss [2], while Baldwin et al. [3] undertook the experimental verification of the existence of crossdiffusion and also observed that the crossdiffusion coefficients can be quite significant. Since then, various crossdiffusion mathematical models have been suggested to interpret and predict many interesting features of natural multicomponent dynamics [4–10].For example, Darcy’s law implies that the velocity is proportional to the negative pressure gradient, and the pressure is defined by a state equation imposed by the volume extension of the mixture. Druet and Jungel [9] considered the convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations. According to the idea that decomposes the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions, they proved the global-in-time existence of classical and weak solutions in a bounded domain with no-penetration boundary conditions. Jungel and Ptashnyk [10] considered crossdiffusion systems defined in a heterogeneous medium, where the heterogeneity is reflected in spatially periodic diffusion coefficients or by the perforated domain. By combining two-scale convergence and the boundedness-by-entropy method, they proved two-scale homogenization limits of parabolic crossdiffusion systems in a heterogeneous medium with no-flux boundary conditions. Practical applications of such approach would investigate the problem of reducing a heterogeneous material to a homogeneous, specifically, predicting the response of refractory concrete to high-temperature exposure in steel furnaces. In nature, the impact of heat and mass transfer processes become particularly evident at high temperatures, where the increased pressure in pores, large temperature gradients, and temperature-induced creep may lead to catastrophic service failures [11]. Benes et al. [12] discussed a nonlinear numerical scheme arising from the implicit time discretization of the Bazant–Thonguthai model (with crossdiffusion) for hygrothermal behavior of concrete at high temperatures. By theoretical analysis and numerical simulation, they found that the model reproduces well the rapid increase of pore pressure in wet concrete due to extreme heating. In particular, there are three different zones which can be predicted: one corresponds to elements failed due to spalling damage, the other one indicates the part of the structure in which the local strength is sufficiently high to sustain the pore pressure, but its stability is lost due the explosive spalling of the former region, and the last one shows the portion of the cross-section is still capable of transmitting stresses due to mechanical loading, which is thereby responsible for the structural safety during fire. This phenomenon is particularly meaningful for the safety assessment of concrete structures prone to thermally induced spalling.In fact, the rapid growth of the field of system biology has further contributed to interest in reaction-diffusion systems. Nowadays, crossdiffusion terms, which are inspired by the model, proposed by Shigesada et al. [13] in 1979 when they have studied the segregation of competing species, also have attracted wide attentions used in reaction-diffusion equations encountered in models from mathematical biology [14–21]. As we all know, the growth of biological population depends not only on time but also on spatial distribution. Spatial species interaction includes the self-diffusion and crossdiffusion. In this paper, we investigate a predator-prey system with both cannibalism and crossdiffusion in the form(1)ut−d1Δu=−u+v+auv+uw,x∈Ω,t>0,vt−Δd2v+d4vε+u2=bu−gv−uv,x∈Ω,t>0,wt−d3Δw=sw−rw2−euw,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,ux,0≥0,vx,0≥0,wx,0≥0,x∈Ω,where ut,vt,andwt are the biomass of an adult predator, a juvenile predator, and prey at time t. a,b,g,s,r,e,andε are positive constants. a denotes the cannibalism rate. The positive constants d1,d2,andd3 are diffusion coefficients, and d4 is the crossdiffusion coefficient. The initial values ux,0,vx,0,andwx,0 are nonnegative smooth functions which are not identically zero. Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω, and v is the outward unit normal vector of the boundary ∂Ω. The homogeneous Neumann boundary condition indicates that this system is self-contained with zero population flux across the boundary. For more details on the backgrounds of crossdiffusion models, one can see [21, 22].The corresponding kinetic system of (1) was introduced by Magnsson in [23]. In this ODE system, Magnusson found that cannibalism, which is used as a means of population control to prevent them from overbreeding [21–31], has a destabilizing effect. When the large prey carries capacity, the juvenile mortality rate is high and adult recruitment rate is low, and stability of the equilibrium will be lost through Hopf bifurcation, the appearance of which is caused by the increasing level of cannibalism. And sustained oscillations can set in for sufficiently high levels of cannibalism. After, Kaewmanee and Tang [32] optimized the results of Magnusson, they obtained that stability of the equilibrium is lost due to the increase of the cannibalism attack rate past a bifurcation point that depends on other parameters. Later, the stability, topological properties, and types of bifurcations of the ODE system have been studied more explicitly by Marlk and Pribylova [33]. The authors proved that both subcritical and supercritical bifurcations may take place and hence the limits’ cycle enclosing the stationary point does not need to be stable. Very recently, Zhang et al. [28] analyzed the effect of cannibalism and illustrated that the cannibalization rate can cause the local stability of the equilibrium changes from stable to unstable to stable, or from unstable to stable to unstable. Moreover, the positive equilibrium must be globally asymptotically stable if the cannibalization rate is large enough. They also obtained that supercritical/subcritical Hopf bifurcation can occur with the rates of two factors (the cannibalization and the benefit from cannibalism) as bifurcation parameters.Recently, Fu and Yang [34] considered the corresponding pure diffusion system of (1) (d4=0) and proved that the positive equilibrium in this system has the same stability properties when it is regarded as equilibrium of the ODE system. It is shown that the decisive factor of destabilization for semilinear reaction-diffusion system is still cannibalism. Therefore, they further discussed the following strongly coupled crossdiffusion system (1).Obviously, ifr≠0 and(2)br+gs−gr>0,s+eg+r1−ab>0,s1−ab+eg−b>0,then (1) has the unique positive equilibrium point u˜=u˜,v˜,w˜, where u˜=α+α2+4eβ/2e, v˜=bu˜/g+u˜,w˜=s−eu˜/r, and α=abr+s−r−eg,β=br+gs−gr. By the linearization analysis, Fu and Yang proved that positive equilibrium can undergo stability switch from stable in the ODE system and semilinear system to unstable in the crossdiffusion system if d4 is sufficiently large. This means that the decisive factor of destabilization for the positive equilibrium point in (1) is the crossdiffusion rate, and cannibalism is an auxiliary destabilizing force. Besides, by using the Leray–Schauder degree theory, they also obtained the existence of nonconstant positive steady states. For completeness, we introduce this existence result, which can be proved by similar arguments as Theorem 5 in [21].Theorem 1.
(see [34]). Let a,b,g,s,r,e,ε,d1,d2, and d3 be fixed positive constants such that 0<ε<ε0, ag<1, r>s, and A1 hold. If μ˜∈μn,μn+1n≥2 and the sum ∑i=2ndimEμi is odd, then there exists a positive constant d4∗ such that the steady state problem corresponding to (1) has at least one nonconstant positive solution for d4≥d4∗. Here, let μn and μn+1 be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition.Our main purpose in this paper is to describe in detail the local structure of the nonconstant steady states and discuss the stability and instability of bifurcation steady states.The rest of the paper is organized as follows. In Section2, the local bifurcation analysis is performed to examine the structure of bifurcating steady states. Furthermore, the stability and instability of bifurcating steady states with small amplitude will be given in Section 3; meanwhile, the paper ends with a brief discussion.


## 2. Local Structure and Formula of Steady State Bifurcation

The existence of nonconstant steady state of system (1) is established under the condition that μ˜∈μn,μn+1n≥2 and the sum ∑i=2ndimEμi is odd in [34]. In this section, using the asymptotic analysis and bifurcation theory similar to those in [35–38], we choose d4 as a bifurcation parameter and fix the rest of the parameter to explore the local structure of nonconstant steady states of (1) bifurcating from the constant steady state u˜ in one dimension.Before proceeding, we present some properties about the negative Laplace operator. Let0=μ1<μ2<μ3<⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition, and let Eμi be the eigenspace corresponding to μi in H1Ω. Let X be the closure of C1Ω¯3 in H1Ω3, ϕij:j=1,2,…,dimEμi be an orthonormal basis of Eμi, and Xij=cϕij:c∈ℝ3. Then,(3)X=⊕i=1+∞Xi,Xi=⊕j=1dimEμiXij.In particular, forΩ=0,l, it is well known that the problem(4)−ω″x=μωx,x∈0,l,ω′x=0,x=0,l,has a sequence of simple eigenvalues(5)μj=πjl2,j=0,1,2,…,whose corresponding eigenfunctions are given by(6)ωjx=1,j=0,cosπjxl,j>0.This set of eigenfunctions is an orthogonal basis inL20,l. For later use, we now define a Banach space X by(7)X=u,v,w:u,v,w∈C20,l,u′=v′=w′=0atx=0,l,equipped with usual C2 norm, and a Hilbert space Y(8)Y=L20,l×L20,l×L20,l,with the inner product(9)ρ1,ρ2,ρ3Y=u1,u2,u3L20,l+v1,v2,v3L20,l+w1,w2,w3L20,l,for ρi=ui,vi,wi,i=1,2,3.For the sake of simplify, we investigate the structure of nonconstant positive steady state of (1) in one-dimensional interval Ω=0,l, i.e., consider the associated elliptic problem:(10)−d1u″=−u+v+auv+uw,x∈0,l,−d2v+d4vε+u2″=bu−gv−uv,x∈0,l,−d3w″=sw−rw2−euw,x∈0,l,u′=v′=w′=0,x=0,l.Define the mapF:0,∞×X⟶Y as(11)Fd4,u=d1Δu−u+v+auv+uwΔd2v+d4vε+u2+bu−gv−uvd3Δw+sw−rw2−euw.Then,u=u,v,w is a solution of (10), equivalent to it is a zero-point of the map F. Clearly,(12)Fd4,u˜=0,d4>0.Notice that(13)fu,v≡vε+u2=fu˜,u˜+fuu˜,u˜u−u˜+fvu˜,u˜v−u˜+⋯=u˜ε+u˜2−2u˜2ε+u˜22u−u˜+1ε+u˜2v−u˜+⋯.Assume(14)d4=d40+∑k=1∞τkd4k.For nonconstant solutionu¯s=us,vs,ws of (10) bifurcating from u˜ with small amplitude, let(15)us=u˜+∑k=1∞τkuk,vs=u˜+∑k=1∞τkvk,ws=w˜+∑k=1∞τkwk,(16)ℒ3v,u,w≔d2+d40hu˜v″−g+u˜+d40h′u˜v˜1+au˜d1v+b−v˜+d40h′u˜v˜2d1u˜u−d40h′u˜u˜v˜d1w,where 0<τ≪1. Substituting (14) and (15) into (10) and equating the Oτ and Oτ2 terms, respectively, we derive two systems(17)d1u1″+−1+av˜+w˜u1+1+au˜v1+u˜w1=0,x∈0,l,ℒ3v1,u1,w1=0,x∈0,l,d3w1″−ew˜u1−rw˜w1=0,x∈0,l,u1′=v1′=w1′=0,x=0,l,(18)d1u2″+−1+av˜+w˜u2+1+au˜v2+u˜w2=F1,x∈0,l,ℒ3v2,u2,w2=F2,x∈0,l,d3w2″−ew˜u2−rw˜w2=F3,x∈0,l,u2′=v2′=w2′=0,x=0,l,where(19)F1=−au1v1−u1w1,F3=rw12+eu1w1,hu˜=1ε+u˜2,h′u˜=−2u˜ε+u˜22,h″u˜=23u˜2−εε+u˜23,F2=−d40h′u˜u1v1″+h″u˜v˜u1u1′′−d41hu˜v1″+h′u˜v˜u1″+u1v1.It is easy to get nonzero solutions for (unique up to a constant multiple for any given positive integerj, and this constant can be absorbed into τ in (15)) for (17) as(20)u1=ωj,v1=e1jωj,e1j=11+au˜v˜u˜+eu˜w˜rw˜+d3μj+d1μj>0,w1=h1jωj,h1j=−ew˜rw˜+d3μj<0,as long as d40 is given by(21)d40=g+u˜+d2μje1−b−v˜−hu˜μje1−h′u˜v˜μj≔d40j,j=1,2,….It is clearly known thatd40>0 if(22)g+u˜v˜<b−v˜1+au˜u˜,e1<2u˜v˜ε+u˜2.Here, in order to get the uniqueness of solution, we need assume thatd40j≠d40k for any integer j≠k.Set(23)d4min=minj∈ℤ+d40j=minj∈ℤ+g+u˜+d2μje1−b−v˜−hu˜μje1−h′u˜v˜μj=d40j0,for a positive integer j0. Since d4 is regarded as the bifurcation parameter, d40j,j=1,2,…, is called the possible bifurcation location for the formation of new patterns [38]. This means that the first bifurcation occurs when the parameter d4 crosses the bifurcation value d4min. If the bifurcation is stable, it will be the pattern us,vs,ws with formulae given in (15) and (20).In the following, we find the formula ofd41. The adjoint system of the homogeneous system associated with (18) is(24)d1u¯2″+−1+av˜+w˜u¯2+b−v˜+d40h′u˜v˜2d1u˜v¯2−ew˜w¯2=0,x∈0,l,d2+d40hu˜v¯2″+1+au˜u¯2−g+u˜+d40h′u˜v˜1+au˜d1v¯2=0,x∈0,l,d3w¯2″+u˜u¯2−d40h′u˜u˜v˜d1v¯2−rw˜w¯2=0,x∈0,l,u¯2′=v¯2′=w¯2′=0,x=0,l,which has one solution(25)u¯2=e2jωj,e2j=11+au˜d2μj+d40hu˜μj+g+u˜+d40h′u˜v˜1+au˜d1,v¯2=ωj,w¯2=h2jωj,h2j=u˜rw˜+d3μj1+au˜d2μj+d40hu˜μj+g+u˜>0,where ωj is defined as in (6) for j=1,2,…. By the solvable condition for (18), that is, the vectors F1,F2,F3 and u¯2,v¯2,w¯2 should be orthogonal in L20,l, we have the solvability equation for d41 as follows:(26)∫0lu¯2F1+v¯2F2+w¯2F3dx=0.A direct computation gives(27)d41=d41j=0,for allj=1,2,….Whend41=0, for each j, F2 in (19) can be simplified to(28)F2=e12+2d40h′u˜e1μj+d40h″u˜v˜μj+e12cos2μjx.From this, a particular solutionu2,v2,w2 can be found as(29)u2=a1j+a2jcos2μjx,v2=a3j+a4jcos2μjx,w2=a5j+a6jcos2μjx,where(30)A=−1+av˜+w˜,B=1+au˜,C=d2+d40hu˜,D=b−v˜+d40h′u˜v˜2d1u˜,E=g+u˜+d40h′u˜v˜1+au˜d1,F=d40h′u˜u˜v˜d1,G=2d40h′u˜e1μj+d40h″u˜v˜μj+e12,N1=u˜E−BFrh12+eu˜E−eBF−rw˜Eh1+e1w˜rB−aE,M1=2w˜AE+BDr+BF−u˜Ee,M2=64d1d3Cμj3+16rw˜d1C−d3AC+d1d3Eμj2+4−d3BD−d3AE−rw˜AC+rw˜d1E+eu˜w˜Cμ−rw˜AE+BD+u˜E−FBew˜,N2=16d3Cae1+h1μj2+4−ru˜h12C+−eu˜C+rw˜C+d3Eh1+ae1w˜rC+d3E−2d3GBμj+BF−u˜Erh12+eBF−eu˜E+rw˜Eh1+ae1E−2GBrw˜,a1j=N1M1,a2j=−N22M2,a3j=−1BA−u˜era1−u˜rh12+eh12rw˜+ae1+h12,a4=1B4d1μj−A+eu˜w˜4d1μj+rw˜a2+rh12+eh1u˜24d1μj+rw˜−ae1+h12,a5=−era1−rh12+eh12rw˜,a6=−ew˜a24d3μj+rw˜−rh12+eh124d3μj+rw˜.Sinced41=0, it is necessary for further analysis to obtain d42. Following the process of getting (17) and (18), further computation up to the order Oτ3 and equating the Oτ3 term yields(31)d1u3″+−1+av˜+w˜u3+1+au˜v3+u˜w3=F4,x∈0,l,ℒ3v3,u3,w3=F5,x∈0,l,d3w3″−ew˜u3−rw˜w3=F6,x∈0,l,u2′=v2′=w2′=0,x=0,l,where(32)F4=−au1v2−av1u2−u1w2−w1u2,F6=2rw1w2+eu1w2+eu2w1,h‴u˜=−24u˜3−u˜εε+u˜24,F5=−d40h′u˜u1v2+u2v1″+h″u˜v˜u1u2″+12u12v1′+u1v1u1′′+12h‴u˜v˜u12u1′′−d42hu˜v1″+h′u˜v˜u1″+u1v2+u2v1.The solvability condition can be simplified as(33)∫0lF4e2cosπjxl+F5cosπjxl+F6h2cosπjxldx=0.Substituting (32) into (33) leads to(34)d42j=−d40j4hu˜e1+h′u˜u˜2h′u˜2e1a1j+e1a2j+2a3j+a4j+h″u˜4a1j+2a2j+3e1+2h‴u˜u˜+12hu˜e1+h′u˜v˜μje2ae1+h12a1j+a2j+a2a3+a4+2a5j+a6j−2a3j+a4j+e12a1j+a2j−h22a5j+a6j2rh1+e+eh12a1j+a2j.According to the above computation and bifurcation theorem [38], the local bifurcation of (10) is given as follows.Theorem 2.
If for any positive integerj≠k, d40j≠d40k, then d40j is a bifurcation value of the equation Fd4,u=0 with respect to the curve d4,u˜,d4>0. Moreover, there is a one-parameter family of nontrivial solutions Γτ=d4τ,u∗τ,v∗τ,w∗τ of (10) for τ sufficiently small, where d4τ,u∗τ,v∗τ, and w∗τ are continuous functions such that d40=d40j and(35)u∗τ=u˜+τu1+τ2u2+oτ2,v∗τ=u˜+τv1+τ2v2+oτ2,w∗τ=w˜+τw2+τ2w2+oτ2.The zero-point set ofFd4,u constitutes two curves d4,u˜ and Γτ in a neighborhood of the bifurcation point d40j,u˜.To clarify the relationship between the solutionus,vs,ws and its bifurcation location d40j, we may relabel us,vs,ws as usj,vsj,wsj, i.e.,(36)usj=u˜+τu1+τ2u2+⋯+τkuk+⋯,vsj=v˜+τv1+τ2v2+⋯+τkvk+⋯,wsj=w˜+τw1+τ2w2+⋯+τkwk+⋯.As specified in [27], u˜,v˜,w˜ is called the base term of the pattern, and the pattern shape and its amplitude are primarily determined by the leading term u1,v1,w1 when τ is small.


## 3. Stability of Steady State Bifurcation

This section is devoted to study the stability of the pattern solutionusj,vsj,wsj bifurcated from d40j,u˜ by analyzing the sign of the principal eigenvalue.For (10), set(37)u=usj+φeλt,v=vsj+ξeλt,w=wsj+ηeλt.Then,(38)d1φ″+−1+avsj+wsjφ+1+ausjξ+usjη=λφ,x∈0,l,d2+d4husjξ″+ℋφ″+ℋξ′+ℋφ′+ℬξ+ℋφ=λξ,x∈0,l,d3η″−ewsjφ+s−2rwsj−eusjη=λη,x∈0,l,φ′=ξ′=η′=0,x=0,l,where(39)P=d4h′usjvsj,ℋ=2d4h′usjusj′,ℋ=2d4h′usjvsj′+h″usjvsjusj′,ℬ=d4h′usjusj″+h″usjusj′2−usj−g,ℋ=d4h′usjvsj″+h″usjvsjusj″+h″usjusj′2vsj+2h″usjusj′vsj′+b−vsj.Setting(40)φ=φ0+τφ1+τ2φ2+⋯,ξ=ξ0+τξ1+⋯,η=η0+τη1+⋯,λ=λ0+τλ1+⋯,(41)usj=u˜+τu1+τ2u2+⋯,vsj=v˜+τv1+τ2v2+⋯,wsj=w˜+τw1+τ2w2+⋯,d4=d40j+τd41j+τ2d42j+⋯,and substituting them into (38), we can obtain a system by equating the O1 terms(42)d1φ0″+−1+av˜+w˜φ0+1+au˜ξ0+u˜η0=λ0φ0,x∈0,l,ℒ3jξ0,φ0,η0=λ0ξ0−λ0d40jh′u˜v˜d1φ0,x∈0,l,d3η0″−ew˜φ0−rw˜η0=λ0η0,x∈0,l,φ0′=ξ0′=η0′=0,x=0,l,where(43)ℒ3ξ,φ,η≔d2+d40hu˜ξ″−g+u˜+d40h′u˜v˜1+au˜d1ξ+b−v˜+d40h′u˜v˜2d1u˜φ−d40h′u˜u˜v˜d1η.It is obvious that the sign ofλ0 determines the stability of the stationary solution usj,vsj,wsj. To solve the eigenvalue problem (42), we can use −μkφ0,ξ0,η0 to replace φ0″,ξ0″,η0″ for some integer k≥0. As to the existence of nonzero solution φ0,ξ0,η0, we have(44)λ03+k1kλ02+k2kλ0+k3k=0,where(45)k1k=d1+d2+d3+d40jhu˜μk+rw˜+g+u˜+v˜u˜>0,k2k=d1+d3d40jhu˜+d1d2+d1d3+d2d3μk2+g+u˜+rw˜d1+v˜u˜+rw˜d2+v˜u˜+g+u˜d3+v˜u˜+rw˜hu˜+1+au˜v˜h′u˜d40jμk+g+u˜rw˜+u˜+v˜+e−1u˜w˜−b1+au˜+rw˜+gv˜u˜−d0jh′u˜d1u˜v˜+au˜v˜2+u˜v˜w˜−11+au˜,k3k=d1d3d40jhu˜+d1d2d3μk3+v˜d3u˜+rw˜d1hu˜+v˜1+au˜d3h′u˜d40j+rw˜d2+g+u˜d1+v˜u˜d2d3μk2+rv˜w˜u˜+eu˜w˜d40jhu˜−d40jh′u˜d1−rv˜w˜d11+au˜+d3v˜av˜1+au˜+au˜w˜−1+rw˜g+u˜d1+rv˜w˜u˜+eu˜w˜d2+v˜gu˜−b1+au˜+u˜+v˜−u˜w˜d3μk−d40jh′u˜arv˜w˜d1v˜1+au˜+u˜w˜−1+w˜v˜gru˜−br1+au˜+ru˜+v˜−u˜w˜+eu˜g+u˜=d1d2d3μk3+rw˜d1d2+g+u˜d1d3+v˜u˜d2d3μk2+rw˜g+u˜d1+rv˜w˜u˜+eu˜w˜d2+v˜gu˜−b1+au˜+u˜+v˜−u˜w˜d3μk+w˜v˜gru˜−br1+au˜+ru˜+v˜−u˜w˜+eu˜g+u˜−d40jhu˜d1d3μk3+rw˜d1+v˜u˜d3μk2+rw˜v˜u˜+eu˜w˜μk+h′u˜v˜1+au˜d3μk2+rw˜μk=hu˜d1d3μk3+rw˜d1+v˜u˜d3μk2+rw˜v˜u˜+eu˜w˜μk+h′u˜v˜1+au˜d3μk2+rw˜μkd40k−d40j,and d40j is given by (21).It follows from the Routh–Hurwitz stability criteria that all the eigenvalues have negative real part (usj,vsj,wsj is locally asymptotically stable) if and only if k3k>0,k1kk2k−k3k>0 for all k∈ℕ+. Hence, there exist some eigenvalues with positive real parts if one of the conditions above fails for some k∈ℕ+.We setd4min=d40j0 for a positive integer j0 which is called the principle wave mode. This means that the first bifurcation will occur when the parameter d4 crosses the bifurcation value d4min. Hence, d40j≠d4min if j≠j0 and there exists an integer k=j0 such that k3k<0. As such, equation (44) at least has a root with positive real part. Therefore, a necessary condition for the stability of usj,vsj,wsj is stated as below.Theorem 3.
(stability criterion). Ifj≠j0, then d40j≠d4min and the steady state usj,vsj,wsj in (15) is unstable. In other words, if usj,vsj,wsj is stable, then j=j0.Now, we determine the stability of the steady stateusj0,vsj0,wsj0. When j=j0, we have d40j=d4min. Obviously, the principal eigenvalue for (42) is λ0=0 with eigenvector:(46)φ0,ξ0,η0=ωj0,11+au˜v˜u˜+eu˜w˜rw˜+d3μj+d1μjωj0,−ew˜rw˜+d3μjωj0.In order to obtain the stability ofusj0,vsj0,wsj0, we need to evaluate λ1 and λ2. Again substituting (40) and (41) into (38) and equating the Oτ terms, we have(47)d1φ1″+−1+av˜+w˜φ1+1+au˜ξ1+u˜η1=λ1φ0+G1,x∈0,l,ℒ3jξ0,φ0,η0=λ1ξ0−λ1d40jh′u˜v˜d1φ0+G2,x∈0,l,d3η1″−ew˜φ1−rw˜η1=λ1η0+G3,x∈0,l,φ1′=ξ1′=η1′=0,x=0,l,where(48)G1=−av1+w1φ0−au1ξ0−u1η0,G2=−d40jh′u˜v1φ0+h″u˜v˜u1φ0+h′u˜u1ξ0″+v1φ0+u1ξ0,G3=ew1φ0+2rw1+eu1η0.The solvability condition for equation (47) gives(49)∫0lλ1φ0+G1u¯2dx+∫0lλ1ξ0−λ1d40jh′u˜v˜d1φ0+G2v¯2dx+∫0lλ1η0+G3w¯2dx=0,where u¯2,v¯2,w¯2 is given in (24) with j=j0. Therefore,(50)λ1=−∫0lG1u¯2+G2v¯2+G3w¯2dx∫0lφ0u¯2+ξ0−d40jh′u˜v˜/d1φ0v¯2+η0w¯2dx,where(51)∫0lφ0u¯2+ξ0−d40jh′u˜v˜d1φ0v¯2+η0w¯2dx=12l11+au˜d1+d2+d40j0hu˜μj0+eu˜w˜rw˜+d3μj0+g+u˜+v˜u˜−d40j0h′u˜v˜d1−ew˜rw˜+d3μj021+au˜d2μj+d40hu˜μj0+g+u˜.A simple calculation gives(52)∫0lG1u¯2+G2v¯2+G3w¯2dx=0.Thus,λ1=0. For this reason, we need further to compute λ2. Since G2 can be expressed as(53)G2=e1+4d40h′u˜e1μj+2d40h″u˜v˜μj+e1cos2μjx,and we can find a particular solution φ1,ξ1,η1 of (51) as(54)φ1=a¯1+a¯2cos2μj0x,ξ1=a¯3+a¯4cos2μj0x,η1=a¯5+a¯6cos2μj0x,where(55)a¯i=2aij0,i=1,…,6,with aij0,i=1,…,6, as defined in (30). A further similar computation by equating the Oτ2 term gives the following system:(56)d1φ2″+−1+av˜+w˜φ2+1+au˜ξ2+u˜η2=λ2φ0+G4,x∈0,l,ℒ3jξ0,φ0,η0=λ0ξ0−λ0d40jh′u˜v˜d1φ0+G5,x∈0,l,d3η2″−ew˜φ2−rw˜η2=λ2η0+G6,x∈0,l,φ2′=ξ2′=η2′=0,x=0,l,where(57)G4=−av2+w2φ0−av1+w1φ1−au2ξ0−au1ξ1−u2η0−u1η1,G5=−d40j0h′u˜u1ξ1+v1φ1+u2ξ0+v2φ0″+h″u˜v˜u1φ1+v˜u2φ0+u1v1φ0″+12u12ξ0′+u1′u1ξ0″+h‴u˜v˜12u12φ0′+u1′u1φ0′−d42j0hu˜ξ0″+h′u˜v˜φ0″+u1ξ1+u2ξ0+v1φ1+v2φ0,G6=ew2φ0+ew1φ1+2rw2+eu2η0+2rw1+eu1η1.Similarly, by applying the solvability condition of (56), we have(58)λ2=−∫0lG4u¯2+G5v¯2+G6w¯2dx∫0lφ0u¯2+ξ0−d40jh′u˜v˜/d1φ0v¯2+η0w¯2dx.A tedious computation leads to(59)∫0lG4u¯2+G5v¯2+G6w¯2dx=−2lμj0ϒ,with(60)ϒ=hu˜e1+h′u˜v˜d42j0.Owing to the stability of nonconstant steady state,usj0,vsj0,wsj0 depends on the sign of λ2. Furthermore, ϒ<0 if e1<2u˜v˜/ε+u˜2, the denominator of (58) is positive since ew˜<rw˜+d3μj021+au˜ and λ2<0. Therefore, we have the following result.Theorem 4.
(stability). Letj0>0 be a positive integer such that d40j0=d4min. Then, the small-amplitude steady state usj,vsj,wsj is stable provided that j=j0 and λ2<0.
In particular, ifmaxew˜/rw˜+d3μj02,ε+u˜2/2u˜v˜v˜/u˜+eu˜w˜/rw˜+d3μj0+d1μj0<1+au˜, then λ2<0, and the small-amplitude steady state usj0,vsj0,wsj0 is stable.The following example can illustrate the result of Theorem4.Example 1.
For (1), in the interval l=0,10π, set a=1,b=0.5,g=0.6,s=1.5,r=1.6,e=1,d1=0.3,d2=0.2,d3=0.2,andε=0.01, and choose d4 as the variable parameter. It is not hard to verify that the equilibrium point u˜=0.91,0.3,0.37, d40j0=6.02j0=9, and max0.9,1.19<1.91. From Theorem 4, usj,vsj,wsj is stable.
In conclusion, we study a predator-prey crossdiffusion model with cannibalism and discuss the effects of crossdiffusion on steady-state solutions of (1). We obtain the local structure of steady state bifurcation from the homogeneous steady states u˜,v˜,w˜ in one dimension by treating the crossdiffusion coefficient d4 as a bifurcation parameter. Moreover, we establish the stability criteria and find a selection mechanism of the principal eigenvalue. More specifically, if the stationary pattern is stable, then its principal wave mode j is a positive integer j0 at which d4j0 is minimized. And all the rest bifurcations are unstable if j≠j0. In addition, the stable patterns can be derived if j=j0 and λ2<0.



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