# Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

Authors: Meng Lin; Yanyou Chai; Xuguang Yang; Yufeng Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.


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## Body

## 1. Introduction

Spatiotemporal pattern formations are one of the core problems arising in biology and ecology. Hence, it has attracted the extensive attentions from a huge number of investigations, including biologists, chemists, and mathematicians. It is well known that diffusion not only can induce spatial patterns, but also can drive spatiotemporal patterns, including oscillatory patterns. In this paper, we are mainly concerned with the spatiotemporal oscillatory patterns of a particular reaction-diffusion predator-prey system arising from ecology. In the model of our concern, the prey is assumed to exhibit herd behavior, so that the predator interacts with the prey along the outer corridor of the herd of prey, and the interaction terms are supposed to use the square root of the prey population rather than simply the prey population, where the use of the square root properly represents the assumption that the interactions occur along the boundary of the population (see [1–15]). The reaction-diffusion system takes in the following form:(1)∂u∂t-d1Δu=u1-u-uv1+αu,x∈0,lπ,t>0∂v∂t-d2Δv=βv-γ+u1+αu,x∈0,lπ,t>0ux0,t=vx0,t=uxlπ,t=vxlπ,t=0,t>0ux,0=u0x≥0,vx,0=v0x≥0,x∈0,lπwhere u(x,t) and v(x,t) stand for the population densities of the prey and predator at time t and the position x, respectively; β represents the interaction strength between the predator and the prey; d1 and d2 are the diffusion rates of u and v, respectively. The spatial domain is supposed to be one dimensional, Ω=(0,lπ), with l∈R+; βγ stands for the death rate of v; α is the saturation rate. The boundary conditions are supposed to be the no flux boundary conditions.System (1) has been studied extensively in the past few years (see [16–25]). For example, in [16], by using s≔βγ as the bifurcation parameter, the authors showed that, as the parameter s changes, the stability of the constant positive steady state varies, which induces the occurrence of Hopf bifurcating periodic solutions; however, their works tend to be numerical, rather than analytical. In [22], in the special case when α=0, Tang and Song introduced the time delay into the system. By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the time delay is used to be the bifurcation parameter. In [25], in the special case when α=0, Xu and Song introduced the time delay into the system. By using the stability analysis, center manifold theory, and normal form methods, they were capable of showing the existence of Hopf bifurcations, where the parameter θ is chosen as the bifurcation parameter. In [17], G. Gimmelli, Bob W. Kooi, and E. Venturino considered the ecoepidemic models with prey group defense and feeding saturation in the form of ordinary differential equations. The Hopf bifurcation, zero-Hopf bifurcation, and global bifurcations are considered.In this paper, however, we do not impose the special case whenα=0; instead, we consider the more general case when α>0. This is the first highlight of our paper. Secondly, in the existing literatures, the bifurcation parameter is to be chosen as either s, or the diffusion coefficient, while, in our paper, use μ, or equiv. the first component u∗ of the positive equilibrium (u∗,v∗), as the bifurcation parameter. The different choice of the bifurcation parameter makes our theoretical analysis quite different. This is the second highlight of our paper; finally, the Hopf bifurcation analysis in this paper is mainly based on the theoretical framework of the abstract Hopf bifurcations due to Yi, Wei, and Shi in [26]. In [26], the authors used the same methods to consider a kind of homogeneous diffusive predator-prey system with Holling type-II functional response subject to the homogeneous Neumann boundary conditions. They found that, under suitable conditions, for any eigenmode cos(nx/l), the predator-prey system in [26] has 2n Hopf bifurcation points. In this paper, however, we will show that the Hopf bifurcation points of the system are much more complex than those in [26]. This is the third highlight of our paper.The main analytical tools we used in the discussions of the paper include center manifold theory, normal form methods, and the basic theory in reaction diffusion equations. To better illustrate and support our theoretical bifurcation analysis, we include some delicate numerical simulations in the paper.The remaining parts of this paper are organized as follows: in Section2, we not only prove the existence of Hopf bifurcations, but also consider the detailed properties of Hopf bifurcations by using center manifold methods and normal form methods; in Section 3, we include numerical simulations to support our theoretical analysis; in Section 4, we end up our discussions by drawing some conclusions.


## 2. Hopf Bifurcation Analysis

System (1) has three nonnegative constant steady state solutions: (0,0), (1,0), and (u∗,v∗)≔(μ2,vμ), where (2)μ≔γ1-αγ,vμ≔μ1-μ21+αμ=γαγ-γ-1αγ+γ-1αγ-14.Clearly,(u∗,v∗) is a positive equilibrium solution if and only if αγ+γ<1. In the following, we will always assume that αγ+γ<1 holds and will choose μ as the main bifurcation parameter (or equivalently γ as a parameter) by fixing α and β.The linearized operator of system (1) evaluated at (μ2,vμ) is given by(3)Lμ≔d1∂2∂x2+AμBμCμd2∂2∂x2,where(4)Aμ≔-4αμ3+2αμ-3μ2+12αμ+1,Bμ≔-μαμ+1andCμ≔β1-μ22αμ+1.It follows from [26] that the eigenvalues of the operator L(μ) are determined by the eigenvalues of Ln(μ), with (5)Lnμ≔Aμ-d1n2l2BμCμ-d2n2l2,n∈N0≔0,1,2,⋯.The characteristic equation ofLn(μ) is (6)η2-ηTnμ+Dnμ=0,n∈N0≔0,1,2,⋯,where(7)Tnμ=-4αμ3+2αμ-3μ2+12αμ+1-d1+d2n2l2,Dnμ=βμ1-μ22αμ+12-d2-4αμ3+2αμ-3μ2+12αμ+1n2l2+d1d2n4l4.To consider the stability/instability of the positive constant steady state solution(u∗,v∗)≔(μ2,vμ), as well as the Hopf bifurcations bifurcating from (u∗,v∗)≔(μ2,vμ), we need to know the signs of Tn(μ) and Dn(μ).Lemma 1.
Define(8)f1μ≔-4αμ3+2αμ-3μ2+1.Then, f1(μ)=0 has three zero points, denoted by μ1,μ2,μ∗, satisfying μ1<μ2<0<μ∗, such that (9)f1μ1=f1μ2=f1μ∗=0,f1′μ1<0,f1′μ2>0,f1′μ∗<0and (10)f1μ>0,μ∈-∞,μ1∪μ2,μ∗,<0,μ∈μ1,μ2∪μ∗,∞.And in particular, for the largest zero point μ∗, it satisfies that μ∗∈(-3+9+32α2/8α,+∞).Proof.
By (8), we have f1′(μ)=2α-6μ-12αμ2,f1′′(μ)=-6-24αμ. Solving f1′(μ)=0, we have (11)μ1′=-3-33+8α212α<0,μ2′=-3+33+8α212α>0.At μ=μ1′, we have (12)f1′′μ1′=29+24α2>0,f1μ1′=36α2-9-8α2+39+24α272α2.
We argue thatf1(μ1′)<0. In fact, letting 9+24α2=t, we have f1(μ1′)=-2t2+15t-45/18(t-3). Since t>3 and -2t2+15t-45<0, we have f(μ1′)<0. Since f1(μ)=μ(-4αμ2-3μ+2α)+1, we have f1(-∞)=+∞. Then, there exists a μ1∈(-∞,μ1′), such that f1(μ1)=0 and f1′(μ1)<0.
On the other hand, byf1(0)=1>0 and f1(μ1′)<0, there exists a μ2∈(μ1′,0) such that f1(μ2)=0 and f1′(μ2)>0. Clearly, μ1<μ2<0.
Sincef1(-3+9+32α2/8α)=1=f1(0), we can conclude that f1′(μ)=0 at a critical point in (0,-3+9+32α2/8α). Thus, we have μ2′∈(0,-3+9+32α2/8α).
Atμ=μ2′, we have (13)f1′′μ2′=-29+24α2<0,f1μ2′=-9+39+24α2+4α29+29+24α272α2.
We argue thatf1(μ2′)>0. In fact, letting 9+24α2=t, we have f1(μ2′)=2t2+15t+45/18(t+3)>0. Noticing that f1(+∞)=-∞, we can conclude that there exists a μ∗∈(μ2′,∞), such that f1(μ∗)=0. Since f1(-3+9+32α2/8α)=1>0, we have μ∗∈(-3+9+32α2/8α,+∞). In particular, (14)f1μ>0,μ∈-∞,μ1∪μ2,μ∗,<0,μ∈μ1,μ2∪μ∗,∞.So far, we have completed the proof.We are now identifying Hopf bifurcation valuesμH which satisfy the Hopf bifurcation condition (H1) (due to [26]), taking the following form: there exists a n∈N0≔{0,1,2,…,}, such that(15)TnμH=0,DnμH>0,andTjμH≠0,DjμH≠0forj≠nand for the unique pair of complex eigenvalues near the imaginary axis λ(μ)±iω(μ),(16)λ′μH≠0.From (7), it follows that Tn(μ)<0 for μ∈(μ∗,+∞). Hence any potential bifurcation point μH must be in the interval (0,μ∗]. Let λ(μ)±iω(μ) be the eigenvalues of Ln(μ) for μ close to the Hopf bifurcation value μH. Then, (17)λμ=Aμ2-d1+d2n22l2,ωμ=Dnμ-λ2μ,and (18)λ′μH=A′μ2μ=μH=g1μH21+αμH2,where(19)g1μ≔α-6μ-15αμ2-8α2μ3.Lemma 2.
Letg1(μ) be defined in (19). Then for any μ>0, g1′(μ)>0 and there exists a unique μ#∈(0,-3+9+32α2/8α), such that g1(μ#)=0.Proof.
Byg1(μ)=α-6μ-15αμ2-8α2μ3, we have g′(μ)=-6-30αμ-24α2μ2<0, where μ>0. At μ=-3+9+32α2/8α, we have (20)g1-3+9+32α28α=-9+64α2-39+32α2+16α29+32α232α.
Again, lettingt=9+32α2, we have (21)g1-3+9+32α28α=--3+t2+t9+t82-9+t2<0,where we use the fact that t>3. By g1(0)=α>0, we conclude that there exists a μ#∈(0,-3+9+32α2/8α) such that g1(μ#)=0. We thus complete the proof.According to Lemmas1 and 2, we can draw the graphs of the functions A(μ) and A′(μ) in a coordinate system (Figure 1). Thus, the transversality condition (16) is always satisfied as long as μH≠μ#. Moreover when μ#<μH<-3+9+32α2/8α, the real part of one pair of complex eigenvalues of λ(μ) becomes positive when μ decreases crossing μH, and when 0<μH<μ#, the real part of one pair of complex eigenvalues of λ(μ) becomes negative when μ decreases crossing μH. That is, in (μ#,-3+9+32α2/8α), the constant steady state loses stability when μ decreases across a bifurcation point, but, in (0,μ#), it regains the stability when μ decreases across a bifurcation point.Figure 1
The graphs of the functionsA(μ) and A′(μ).From the aforementioned discussions, the determination of Hopf bifurcation points reduces to describing the set(22)Λ≔μ∈0,μ∗∖μ#:for  somen∈N0is  satisfiedwhen a set of parameters (l,d1,d2,α,β) is given. In the following we fix d1,d2,α>0 and 0<γ/(1-αγ)<1, but we choose l appropriately.On one hand, it is clear thatμ0H≔μ∗ is always an element of Λ for any l>0 since T0(μ0H)=0,Tj(μ0H)<0 for any j≥1, and Dm(μ0H)>0 for any m∈N0. This corresponds to the Hopf bifurcation of spatially homogeneous periodic solution which has been known from the studies of ODE model. Clearly, μ0H is also the unique value μ for the Hopf bifurcation of spatially homogeneous periodic solution for any l>0.In the following, we will look for the spatially nonhomogeneous Hopf bifurcation forn≥1. By the properties of f1(u) (stated in Lemma 1), it follows that A(μ1)=A(μ2)=A(μ∗)=0, and in particular, A(μ) has a unique critical point μ=μ#, with μ2<μ#<μ∗, at which A(μ) achieves a local maximum A(μ#)≔M∗>0. In particular, by Lemma 2, μ#∈(0,-3+9+32α2/8α).Theorem 3.
Letμ# be the critical point of A(μ), with μ2<μ#<μ∗, at which A(μ) achieves a local maximum A(μ#)≔M∗>0. (1)
Suppose thatμ#≤0 and that, for sufficiently small ϵ>0, (23)d2A022βd1<ϵ-ϵ3αϵ+12holds. Define(24)l¯n=nd1+d2A0,n∈N.Forl¯n<l≤l¯n+1 and 1≤j≤n, there exist n points μjH, with 1≤j≤n, satisfying (25)0<μnH<⋯<μ2H<μ1H<μ0H≔μ∗,such that the system (1) undergoes a Hopf bifurcation at μ=μjH or μ=μ0H≔μ∗.(2)
Suppose thatμ#>0 and that, for sufficiently small ϵ>0, (26)d2A022βd1<ϵ-ϵ3αϵ+12holds. Define(27)l_n=nd1+d2M∗,n∈N.(a)
Then forl_n<l≤l_n+1 and l¯n-k<l<l¯n-k+1, with k≤j≤n for some k∈N0, there exist 2k points μj,±H, with n-k≤j≤n satisfying (28)0<μ1,-H<μ2,-H<⋯<μk,-H<μ∗<μn,+H<⋯<μ2,+H<μk,+H<μ0H≔μ∗such that the system (1) undergoes a Hopf bifurcation at μ=μjH or μ=μ0H≔μ∗.(b)
Forl_n<l≤l_n+1 and 1≤j≤n, there exist n points μjH, with 1≤j≤n, satisfying (29)0<μnH<⋯<μ2H<μ1H<μ0H≔μ∗.Then for anyl∈(l_n,l_n+1], there exist n points μjH(l),1≤j≤n, satisfying 0<μnH<⋯<μ2H<μ1H<μ0H, such that the system (1) undergoes a Hopf bifurcation at μ=μjH or μ=μ0H≔μ∗.
Moreover:
(1) The bifurcating periodic solutions from μ=μ0H≔μ∗ are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system.
(2) The bifurcating periodic solutions from μ=μj,±H (or μjH) are spatially nonhomogeneous.Proof.
(1)
Suppose thatμ#≤0 holds. Then, A(0)=1=maxμ∈[0,+∞)A(μ) and A′(μ)<0 for any μ∈[0,+∞). In this case, define(30)l¯n=nd1+d2A0,n∈N. Forl¯n<l≤l¯n+1 and 1≤j≤n, we define μjH to be the unique root of A(μ)=(d1+d2)j2/l2 satisfying 0<μjH<μ∗<μ0H≔μ∗. These points satisfy (31)0<μnH<⋯<μ2H<μ1H<μ0H. ClearlyTj(μjH)=0 and Ti(μjH)≠0 for i≠j. In this case, the transversality condition always holds true. Since μ=γ/(1-αγ) and αγ+γ<1, we can assume that μ>ϵ, where ϵ is an arbitrary positive number. Indeed if μ∈(0,μ0H], then we have(32)Diμ≥βϵ-ϵ32αϵ+12-d2A0i2l2+d1d2i4l4≔hi2l2. The quadratic functionh(y)≔d1d2y2-d2A(0)y+βϵ-ϵ3/2(αϵ+1)2 is positive for all y∈R if(33)d2A022βd1<ϵ-ϵ3αϵ+12. (2)
Suppose thatμ#>0 holds. Then, A′(μ)>0 for any μ∈[0,+μ∗∗) and A′(μ)<0 for any μ∈(μ∗∗,+∞). Define(34)l_n=nd1+d2M∗,n∈N. Then forl_n<l≤l_n+1 and l¯n-k<l<l¯n-k+1, with k≤j≤n for some k∈N0, we define μj,-H and μj,+H to be the roots of A(μ)=(d1+d2)j2/l2 satisfying 0<μj,-H<μ∗∗<μj,+H<μ0H=μ∗. These points satisfy (35)0<μ1,-H<μ2,-H<⋯<μk,-H<μ∗<μn,+H<⋯<μ2,+H<μk,+H<μ0H ClearlyTj(μj,±H)=0 and Ti(μj,±H)≠0 for i≠j. In this case, the transversality condition always holds true. Forl_n<l≤l_n+1 and 1≤j≤n, we define μjH to be the unique root of A(μ)=(d1+d2)j2/l2 satisfying 0<μjH<μ0H≔μ∗. These points satisfy (36)0<μnH<⋯<μ2H<μ1H<μ0H. ClearlyTj(μjH)=0 and Ti(μjH)≠0 for i≠j. In this case, the transversality condition always holds true. Now we only need to verify whetherDi(μH)≠0 for all i∈N0 for the aforementioned three cases, and, in particular, Di(μH)>0. Here we derive a condition on the parameters so that Di(μ)>0 for all μ∈(0,μ0H] so that Di(μj,±H)>0. Since μ=γ/(αγ-1) and αγ>γ+1, we can assume that μ>ϵ, where ϵ is an arbitrary positive number. Indeed if μ∈(0,μ0H], then we have(37)Diμ≥βϵ-ϵ32αϵ+12-d2M∗i2l2+d1d2i4l4≔hi2l2. The quadratic functionh(y)≔d1d2y2-d2M∗y+βϵ-ϵ3/2(αϵ+1)2 is positive for all y∈R if(38)d2M∗22βd1<ϵ-ϵ3αϵ+12. Then, applying Theorem 2.1 in [27], we have completed the proof.In the following, we will consider the bifurcation direction and the stability of the bifurcating periodic solutions. We have the following results.Theorem 4.
For the Hopf bifurcation pointsμ0H=μ∗ obtained in Theorem 3, the following conclusions hold: (1)
The Hopf bifurcation atμ∗ is supercritical if Γ0(μ∗)<0. It is subcritical if Γ0(μ∗)>0.(2)
The spatially homogeneous periodic solutions bifurcating from the constant positive steady state(u∗,v∗) at μ=μ∗ are locally orbitally stable if Γ0(μ∗)<0, and they are unstable if Γ0(μ∗)>0,
whereΓ0(μ∗) is precisely defined in (63).Proof.
We rely on the abstract results obtained in [26] to determine the stability and bifurcation direction of the bifurcating periodic solution. To that end, we calculate Re(c1(μ∗) in the context of [26]. We use μ∗ to stand for μ∗. When μ=μ∗, we let (39)
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∗
1
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L
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.
As in [26], we define (42)fμ,u,v≔u+u∗1-u-u∗-u+u∗v+v∗1+αu+u∗,gμ,u,v≔βv+v∗-γ+u+u∗1+αu+u∗.
We rewrite the original system in the following abstract equation:(43)dUdt=LμU+Fμ,U,where(44)Fμ,U≔fμ,u,v-Aμu-Bμvgμ,u,v-Cμu,with U=(u,v)T∈X, where A(μ),B(μ) and C(μ) are shown in (4). At μ=μ∗, the system (43) reduces to(45)dUdt=Lμ∗U+F0U,where F0(U)≔F(μ,U)|μ=μ∗.
Let(46)X=u,v∈H20,lπ×H20,lπ∣ux,vxx=0,lπ=0and decompose X=Xc⊕Xs, with Xc≔{zq+z¯q¯∣z∈C}, Xs≔{u∈X∣〈q∗,u〉=0}. For any (u,v)∈X, there exist z∈C and w=(w1,w2)∈Xs such that (47)uv=zq+z¯q¯+w1w2.
Thus the system (45) is reduced to the following system in (z,w) coordinates:(48)dzdt=iω0z+q∗,F0,dwdt=Lλ0w+Hz,z¯,w,where(49)Hz,z¯,w≔F0-q∗,F0q-q¯∗,F0q¯,andF0≔F0zq+z¯q¯+w.
As in [26], we write F0 in the following form:(50)F0U≔12QU,U+16CU,U,U+OU4,whereU=u,v,where Q,C are symmetric multilinear forms. For simplicity, we write QXY=Q(X,Y) and CXYZ=C(X,Y,Z).
Then, the reaction-diffusion system restricted to the center manifold is given by(51)dzdt=iω0z+q∗,F0=iω0z+∑2≤i+j≤3giji!j!ziz¯j+Oz4,where g20=〈q∗,Qqq〉, g11=〈q∗,Qqq¯〉, g02=〈q∗,Qq¯q¯〉, and (52)g21=2q∗,Qw11,q+q∗,Qw20,q¯+q∗,Cq,q,q¯.The dynamics of (48) can be determined by the dynamics of (51).
We write the Poincaré normal form of (43) (for μ in a neighborhood of μ∗) in the following form: (53)z˙=αμ+iωμz+z∑j=1Mcjμzz¯j,where z is a complex variable, M≥1, and cj(μ) are complex-valued coefficients.
Then, we have(54)c1μ=g20g113αμ+iωμ2α2μ+ω2μ+g112αμ+iωμ+g0222αμ+3iωμ+g212.
Thus,(55)c1μ∗=i2ω0g20g11-2g112-13g022+g212=i2ω0q∗,Qqq·q∗,Qqq¯+q∗,Qw11,q+12q∗,Qw20,q¯+12q∗,Cq,q,q¯.Then, it remains to calculate 〈q∗,Qqq〉,〈q∗,Qqq¯〉, 〈q¯∗,Qqq〉, 〈q∗,Cqqq¯〉, and 〈q∗,Cqqq¯〉.
After direct calculations, we observe that all of thefvv(μ∗,0,0), fuvv(μ∗,0,0), fvvv(μ∗,0,0), gvv(μ∗,0,0), guvv(μ∗,0,0), and gvvv(μ∗,0,0) are all zeros. And in particular, (56)fuuμ∗,,0,0=-2+1-μ∗23αμ∗+14μ∗2αμ∗+12,fuvμ∗,0,0=-12μ∗αμ∗+12,fuuuμ∗,0,0=3μ∗2-15α2μ∗2+4αμ∗+18μ∗4αμ∗+13,fuuvμ∗,0,0=3αμ∗+14μ∗3αμ∗+13,guuμ∗,0,0=βμ∗2-13αμ∗+14μ∗2αμ∗+12,guvμ∗,0,0=β2μ∗αμ∗+12,guuuμ∗,0,0=-3βμ∗2-15α2μ∗2+4αμ∗+18μ∗4αμ∗+13,guuvμ∗,0,0=-β3αμ∗+14μ∗3αμ∗+13.
Then, from [26], we have (57)c0≔fuua02+2fuva0b0=8α2μ∗4+19αμ∗3-3αμ∗+9μ∗2-1+4iω0αμ∗+14ω02αμ∗+14,d0≔guua02+2guva0b0=β-3αμ∗3-μ∗2+3αμ∗+1+4iω0αμ∗+14ω02αμ∗+14,e0≔fuua02+fuva0b¯0+a¯0b0=-8α2μ∗4-19αμ∗3+3αμ∗-9μ∗2+14ω02αμ∗+14,and (58)f0≔guua02+guva0b¯0+a¯0b0=βμ∗2-13αμ∗+14ω02αμ∗+14,g0≔fuuua02a0+fuuv2a02b¯0+a02b¯0=2ω03α2μ∗2+4αμ∗+1-3iμ∗2-15α2μ∗2+4αμ∗+18ω03μ∗αμ∗+16,h0≔guuua02a0+guuv2a02b¯0+a02b¯0=-2βω0αμ∗+13αμ∗+1+3iβμ∗2-1αμ∗5αμ∗+4+18ω03μ∗αμ∗+16.
Again, by [26], we have (59)q∗,Qqq=∫0lπq∗¯Tc0,d0Tdx=-βμ∗μ∗2-13αμ∗+1+4ω02αμ∗+128ω02μ∗αμ∗+14+iμ∗8α2μ∗3+19αμ∗2+9μ∗-3α-4β-18ω0μ∗αμ∗+13,q∗,Qqq¯=∫0lπq∗¯Te0,f0Tdx=βμ∗2-13αμ∗+18ω02αμ∗+14+i1-μ∗8α2μ∗3+19αμ∗2-3α+9μ∗8ω0μ∗αμ∗+13,q¯∗,Qqq=∫0lπq∗Tc0,d0Tdx=4ω02αμ∗+12-βμ0μ∗2-13αμ∗+18ω02μ∗αμ∗+14+i1-μ∗8α2μ∗3+19αμ∗2+9μ∗-3α+4β8ω0μ∗αμ∗+13,and (60)q¯∗,Qqq¯=∫0lπq∗Te0,f0Tdx=βμ∗2-13αμ∗+18ω02αμ∗+14+iμ∗8α2μ∗3+19αμ∗2-3α+9μ∗-18ω0μ∗αμ∗+13,q∗,Cqqq¯=∫0lπq∗¯Tg0,h0Tdx=μ∗15α2μ∗3+12αμ∗2+μ∗-15α2-6αβ+3-12α-2β-316ω02μ∗2αμ∗+15+i3βμ∗μ∗2-1αμ∗5αμ∗+4+1+2ω02αμ∗+123αμ∗+116ω03μ∗2αμ∗+16.
Then, from [26], we have (61)H20≔c0d0-q∗,Qqqa0b0-q¯∗,Qqqa0¯b0¯=00,H11≔e0f0-q∗,Qqq¯a0b0-q¯∗,Qqq¯a0¯b0¯=00,which implies that w20=0,w11=0. Therefore, (62)q∗,Qw11q=q∗,Qw20q¯=0.
So far, based on the aforementioned computations, we have(63)Rec1μ∗=Γ0μ∗=Γ0μ∗≔Rei2ω0q∗,Qqq·q∗,Qqq¯+12q∗,Cqqq¯=-17α2μ∗4-15α2μ∗2-6αβμ∗2-64αμ∗3-2βμ∗-33μ∗2+132ωH2μ∗2αμ∗+15-βμ∗2-13αμ∗+18α2μ∗4+19αμ∗3-3αμ∗-2βμ∗+9μ∗2-116ω04μ∗αμ∗+17.
Then, from [26], we complete the proof.


## 3. Numerical Simulations

In this section, we include some numerical simulations to support our theoretical analysis. We use the symbolic mathematical softwareMatlab (Version 7.0) to plot numerical graphs.The model (1) has five different system parameters, which are: α, β, γ, d1, and d2. On one hand, we choose the system parameters in the following way:(64)α=1.7,β=0.9,γ=0.3,d1=1,d2=3.Under (64) and after direct calculations, we have (u∗,v∗)=(0.3748,0.7811). In this case, the positive equilibrium solution (u∗,v∗) is locally asymptotically stable, and no Hopf bifurcations can be observed. This is shown in Figure 2, where the initial condition is taken by u(0)=0.3748+0.01cos⁡(2πx/3), v0=0.7811+0.01cos⁡2πx/3 (Figure 2).Figure 2
Numerical simulations of a stable homogeneous equilibrium solution of system with parameters in (64). The left: component u; the right: component v.On the other hand, we choose the system parameters in the following way:(65)α=2,β=9,γ=0.2851,d1=1,d2=3.Under (65) and after direct calculations, we have (u∗,v∗)=(0.6632,0.8643). In this case, (u∗,v∗) is unstable and a family of periodic solutions bifurcating from (u∗,v∗) can be observed. This is shown in Figure 3, where the initial condition is taken at u(0)=0.6632+0.01cos⁡(2πx/3), v(0)=0.8643+0.01cos⁡(2πx/3) (Figure 3).Figure 3
Numerical simulations of the Hopf bifurcating spatially homogeneous periodic solutions of system (1) with parameters in (65). The left: component u; the right: component v.


## 4. Conclusions

In this paper, we considered a new homogeneous diffusive predator-prey system with herd behavior in terms of the square root of the prey population. By using the classical Hopf bifurcation theorem applicable to the general reaction-diffusion equations due to [26], we are capable of showing the existence of both spatially homogeneous/inhomogeneous periodic solutions bifurcating from the positive constant steady state solutions. For the homogeneous periodic solutions, we derive the formulae to determine their stability and instability. In this paper, there are three highlights: (1) we consider the more general case when α>0, instead of α=0. (2) We use μ, or equiv. the first component u∗ of the positive equilibrium (u∗,v∗), as the bifurcation parameter, instead of s and the diffusion coefficient; (3) we show that the Hopf bifurcation points of the system are much more complex than those in [26]. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.



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*Source: 3907453-2019-06-13.xml*