# Hopf Bifurcations and Oscillatory Patterns of a Homogeneous

Authors: Zhenhua Bao; He Liu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

A kind of homogeneous reaction-diffusion singular predator-prey model with no-flux boundary condition is considered. By using the abstract simplified Hopf bifurcation theorem due to Yi et al. 2009, we performed detailed Hopf bifurcation analysis of this particular pattern formation system. These results suggest the existence of oscillatory patterns if the system parameters fall into certain parameter ranges. And all these oscillatory patterns are proved to be unstable.


---

## Body

## 1. Introduction

In this paper, we consider the following reaction-diffusion singular predator-prey model:(1)ut=d1uxx+α(1-u)u-v,x∈(0,ℓπ),t>0,vt=d2vxx+βv(1-vu),x∈(0,ℓπ),t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈(0,ℓπ),ux′(x,t)=vx′(x,t)=0,x=0,ℓπ,t>0,
where u(x,t) and v(x,t) are the population densities of the prey and predator at time t and position x,  α and β are dimensionless positive parameters; ℓ∈(0,∞). The underlying spatially homogeneous problem was derived in [1] to model prey-predator interactions in fragile (insular) environments. The spatially structured system (1) supplemented with initial data and no-flux boundary conditions was introduced in [2]. For more information on the system (1), we refer the reader to [1, 2] for great details.Mathematically, the model was considered by Ducrot and Langlais [3], where the authors first provided a suitable notion of global travelling wave weak solution. They mainly studied the existence of travelling waves solutions for predator invasion in such environments. Under suitable conditions on the diffusion coefficients and on species growth rates they were able to prove that the travelling wave solutions were actually positive on a half line and identically zero elsewhere.The present paper is targeting to consider the Hopf bifurcations of the reaction-diffusion system (1), by using the simplified Hopf bifurcation theorem due to [4], which is widely used to prove the existence of oscillatory patterns of different kind of pattern formation systems, including Gierer-Minhardt model [5], Degn-Harrison model [6], bimolecular model [7], hair growth controlling model [8], and the Sel’kov model [9].Our results show that, under suitable choice of system parameters, system (1) will undergo spatially homogeneous and nonhomogeneous oscillatory phenomena. And once the oscillatory patterns exist, they are always unstable. These unstable patterns cannot be observed by numerical simulations; thus, numerical simulations corresponding to our analytical analysis are unavailable in the paper, even though, the analytical results we obtained allow for the clearer understanding of the rich dynamics of the system.The rest of this paper is structured in the following way. In Section2, we perform Hopf bifurcation analysis of the system. In Section 3, we draw some concluding remarks to end up our discussion. Throughout the paper, we denote by ℕ the set of all the positive integers and ℕ0=ℕ∪{0}.


## 2. Stability and Hopf Bifurcation Analysis

To begin with, for convenience of our discussion, we copy (1) in the following:
(2)ut=d1uxx+α(1-u)u-v,x∈(0,ℓπ),t>0,vt=d2vxx+βv(1-vu),x∈(0,ℓπ),t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈(0,ℓπ),ux′(x,t)=vx′(x,t)=0,x=0,ℓπ,t>0.System (2) has a unique equilibrium solution (uα,vα), with
(3)uα=vα=:α-1α,
which is in the first quadrant if and only if α>1.In the following, we always assume thatα>1 holds. We fix the parameter β and choose α as the bifurcation parameter.The linearized operator of system (2) evaluated at (uα,vα) is given by
(4)L(α):=(d1∂2∂x2+(2-α)-1βd2∂2∂x2-β).It follows from [4, 10] that the eigenvalues of L(α) are given by these of the operator Ln(α), which is defined by
(5)Ln(α):=(-d1n2ℓ2+(2-α)-1β-d2n2ℓ2-β),
whose characteristic equation is
(6)λ2-λTn(α)+Dn(α)=0,n=0,1,2,…,
where
(7)Tn(α)=(2-α-β)-(d1+d2)n2ℓ2∶=A(α)-(d1+d2)n2ℓ2,Dn(α)=d1d2n4ℓ4+n2ℓ2[d1β-d2(2-α)]+β(α-1).We have the following theorem on the stability of the unique positive equilibrium solution(uα,vα).Theorem 1.
Suppose that one of the following conditions holds(1)
β≥1 and α>1,(2)
0<β<1 and α>2-βand that d1,d2>0 such that
(8)d1d2≥1β
is satisfied. Then, (uα,vα) is always locally asymptotically stable in the reaction-diffusion equation (2).Proof.
On one hand, for anyα>1, we have
(9)Dn(α)>d1d2n4ℓ4+n2ℓ2(d1β-d2),
which implies that Dn(α)>0 always holds for any α∈(1,∞) if
(10)d1d2≥1β
holds.
On the other hand, ifβ≥1 and α>1, or 0<β<1 but α>2-β holds, then we can obtain that, for any n∈ℕ0 and α>0, we always have Tn(α)<0. Thus, (uα,vα) is always locally asymptotically stable in the reaction-diffusion equation (2).Now we consider the Hopf bifurcations of the system. According to [4], a point αH∈(1,∞) is a Hopf bifurcation point if and only if there exists n∈ℕ0, such that
(11)Tn(αH)=0,Dn(αH)>0;Tj(αH)≠0,Dj(αH)≠0forj≠n;
and τ′(αH)≠0, where τ(α) is the real parts of the unique pair of complex eigenvalues τ(α)±iω(α) near the imaginary axis.Thus, any potential Hopf bifurcation pointαH must be in the interval (1,2-β], where we assume that 0<β<1.For any Hopf bifurcation pointαH∈(1,2-β], let τ(α)±iω(α) be the eigenvalues of Ln(α) near α=αH then we have
(12)τ(α)=A(α)2-(d1+d2)n22ℓ2,ω(α)=Dn(α)-τ2(α).Then, for anyα∈(1,2-β], we have
(13)τ′(α)=-12.
This implies that the transversality condition τ′(α)≠0 is always satisfied for any α∈(1,2-β].Suppose that0<β<1 holds and define
(14)ℓn=n(1-β)/(d1+d2),n∈ℕ0.Then, for anyℓn<ℓ≤ℓn+1 and 1≤j≤n, we define αjH as the roots of
(15)A(α)=(d1+d2)j2ℓ2
satisfying 1<αjH<α0H:=2-β, and these points satisfy
(16)1<αnH<αn-1H<⋯<α1H<α0H=2-β
and limj→∞αjH=1. Clearly Tj(αjH)=0 and Ti(αjH)≠0 for i≠j.Now we derive a condition from the parameters so thatDn(α)>0 for all α∈(1,2-β] and n∈ℕ0. In fact, from (8) in Theorem 1, it follows that Dn(α)>0 always holds.Based on the discussions above, we have the following Hopf bifurcation results for the reaction-diffusion model (2).Theorem 2.
Suppose that the constantsd1,d2,β satisfying the condition (8) and ℓn are defined as in (14). Then, for any ℓ in (ℓn,ℓn+1], there exist n points αjH(ℓ), 1≤j≤n, satisfying
(17)1<αnH<αn-1H<⋯<α1H<α0H=2-β,
such that the system (1) undergoes a Hopf bifurcation at α=αjH or α=α0H. Moreover,(1)
the bifurcating periodic solutions fromα=α0H are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system;(2)
the bifurcating periodic solutions fromα=αjH are spatially nonhomogeneous.Next we consider the bifurcation direction and stability of the bifurcating periodic solutions.Theorem 3.
For the system (2), the direction of the Hopf bifurcation at α=α0H is forward, and the bifurcating (spatial homogeneous) periodic solutions are unstable.Proof.
By Theorem 2.1 of [4], in order to determine the stability and bifurcation direction of the bifurcating periodic solution, we need to calculate Re(c1(α0H)). When α=α0H, we put
(18)q=(a0b0)=(1β-ω0i),q*=(a0*b0*)=(ω0+βi2ω0ℓπ-i2ω0ℓπ),
where ω0=β(1-β).
Now translate (2) into the following system by the translation u^=u-uα and v^=v-vα, and still let u and v denote u^ and v^ for the convenience of notation. We have
(19)ut-d1uxx=α[1-(u+uα)](u+uα)-(v+vα),x∈(0,ℓπ),t>0,vt-d2vxx=β(v+vα)(1-v+vαu+uα),x∈(0,ℓπ),t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈(0,ℓπ),ux′(x,t)=vx′(x,t)=0,x=0,ℓπ,t>0.
Following [4], we define
(20)f(α,u,v)=α[1-(u+uα)](u+uα)-(v+vα),g(α,u,v)=β(v+vα)(1-v+vαu+uα).
By (2.19) of [4], we have
(21)c0=e0=2(β-2),d0=2β(β-2)1-β(1-β+ω0i)2,f0=β(β-2),g0=0,h0=2β(2-β)21-β(3-2β+2ω0i).
Following [4], we define
(22)Qqq=(c0d0),Qqq¯=(e0f0),Cqqq¯=(g0f0).
Note that〈·,·〉 is the complex-value inner product defined as
(23)〈U1,U2〉=∫0ℓπ‍(u¯1u2+v¯1v2)dx
with Ui=(ui,vi)T,  i=1,2.
Then we obtain easily that(24)〈q*,Qqq〉=β-2ω0[(1-2β)ω0-2β2i],〈q¯*,Qqq〉=β-2ω0[(1+2β)ω0+2β2i],〈q*,Qqq¯〉=〈q¯*,Qqq¯〉=β-2,〈q*,Cqqq¯〉=β(2-β)2ω0(1-β)[-2ω0+(3-2β)].
Hence it is straightforward to calculate that(25)H20=(c0d0)-〈q*,Qqq〉(a0b0)-〈q¯*,Qqq〉(a0¯b0¯)=(00),H11=(e0f0)-〈q*,Qqq¯〉(a0b0)-〈q¯*,Qqq¯〉(a0¯b0¯)=(00).
By (25), we have
(26)w20=[2iω0I-L(α0H)]-1H20=(00),w11=-[L(α0H)]-1H11=(00),
where I is the identity matrix. The equalities in (26) lead to
(27)〈q*,Qw11q〉=〈q*,Qw20q¯〉=0.
Therefore, we can calculate that(28)Re(c1(α0H))=Re{i2ω0〈q*,Qqq〉·〈q*,Qqq¯〉+〈q*,Qw11q〉+12〈q*,Cqqq¯〉+12〈q*,Qw20q¯〉i2ω0}=Re{(2-β)22ω02[2β2+(1-2β)ω0i]+β(2-β)2ω0(1-β)[-2ω0+(3-2β)i]}=β(2-β)>0.
From (13), it follows that τ′(α0H)=-1/2<0, and then by Theorem 2.2 of [4], the direction of the Hopf bifurcation is forward and the bifurcating periodic solutions are unstable since Re(c1(α0H))>0.Remark 4.
(1) The direction of Hopf bifurcations atα=αjH with j≥1 can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. Thus, here we did not calculate the bifurcation direction of Hopf bifurcations at α=αjH with j≥1.
(2) The spatial nonhomogeneous periodic solutions atα=αjH with j≥1 found in Theorem 2 are clearly unstable since the steady state (uα,vα) is unstable.
(3) Since all the periodic solutions are unstable, the simulations of the oscillatory patterns are unavailable here.


## 3. Concluding Remarks

In this paper, we considered a kind of diffusive Gaucel-Langleis model. By using the simplified Hopf bifurcation theory due to [4], we were capable of investigating the existence of Hopf bifurcations of the system, which indicates the existence of oscillatory patterns of the system. Our main bifurcation and stability analysis results in the paper can be summarized as follows.(1)
Ifβ is sufficiently large, say, if β≥1 holds, then Hopf bifurcations will never be possible. In fact, we can show that if one of the following conditions holds
(a)
β≥1 and α>1,(b)
0<β<1 and α>2-β,andd1,d2>0 such that d1/d2≥1/β is satisfied, then, (uα,vα) is always locally asymptotically stable in the reaction-diffusion equation (2) (Theorem 1).(2)
However, ifβ is not that large, say, 0<β<1, then, Hopf bifurcation is possible in suitable parameter ranges. In fact, by choosing α as the bifurcation parameter, we can obtain that if the constants d1,d2,β satisfying the condition (14) and ℓn are defined as in (14), then, for any ℓ in (ℓn,ℓn+1], there exist n points αjH(ℓ),  1≤j≤n, satisfying
(29)1<αnH<αn-1H<⋯<α1H<α0H=2-β,
such that the system (1) undergoes a Hopf bifurcation at α=αjH or α=α0H, (Theorem 2). Moreover,
(a)
the bifurcation direction of Hop bifurcations atα=α0H is forward and the bifurcating periodic solutions are unstable (Theorem 3),(b)
the bifurcation direction of Hopf bifurcations atα=αjH with j≥1 can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. This is beyond the scope of this paper,(c)
the spatial nonhomogeneous periodic solutions atα=αjH with j≥1 found in Theorem 2 are clearly unstable since the steady state (uα,vα) is unstable (Remark 4).



---

*Source: 547425-2013-09-25.xml*