# Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

Authors: Wenjie Zuo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.


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## Body

## 1. Introduction

Predator-prey systems describing the interaction between two species with positive-negative feedbacks continue to arouse the interest of applied mathematicians and ecologists. The functional response is the important component depicting the predator-prey relationship [1]. In order to better characterize the ecological interaction such as lynx and hare and sparrow and sparrow hawk especially by Tanner [2] and Wollkind et al. [3], May developed the so-called Holling-Tanner predator-prey model [4] which takes the following form:
(1)dUdT=rU(T)(1-U(T)K)-mU(T)V(T)u(T)+q,dVdT=sV(T)(1-hV(T)U(T)),
where U(T), V(T) stand for the prey and the predator densities, respectively. And in ecological senses, r, K, m, q, s, h are positive constants which denote prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predator, respectively. The functional response mU/(U+q) is of Michaelis-Menten type in enzyme-substrate kinetics, proposed by Holling [5].Hsu and Huang [6] proved the global stability of the positive equilibrium of System (1) by applying the Dulac’s criterion and constructing Lyapunov function. Gasull et al. [7] showed that the stable positive equilibrium is surrounded by two limit cycles. Thus, the local stability does not imply the global stability.Spatial diffusion is ubiquitous. Assuming the preys and the predators are in an isolate patch, I neglect the impact of migration, including immigration and emigration, and only consider the diffusion of the spatial domain. It has been shown that the diffusion can generate the more rich spatiotemporal complex [8–10]. A diffusive Holling-Tanner predator-prey system is as follows:
(2)∂U∂T=D1Uxx+rU(x,T)(1-U(x,T)K)-mU(x,T)V(x,T)U(x,T)+q,x∈(0,lπ),T>0,∂V∂T=D2Vxx+sV(x,T)(1-hV(x,T)U(x,T)),000000x∈(0,lπ),T>0,Ux(0,T)=Ux(lπ,T)=Vx(0,T)=Vx(lπ,T)=0,T≥0,U(x,0)=U0(x)≥0,V(x,0)=V0(x)≥0,x∈[0,lπ].
Ma and Li [11] investigated the existence of Hopf bifurcation and the steady state bifurcations of simple and double eigenvalues of System (2). Peng and Wang [12] proved the global stability of the unique positive equilibrium of System (2) under certain hypothesis by constructing Lyapunov function. Chen and Shi [13] improved the results given in [12].Some biologists have questioned the functional response solely depending on the prey density [14–16], especially when predators have to search for food, (and therefore have to share or compete for food). Numerous fields and laboratory experiments have revealed that a more reasonable functional response should be the so-called ratio-dependent predator-prey theory [14, 15], which can be strongly stated as that the per capita predator growth rate should be the function of the ratio of predator abundance. A predator-prey system with the ratio-dependent functional response can exhibit rich dynamics [17, 18].In this paper, we consider the following diffusive ratio-dependent Holling-Tanner predator-prey system:(3)∂U∂T=D1Uxx+rU(x,T)(1-U(x,T)K)-mU(x,T)V(x,T)U(x,T)+qV(x,T),x∈(0,lπ),T>0,∂V∂T=D2Vxx+sV(x,T)(1-hV(x,T)U(x,T)),000000x∈(0,lπ),T>0,Ux(0,T)=Ux(lπ,T)=Vx(0,T)=Vx(lπ,T)=0,T≥0,U(x,0)=U0(x)≥0,V(x,0)=V0(x)≥0,x∈[0,lπ].
M. Banerjee and S. Banerjee [19] examined the existence of Turing and non-Turing patterns of System (3) by numerical simulations and revealed the fact that Hopf bifurcation is essential for spatiotemporal chaos. Liang and Pan [20] established the global stability of positive equilibrium of System (3) without diffusion by constructing Lyapunov function and obtained the uniqueness of the limit cycle.My work is the extension of [11, 19]. In addition, what is more important in ecosystem is whether the species would coexist in the long run; that is, whether the positive equilibrium is globally asymptotically stable. In this paper, we aim to study Hopf bifurcations and steady state bifurcations induced by intrinsic growth rate and Turing instability induced by diffusion. Particularly, the global stability of the unique positive equilibrium is obtained by the upper-lower solutions method under certain conditions. Simulations are carried out to strongly support the theoretical results.The rest of the paper is organized as follows. In Section2, the instability of the semitrivial equilibrium and the local stability of the positive coexistence of system (3) are studied by the distribution of eigenvalues. And the existence, stability, and direction of Hopf bifurcations are investigated in detail as the intrinsic growth rate λ crosses the critical value. Turing bifurcation induced by diffusion takes place at the certain critical values. In Section 3, the steady state bifurcations are considered with λ as the bifurcation parameter and the interaction of Hopf bifurcation and steady state bifurcation are discussed. In Section 4, the global stability of the unique positive coexistence is proved by the upper-lower solutions method.


## 2.  Hopf Bifurcations and Turing Bifurcations

For simplicity of later discussion, we suppose thatD1=r. Introducing the following dimensionless variables:
(4)u=UK,v=mVrK,t=rT,
System (3) can be transformed into
(5)∂u∂t=uxx+u(x,t)(1-u(x,t))-u(x,t)v(x,t)u(x,t)+αv(x,t),x∈(0,lπ),t>0,∂v∂t=dvxx+λv(x,t)(β-v(x,t)u(x,t)),000x∈(0,lπ),t>0,ux(0,t)=ux(lπ,t)=vx(0,t)=vx(lπ,t)=0,t≥0,u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈[0,lπ],
where d=D2/D1, α=qr/m, λ=sh/m, β=m/hr.It is easy to check that System (5) has always a boundary equilibrium E0(1,0) and has a positive constant coexistence E(u*,v*) if and only if αβ+1>β, where
(6)u*=1-βαβ+1>0,v*=βu*.
The characteristic equations of the linearization of (5) at E0(1,0) are given by
(7)(μ+n2l2+1)(μ+dn2l2-λβ)=0,n=0,1,2,….
Thus,
(8)μ1=-n2l2-1,μ2=λβ-dn2l2,n=0,1,….
Hence, E0(1,0) is always unstable since (7) has at least a positive root μ=λβ. Next we study the local stability of the positive steady state E(u*,v*) and the existence of bifurcations.


### 2.1. Stability of the Positive Coexistence and Hopf Bifurcations

The linearized operator of System (5) at E(u*,v*) is as follows:
(9)L(λ)=(∂2∂x2+αβ2+2β(αβ+1)2-1-1(αβ+1)2λβ2d∂2∂x2-λβ),Ln(λ)=(-n2l2+αβ2+2β(αβ+1)2-1-1(αβ+1)2λβ2-dn2l2-λβ).
The characteristic equations corresponding to Ln(λ) are
(10)μ2-Tn(λ)μ+Dn(λ)=0,n=0,1,2,…,
where
(11)Tn(λ)=-(1+d)n2l2-λβ+αβ2+2β(αβ+1)2-1=-(1+d)n2l2-(λ-λ0H)β,Dn(λ)=dn4l4+n2l2(d(1-αβ2+2β(αβ+1)2)+λβ)+λβ(1-β1+αβ)=dn4l4+n2l2(λ-dλ0H)β+λβu*,
where λ0H=(1/β)((αβ2+2β)/(αβ+1)2-1).Obviously, ifαβ2+2β≤(αβ+1)2, then Tn(λ)<0, Dn(λ)>0, which implies that all the roots of (10) are negative real-part. Thus the unique positive constant equilibrium E(u*,v*) is locally asymptotically stable.Indeed, in Section4, we can prove the global stability of the unique positive coexistence under the stronger conditions. Next, we will seek all the possible Hopf bifurcation values λ0, so we assume αβ2+2β>(1+αβ)2 holds.We will identify Hopf bifurcation valueλ0 which satisfies
(12)Tn(λ0)=0,Dn(λ0)>0,Tj(λ0)≠0,Dj(λ0)≠0,∀j≠n.
From (11), Tn(λ)<0, for all λ∈(λ0H,+∞), which means that the possible Hopf bifurcation points belong to the interval (0,λ0H]. For any Hopf bifurcation point λ*∈(0,λ0H], α(λ)±iω(λ) are a pair of conjugate eigenvalues of Ln(λ) near λ=λ*, where
(13)α(λ)=Tn(λ)2,ω(λ)=124Dn(λ)-Tn2(λ),
and α′(λ*)=-β/2<0.Hence, all Hopf bifurcation points reduce to the following set:(14)Λ1={λ∈(0,λ0H],forsomen∈N,0thecondition(12)issatisfied}.
Obviously, Tn(λ)=0 is equivalent to
(15)λn=λ0H-(1+d)n2βl2,(16)Dn(λn)=-n4l4-n2l2B0+λ0Hβu*,
where B0=2-(2αβ2+3β-dβ)/(1+αβ)2. Since D0(λ0)=λ0Hβu*>0, there exists N0∈N+ such that Dn(λn)>0, n=0,1,2,…,N0, and Dn(λn)≤0, n=N0+1, N0+2,….Clearly,Ti(λn)=((1+d)/l2)(n2-i2)≠0, for all i≠n. And for i≠n,
(17)Di(λn)=di4l4+i2l2(λn-dλ0H)β+λnβu*.
We discuss Di(λn)>0, for all i≠n in two cases.Case 1.
We can choose small enoughd>0 such that λn-dλ0H>0, that is, for n=0,1,…,N0, d<d0, where
(18)d0∶=βl2λ0H-N02βl2λ0H+N02>0.Case 2.
Whend>d0,
(19)dλ0H-λn=(d-1)λ0H+(1+d)n2l2β≤(d-1)λ0H+(1+d)N02l2β≜M*,λnu*≥(λ0H-(1+d)N02l2β)u*=(dλ0H-M*)u*.
Therefore,
(20)Di(λn)≥di4l4-i2l2M*β+(dλ0H-M*)u*β=g(i2l2).
In order that g(y)=dy2-M*βy+(dλ0H-M*)u*β>0 for all y∈R, we only need (M*)2β<4d(dλ0H-M*)u*. Thus, Di(λn)>0.
Clearly,λ0H=maxn∈{0,1,…,N0}λn.Summarizing the above analysis results, we draw our main conclusion in this subsection.Theorem 1.
Supposeα,β,d>0 and αβ+1>β, αβ2+2β>(1+αβ)2 hold. If there exists d0>0 such that 0<d<d0, or (M*)2β<4d(dλ0H-M*)u*, where d0 and M* are defined by (18) and (19), respectively. Then for System (5), the following results are true:(1)
The positive constant steady stateE(u*,v*) is locally asymptotically stable for λ∈(λ0H,+∞) and unstable for λ∈(0,λ0H);(2)
System (5) undergoes a Hopf bifurcation at λ=λn,n=0,1,…,N0 (defined by (15)) and the bifurcating periodic solutions can be parameterized into the following forms:
(21)u(s)(x,t)=s(ane2πit/T(s)+an¯e-2πit/T(s))cosnxl+o(s),v(s)(x,t)=s(bne2πit/T(s)+bn¯e-2πit/T(s))cosnxl+o(s).Furthermore, the periodic solutions bifurcating from E(u*,v*) at λ=λ0H are spatially homogeneous. And bifurcating periodic solutions from λ=λn, (n=1,…,N0) are spatially inhomogeneous.Next we only consider the direction and stability of Hopf bifurcation atλ=λ0H.Theorem 2.
For the System (5), the Hopf bifurcations at λ=λ0H are backward, and the bifurcating periodic solutions are locally asymptotically stable.Proof.
Here we apply the notations and calculations in [9, 21], we put
(22)q∶=(a0b0)=(λ0Hβ+iω0λ0Hβ21),q*∶=(a0*b0*)=(λ0Hβ22lπω0iω0-iλ0Hβ2lπω0),
such that 〈q*,q〉=1 and 〈q*,q-〉=0, where ω0=λ0Hβu*.
Recall that in our context,(23)f(λ,u,v)=u(1-u)-uvu+αv,g(λ,u,v)=λv(β-vu).
By direct computation, it follows that(24)c0=-2β2+2ω02(λ0H)2β4-2α(αβ+1)3λ0Hβ-4ω0λ0Hβ3i,d0=2β,e0=-2β2-2u*λ0Hβ3+2α(αβ+1)3λ0Hβ,f0=-2β,g0=αλ0Hβ2u*(αβ+1)4(-2αβ-2-3ω0λ0Hβi),h0=2u*(1β2+3ω0λ0Hβ3i),
and
(25)〈q*,Qqq〉=-1β+i(2λ0Hω0-ω0λ0Hβ2+αβ(αβ+1)3ω0),〈q*,Qqq-〉=〈q*¯,Qqq-〉¯=-1β+iω0(u*β-αβ(αβ+1)3),〈q*¯,Qqq〉=3β-i(2λ0Hω0-ω0λ0Hβ2+αβ(αβ+1)3ω0),〈q*,Cqqq-〉=-(3α2u*λ0Hβ(αβ+1)4+2β2u*)+i(αω0u*(αβ+1)3+λ0Hω0u*β+3ω0λ0Hβ3u*).
Hence, it is direct to calculate, by (24) and (25),
(26)H20=(c0d0)-〈q*,Qqq〉(a0b0)-〈q*¯,Qqq〉(a0¯b0¯)=(00),H11=(e0f0)-〈q*,Qqq-〉(a0b0)-〈q*¯,Qqq-〉(a0¯b0¯)=(00),
which implies that W20=W11=0. Thus, 〈q*,QW11q〉=〈q*,QW20q-〉=0. Therefore, by (25) and ω02=λ0Hβu*, we can obtain that
(27)Re(C1(λ0H))=Re(i2ω0(g20g11-2|g11|2-13|g02|2)+g212)=Re(i2ω0(〈q*,Qqq〉〈q*,Qqq-〉)+12〈q*,Cqqq-〉)=(u*2β2ω02-12λ0Hβ3)+(λ0Hβω02-1β2u*)-3α4u*λ0Hβ(1+αβ)4=-3α4u*λ0Hβ(1+αβ)4<0.
Since α′(λ0H)=-β/2, then the bifurcating periodic solutions are backward and  exist for τ<τ0H. And combining Tn(λ0H)=-((1+d)/l2)n2<0,Dn(λ0H)>0, for all n=1,2,…,N0, we can obtain that the bifurcating periodic solutions are asymptotically stable since Re(C1(λ0H))<0.Next, we give an example to illustrate the theoretical results above.Example 3.
Takingd=0.2,α=0.4,β=1.2,l=2, System (5) has a unique positive constant steady state E(0.1892,0.227). Obviously, αβ+1>β, αβ2+2β>(1+αβ)2. By the formula above, we obtain λ0H=0.2989. And D0(λ0H)<0,D1(λ1)>0. Thus, 0<d<d0=1. By Theorems 1 and 2, E(0.1892,0.227) is asymptotically stable for λ∈(0.2989,+∞) as illustrated in Figures 1(a) and 1(b). When λ crosses through the critical value, E(0.1892,0.227) losses its stability and a family of periodic solutions appear and are asymptotically stable since Re(C1(λ0H))=-0.9215<0 as illustrated in Figures 2(a) and 2(b).E
(
0.1892,0.227
) is asymptotically stable when λ=0.32>λ0H and the initial function u0(x)=0.5+0.2cosx, v0(x)=0.5+0.2cosx.
(a)
(b)Spatially homogenous periodic solutions are asymptotically stable whenλ=0.24<λ0H and the initial function u0(x)=0.5+0.2cosx, v0(x)=0.5+0.2cosx.
(a)
(b)We can compute the direction and stability of bifurcating periodic solutions by the same way as Theorem2 and omit it here. Combining the stability theory, we have the following results.Theorem 4.
For System (5), the bifurcating periodic solutions from λ=λn, n=1,2,…,N0 (defined by (15)) are backward (forward) if Re(C1(λn))<(>)0 and the spatially inhomogeneous Hopf bifurcations are unstable.


### 2.2.  Turing Instability Induced by Diffusion

In this subsection, we consider Turing instability with diffusion effect. As mentioned earlier, whenαβ2+2β≤(αβ+1)2, Tn(λ)<0,Dn(λ)>0, for all n∈{0,1,2,…}. Hence, no Turing instability occurs. Next we always confine αβ2+2β>(αβ+1)2.When System (5) has no diffusion, the characteristic equation (10) at E(u*,v*) can be transformed into
(28)μ2-(λ0H-λ)βμ+λβu*=0.
Then all the roots of (28) are negative real part if λ>λ0H. Combining the stability theory [22], we easily know that the unique positive equilibrium E(u*,v*) of System (5) without diffusion is asymptotically stable for λ∈(λ0H,+∞) and unstable for λ∈[0,λ0H). In the following, we mainly consider the occurrence of Turing instability induced by diffusion. Hence we confine λ∈(λ0H,+∞). That is, E(u*,v*) is locally asymptotically stable without diffusion effect.It is well-known thatE(u*,v*) is unstable if the characteristic equation (10) has at least a positive real-part root. Notice when λ>λ0H, Tn(λ)<0, for all n=0,1,2,…. The necessary condition of the instability of the equlibirium E(u*,v*) of System (5) is Dn(λ)<0 for some n∈N+={1,2,…}.Recall(29)Dn(λ)=dn4l4+n2l2(λ-dλ0H)β+λβu*≜g(n2l2),
where g(x)=dx2-(dλ0H-λ)βx+λβu*.Hence we needλ∈(λ0H,dλ0H),d>1 and Δ≜(dλ0H-λ)2β2-4dλβu*>0. That is, (dλ0H-λ)β>2dλβu*. Otherwise, Di(λ)≥0 for all i∈N.Obviously,g(x)=0 has two positive roots x1,x2 since
(30)x1+x2=(dλ0H-λ)β>0,x1x2>0,
where
(31)x1=12d((dλ0H-λ)β-(dλ0H-λ)2β2-4dλβu*),x2=12d((dλ0H-λ)β+(dλ0H-λ)2β2-4dλβu*).
If we can find some j∈N+ such that x1<j2/l2<x2. Then Dj(λ)<0. That is, x1l<j<x2l. The constant positive coexistence E(u*,v*) is unstable. Hence, we draw the following conclusions.Theorem 5.
Assume thatα, β>0, d>1, and αβ+1>β, αβ2+2β>(αβ+1)2 hold. Then E(u*,v*) is unstable for λ∈(λ0H,dλ0H). That is, Turing instability occurs if
(32)dλ0Hβ-λβ>2dλβu*,x1l<j<x2lforsomej∈N+.


## 2.1. Stability of the Positive Coexistence and Hopf Bifurcations

The linearized operator of System (5) at E(u*,v*) is as follows:
(9)L(λ)=(∂2∂x2+αβ2+2β(αβ+1)2-1-1(αβ+1)2λβ2d∂2∂x2-λβ),Ln(λ)=(-n2l2+αβ2+2β(αβ+1)2-1-1(αβ+1)2λβ2-dn2l2-λβ).
The characteristic equations corresponding to Ln(λ) are
(10)μ2-Tn(λ)μ+Dn(λ)=0,n=0,1,2,…,
where
(11)Tn(λ)=-(1+d)n2l2-λβ+αβ2+2β(αβ+1)2-1=-(1+d)n2l2-(λ-λ0H)β,Dn(λ)=dn4l4+n2l2(d(1-αβ2+2β(αβ+1)2)+λβ)+λβ(1-β1+αβ)=dn4l4+n2l2(λ-dλ0H)β+λβu*,
where λ0H=(1/β)((αβ2+2β)/(αβ+1)2-1).Obviously, ifαβ2+2β≤(αβ+1)2, then Tn(λ)<0, Dn(λ)>0, which implies that all the roots of (10) are negative real-part. Thus the unique positive constant equilibrium E(u*,v*) is locally asymptotically stable.Indeed, in Section4, we can prove the global stability of the unique positive coexistence under the stronger conditions. Next, we will seek all the possible Hopf bifurcation values λ0, so we assume αβ2+2β>(1+αβ)2 holds.We will identify Hopf bifurcation valueλ0 which satisfies
(12)Tn(λ0)=0,Dn(λ0)>0,Tj(λ0)≠0,Dj(λ0)≠0,∀j≠n.
From (11), Tn(λ)<0, for all λ∈(λ0H,+∞), which means that the possible Hopf bifurcation points belong to the interval (0,λ0H]. For any Hopf bifurcation point λ*∈(0,λ0H], α(λ)±iω(λ) are a pair of conjugate eigenvalues of Ln(λ) near λ=λ*, where
(13)α(λ)=Tn(λ)2,ω(λ)=124Dn(λ)-Tn2(λ),
and α′(λ*)=-β/2<0.Hence, all Hopf bifurcation points reduce to the following set:(14)Λ1={λ∈(0,λ0H],forsomen∈N,0thecondition(12)issatisfied}.
Obviously, Tn(λ)=0 is equivalent to
(15)λn=λ0H-(1+d)n2βl2,(16)Dn(λn)=-n4l4-n2l2B0+λ0Hβu*,
where B0=2-(2αβ2+3β-dβ)/(1+αβ)2. Since D0(λ0)=λ0Hβu*>0, there exists N0∈N+ such that Dn(λn)>0, n=0,1,2,…,N0, and Dn(λn)≤0, n=N0+1, N0+2,….Clearly,Ti(λn)=((1+d)/l2)(n2-i2)≠0, for all i≠n. And for i≠n,
(17)Di(λn)=di4l4+i2l2(λn-dλ0H)β+λnβu*.
We discuss Di(λn)>0, for all i≠n in two cases.Case 1.
We can choose small enoughd>0 such that λn-dλ0H>0, that is, for n=0,1,…,N0, d<d0, where
(18)d0∶=βl2λ0H-N02βl2λ0H+N02>0.Case 2.
Whend>d0,
(19)dλ0H-λn=(d-1)λ0H+(1+d)n2l2β≤(d-1)λ0H+(1+d)N02l2β≜M*,λnu*≥(λ0H-(1+d)N02l2β)u*=(dλ0H-M*)u*.
Therefore,
(20)Di(λn)≥di4l4-i2l2M*β+(dλ0H-M*)u*β=g(i2l2).
In order that g(y)=dy2-M*βy+(dλ0H-M*)u*β>0 for all y∈R, we only need (M*)2β<4d(dλ0H-M*)u*. Thus, Di(λn)>0.
Clearly,λ0H=maxn∈{0,1,…,N0}λn.Summarizing the above analysis results, we draw our main conclusion in this subsection.Theorem 1.
Supposeα,β,d>0 and αβ+1>β, αβ2+2β>(1+αβ)2 hold. If there exists d0>0 such that 0<d<d0, or (M*)2β<4d(dλ0H-M*)u*, where d0 and M* are defined by (18) and (19), respectively. Then for System (5), the following results are true:(1)
The positive constant steady stateE(u*,v*) is locally asymptotically stable for λ∈(λ0H,+∞) and unstable for λ∈(0,λ0H);(2)
System (5) undergoes a Hopf bifurcation at λ=λn,n=0,1,…,N0 (defined by (15)) and the bifurcating periodic solutions can be parameterized into the following forms:
(21)u(s)(x,t)=s(ane2πit/T(s)+an¯e-2πit/T(s))cosnxl+o(s),v(s)(x,t)=s(bne2πit/T(s)+bn¯e-2πit/T(s))cosnxl+o(s).Furthermore, the periodic solutions bifurcating from E(u*,v*) at λ=λ0H are spatially homogeneous. And bifurcating periodic solutions from λ=λn, (n=1,…,N0) are spatially inhomogeneous.Next we only consider the direction and stability of Hopf bifurcation atλ=λ0H.Theorem 2.
For the System (5), the Hopf bifurcations at λ=λ0H are backward, and the bifurcating periodic solutions are locally asymptotically stable.Proof.
Here we apply the notations and calculations in [9, 21], we put
(22)q∶=(a0b0)=(λ0Hβ+iω0λ0Hβ21),q*∶=(a0*b0*)=(λ0Hβ22lπω0iω0-iλ0Hβ2lπω0),
such that 〈q*,q〉=1 and 〈q*,q-〉=0, where ω0=λ0Hβu*.
Recall that in our context,(23)f(λ,u,v)=u(1-u)-uvu+αv,g(λ,u,v)=λv(β-vu).
By direct computation, it follows that(24)c0=-2β2+2ω02(λ0H)2β4-2α(αβ+1)3λ0Hβ-4ω0λ0Hβ3i,d0=2β,e0=-2β2-2u*λ0Hβ3+2α(αβ+1)3λ0Hβ,f0=-2β,g0=αλ0Hβ2u*(αβ+1)4(-2αβ-2-3ω0λ0Hβi),h0=2u*(1β2+3ω0λ0Hβ3i),
and
(25)〈q*,Qqq〉=-1β+i(2λ0Hω0-ω0λ0Hβ2+αβ(αβ+1)3ω0),〈q*,Qqq-〉=〈q*¯,Qqq-〉¯=-1β+iω0(u*β-αβ(αβ+1)3),〈q*¯,Qqq〉=3β-i(2λ0Hω0-ω0λ0Hβ2+αβ(αβ+1)3ω0),〈q*,Cqqq-〉=-(3α2u*λ0Hβ(αβ+1)4+2β2u*)+i(αω0u*(αβ+1)3+λ0Hω0u*β+3ω0λ0Hβ3u*).
Hence, it is direct to calculate, by (24) and (25),
(26)H20=(c0d0)-〈q*,Qqq〉(a0b0)-〈q*¯,Qqq〉(a0¯b0¯)=(00),H11=(e0f0)-〈q*,Qqq-〉(a0b0)-〈q*¯,Qqq-〉(a0¯b0¯)=(00),
which implies that W20=W11=0. Thus, 〈q*,QW11q〉=〈q*,QW20q-〉=0. Therefore, by (25) and ω02=λ0Hβu*, we can obtain that
(27)Re(C1(λ0H))=Re(i2ω0(g20g11-2|g11|2-13|g02|2)+g212)=Re(i2ω0(〈q*,Qqq〉〈q*,Qqq-〉)+12〈q*,Cqqq-〉)=(u*2β2ω02-12λ0Hβ3)+(λ0Hβω02-1β2u*)-3α4u*λ0Hβ(1+αβ)4=-3α4u*λ0Hβ(1+αβ)4<0.
Since α′(λ0H)=-β/2, then the bifurcating periodic solutions are backward and  exist for τ<τ0H. And combining Tn(λ0H)=-((1+d)/l2)n2<0,Dn(λ0H)>0, for all n=1,2,…,N0, we can obtain that the bifurcating periodic solutions are asymptotically stable since Re(C1(λ0H))<0.Next, we give an example to illustrate the theoretical results above.Example 3.
Takingd=0.2,α=0.4,β=1.2,l=2, System (5) has a unique positive constant steady state E(0.1892,0.227). Obviously, αβ+1>β, αβ2+2β>(1+αβ)2. By the formula above, we obtain λ0H=0.2989. And D0(λ0H)<0,D1(λ1)>0. Thus, 0<d<d0=1. By Theorems 1 and 2, E(0.1892,0.227) is asymptotically stable for λ∈(0.2989,+∞) as illustrated in Figures 1(a) and 1(b). When λ crosses through the critical value, E(0.1892,0.227) losses its stability and a family of periodic solutions appear and are asymptotically stable since Re(C1(λ0H))=-0.9215<0 as illustrated in Figures 2(a) and 2(b).E
(
0.1892,0.227
) is asymptotically stable when λ=0.32>λ0H and the initial function u0(x)=0.5+0.2cosx, v0(x)=0.5+0.2cosx.
(a)
(b)Spatially homogenous periodic solutions are asymptotically stable whenλ=0.24<λ0H and the initial function u0(x)=0.5+0.2cosx, v0(x)=0.5+0.2cosx.
(a)
(b)We can compute the direction and stability of bifurcating periodic solutions by the same way as Theorem2 and omit it here. Combining the stability theory, we have the following results.Theorem 4.
For System (5), the bifurcating periodic solutions from λ=λn, n=1,2,…,N0 (defined by (15)) are backward (forward) if Re(C1(λn))<(>)0 and the spatially inhomogeneous Hopf bifurcations are unstable.


## 2.2.  Turing Instability Induced by Diffusion

In this subsection, we consider Turing instability with diffusion effect. As mentioned earlier, whenαβ2+2β≤(αβ+1)2, Tn(λ)<0,Dn(λ)>0, for all n∈{0,1,2,…}. Hence, no Turing instability occurs. Next we always confine αβ2+2β>(αβ+1)2.When System (5) has no diffusion, the characteristic equation (10) at E(u*,v*) can be transformed into
(28)μ2-(λ0H-λ)βμ+λβu*=0.
Then all the roots of (28) are negative real part if λ>λ0H. Combining the stability theory [22], we easily know that the unique positive equilibrium E(u*,v*) of System (5) without diffusion is asymptotically stable for λ∈(λ0H,+∞) and unstable for λ∈[0,λ0H). In the following, we mainly consider the occurrence of Turing instability induced by diffusion. Hence we confine λ∈(λ0H,+∞). That is, E(u*,v*) is locally asymptotically stable without diffusion effect.It is well-known thatE(u*,v*) is unstable if the characteristic equation (10) has at least a positive real-part root. Notice when λ>λ0H, Tn(λ)<0, for all n=0,1,2,…. The necessary condition of the instability of the equlibirium E(u*,v*) of System (5) is Dn(λ)<0 for some n∈N+={1,2,…}.Recall(29)Dn(λ)=dn4l4+n2l2(λ-dλ0H)β+λβu*≜g(n2l2),
where g(x)=dx2-(dλ0H-λ)βx+λβu*.Hence we needλ∈(λ0H,dλ0H),d>1 and Δ≜(dλ0H-λ)2β2-4dλβu*>0. That is, (dλ0H-λ)β>2dλβu*. Otherwise, Di(λ)≥0 for all i∈N.Obviously,g(x)=0 has two positive roots x1,x2 since
(30)x1+x2=(dλ0H-λ)β>0,x1x2>0,
where
(31)x1=12d((dλ0H-λ)β-(dλ0H-λ)2β2-4dλβu*),x2=12d((dλ0H-λ)β+(dλ0H-λ)2β2-4dλβu*).
If we can find some j∈N+ such that x1<j2/l2<x2. Then Dj(λ)<0. That is, x1l<j<x2l. The constant positive coexistence E(u*,v*) is unstable. Hence, we draw the following conclusions.Theorem 5.
Assume thatα, β>0, d>1, and αβ+1>β, αβ2+2β>(αβ+1)2 hold. Then E(u*,v*) is unstable for λ∈(λ0H,dλ0H). That is, Turing instability occurs if
(32)dλ0Hβ-λβ>2dλβu*,x1l<j<x2lforsomej∈N+.


## 3. Steady State Bifurcations

In this section, we still chooseλ as the main bifurcation parameter with α,β,d positive constants and study the existence of the steady state bifurcation points. The steady state solutions of System (5) satisfy the following elliptic system:
(33)uxx+u(1-u)-uvu+αv=0,x∈[0,lπ],dvxx+λv(β-vu)=0,x∈[0,lπ],ux(0,t)=ux(lπ,t)=vx(0,t)=vx(lπ,t)=0,t≥0,u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0.
System (33) has a unique positive constant coexistence E(u*,v*) if and only if αβ+1>β. By the same way as Section 2.1, the characteristic equation of the linearization of (33) at E(u*,v*) is
(34)μ2-Tn(λ)μ+Dn(λ)=0,n=0,1,2,…,
where Tn(λ),Dn(λ) are defined by (11).Next we identify all the possible steady state bifurcation points by [9] which satisfy the following conditions:
(35)Dn(λ0)=0,Tn(λ0)≠0,Dm(λ0)≠0,Tm(λ0)≠0,∀m≠n,dDn(λ0)dλ≠0.
For convenience, we rewrite
(36)Dn(λ)=dp2+p(λ-dλ0H)β+λβu*≜g(p),
where p=n2/l2.Obviously,Dn(λ)>0, for all n=0,1,2,… for λ∈[dλ0H,+∞). Hence the potential steady state bifurcation points set is as follows:
(37)∧1={λ∈(0,dλ0H)∣(35)aresatisfied}
with α, β, l, d fixed.ThusDn(λ)=0 has two roots as
(38)p±(λ)=12d((dλ0H-λ)β±(dλ0H-λ)2β2-4dλβu*).
Let
(39)h(λ)=(dλ0H-λ)2β2-4dλβu*=λ2β2-2dλβ(λ0Hβ+2u*)+d2β2(λ0H)2.
Since Δ≜4d2β2(λ0Hβ+2u*)2-4β4d2(λ0H)2=16d2β2u*(λ0Hβ+u*)>0, h(λ)=0 has two positive roots as
(40)λ_=dβ(λ0Hβ+2u*-2u*(λ0Hβ+u*)),λ-=dβ(λ0Hβ+2u*+2u*(λ0Hβ+u*)).
By h(dλ0H)=-4dλβu*<0, hence λ_<dλ0H<λ-. That is, when λ∈(λ_,dλ0H), Dn(λ)>0, for all n∈N. No steady state bifurcation occurs. Then the potential steady state bifurcation points reduce to the following set:
(41)∧2={λ∈(0,λ_]∣(35)aresatisfied},
with α, β, l, d fixed.Next, the properties ofP±(λ),λ∈(0,λ_] are discussed and direct computation can obtain the following results.Lemma 6.
Suppose thatαβ>1+β, αβ2+2β>(αβ+1)2 hold. When λ∈(0,λ_], P+(λ) is decreasing on λ and P-(λ) is increasing on λ. Furthermore,
(42)limλ→0+P+(λ)=λ0Hβ,P+(λ_-)=P-(λ_-)=(dλ0H-λ_)β2d,limλ→0+P-(λ)=0,limλ→λ_-P+′(λ)=-∞,limλ→λ_-P-′(λ)=+∞.From the properties ofP±(λ), if 0<p=n2/l2<λ0Hβ, then there exists a unique
(43)λnS=pβdλ0H-dp2pβ+βu*∈(0,λ_],
such that
(44)P+(λnS)=n2l2orP-(λnS)=n2l2.
Thus, Dn(λnS)=0.Definel~n=n/λ0Hβ. Then for any l>l~n, there exists a unique λiS∈(0,λ_] such that i2/l2=P+(λiS) or i2/l2=P-(λiS).λ
n
S
∈
(
0
,
λ
_
] are potential steady state bifurcation values. But it is possible that for some i,j∈N and i≠j such that P+(λiS)=P-(λjS). In this case, 0 is a double eigenvalue of L(λ) for λ=λiS=λjS. In addition, it is also possible that λiS=λjH. Hence, the dimension of center manifold at E(u*,v*) can be as high as 3.Next we will prove that there are only countable manyl>0, in fact only finitely many l∈(0,M) for any given M>0, such that λiS=λjS, λiS=λjH for some i,j∈N. Let En(λ,l)=l2Tn(λ), Fn(λ,l)=l4Dn(λ) be polynomials of λ and l. Hence, on (λ,l)-plane, qn={(λ,l):En(λ,l)=0}, pn={(λ,l):Fn(λ,l)=0} are finitely many analytic curves. We fix λ∈(0,λ_], so for any M>0, there are only finitely many i,j∈N such that qi⋂‍((0,λ_]×[0,M])≠∅,pj⋂‍((0,λ_]×[0,M])≠∅. So pj,qi have at most only finitely many intersection points. Thus the intersection points of different qi,pj in (0,λ_]×[0,+∞] are at most countable. Define
(45)LE={l>0∣Fi(λ,l)=Fj(λ,l)0orEi(λ,l)=Fj(λ,l)0forsomeλ∈(0,λ_],andi,j∈N}.We assume thatl∈R/LE, and thus 0 is a simple eigenvalue. And
(46)dD(λnS)dλ=pβ+βu*>0.
Obviously, Tm(λnS)≠0,Dm(λnS)≠0, for all m≠n.Summarizing the above, we can obtain the main results in this section.Theorem 7.
Suppose thatα,β,d are positive constants and αβ>1+β, αβ2+2β>(αβ+1)2 hold. P±(λ) are defined as (38) and l~n=n/λ0Hβ.
If for somen∈N, l∈(l~n,+∞)∖LE, there exactly exists one point λnS=(βdλ0Hp-dp2)/(βp+βu*)∈(0,λ_), such that p+(λnS)=n2/l2 or P-(λnS)=n2/l2, where p=n2/l2. Then there is a smooth curve Γn of positive solutions of (33) bifurcating from (λnS,u*,v*). Moreover, near (λnS,u*,v*), Γn={λ(s),u(s),v(s):s∈(-ϵ,+ϵ)}, where
(47)u(s)=u*+sancosnxl+sψ1(s),v(s)=v*+sbncosnxl+sψ2(s),
for s∈(-ϵ,+ϵ) for some C∞ smooth function λ,ψ1,ψ2 such that λ(0)=λnS and ψ1(0)=ψ2(0)=0. Here an,bn satisfy Ln(λnS)(an,bn)T=(0,0)T.In the following, we will discuss the interaction between the Hopf and steady state bifurcation points. All possible Hopf bifurcation pointsλiH are identified in Section 2.1 and all possible steady state bifurcation points λjS can be identified in Section 3. And it has previously shown that at most countable l∈LE such that λjH=λiS or λiS=λjS for some i,j∈N. Hence high-dimensional center manifold from (λ,u*,v*) is still possible. We do not consider it here. We assume l∉LE, Hopf bifurcations, and steady state bifurcation points may occur at this points in the following four cases.Case 1.
NeitherλiH nor λjS exists, and then there is no bifurcation points;Case 2.
λ
j
S exists but not λiH, and then there is one steady state bifurcation point and no Hopf bifurcation point.Case 3.
BothλjS and λiH exist, and then(a)
λiH<λjS; then there is one steady state bifurcation point and no Hopf bifurcation since Di(λiH)<0;(b)
λjS<λiH; then there is one steady state bifurcation point and one Hopf bifurcation.Case 4.
λ
i
H exists but no λjS, then there is no steady state bifurcation point and one Hopf bifurcation.Example 8.
In Figure3, for n≥18 and n∈N, Case 1 occurs, and there are no bifurcation points. For 15≤n≤17, Case 2 occurs, and there exists 3 steady state bifurcation points and no Hopf bifurcation points. For n=14, Case 3 (a) occurs, and there is one steady state bifurcation point λ14S=0.0314 and no Hopf bifurcation points since D14(λ14H)<0. For 1≤n≤13, Case 3 (b) occurs, and there exists 13 Hopf bifurcation points and 13 steady state bifurcation points. For n=0, Case 4 occurs, and there exists one Hopf bifurcation point and no steady state bifurcation points.Figure 3
Graph ofEn(λ,p)Fn(λ,p)=0. Here d=0.5, α=0.4, β=1.2, l=30, u*=0.1892, v*=0.227, and λ0H=0.2989. The horizontal lines are p=n2/l2, where n=0,1,2,….


## 4.  Global Stability of the Positive Coexistence

In this section, we will prove the global stability of the positive constant coexistence by the upper and lower solutions method in Pao [23, 24] and Chen and Shi [13].Theorem 9.
Assume thatd,β,λ are positive constants and α>1. Then for system (5), the positive constant equilibrium E(u*,v*) is globally asymptotically stable if αβ<1 or αβ>1+β. That is, for any initial values u0(x)>0,v0(x)≥0,
(48)limt→∞u(x,t)=u*,limt→∞v(x,t)=v*,uniformlyinx∈[0,lπ]
if
(49)αβ<1orαβ>1+β.Proof.
It is well known that ifc>0 and w(x,t)>0 satisfies
(50)∂w∂t=DΔw+w(c-w),x∈Ω,t>0,∂w∂ν=0,x∈∂Ω,w(x,0)≥(≢)0,x∈Ω,
then limt→∞w(x,t)=c uniformly in x∈Ω¯.
We can chooseϵ0 satisfying
(51)0<ϵ0<β(1-1/α)1+β,sinceα>1.
By the first equation of (5), we have
(52)∂u∂t≤uxx+u(x,t)(1-u(x,t)).
By the comparison principle of the parabolic equation, we obtain
(53)limsupt→+∞maxx∈[0,lπ]u(x,t)≤1.
Then there exists t0>0 such that u(x,t)<1+ϵ0≜u-, for all t>t0, x∈[0,lπ]. Thus, by the second equation of (5),
(54)∂v∂t≤dvxx+λv(β-v1+ϵ0),∀t>t0.
By the comparison principle again, limsupt→+∞maxx∈[0,lπ]v(x,t)≤β(1+ϵ0). There exists t1>t0 such that v(x,t)<β(1+ϵ0)+ϵ0≜v-, for all t>t1,x∈[0,lπ].
On the other hand,(55)∂u∂t=uxx+u(1-u-vu+αv)≥uxx+u(1-1α-u).
By the comparison principle, liminft→+∞minx∈[0,lπ]u(x,t)≥1-1/α>0, since α>1. Thus, there exists t2>t1 such that u(x,t)>1-1/α-ϵ0≜u_>0, for all t>t2. Again in turn this implies
(56)∂v∂t≥dvxx+λv(β-vu_),t>t2.
Hence, liminft→+∞minx∈[0,lπ]v(x,t)≥β(1-1/α-ϵ0). There exists t3>t2 such that
(57)v(x,t)>β(1-1α-ϵ0)-ϵ0≜v_>0,∀t>t3,
since (51) holds. Therefore, for t>t3,u_≤u(x,t)≤u-,  v_≤v(x,t)≤v-, for all x∈[0,lπ] and u-,u_,v-,v_ satisfy
(58)1-u--v_u-+αv_<0,β-v-u-<0,1-u_-v-u_+αv->0,β-v_u_>0.
Equation (58) implies that (u-,v-) and (u_,v_) are a pair of upper and lower solutions of system (5), respectively (see [23, 24]). It is obvious that there exists M>0 such that for any (u_,v_)≤(u1,v1), (u2,v2)≤(u-,v-),
(59)|u1(1-u1)-u1v1u1+αv1-u2(1-u2)+u2v2u2+αv2|≤M(|u1-u2|+|v1-v2|),|λv1(β-v1u1)-λv2(β-v2u2)|≤M(|u1-u2|+|v1-v2|).
We construct the iteration sequences (u-(m),v-(m)) and (u_(m),v_(m)), m≥1 satisfy
(60)u-(m)=u-(m-1)+1Mu-(m-1)×(1-u-(m-1)-v_(m-1)u-(m-1)+αv_(m-1)),v-(m)=v-(m-1)+1Mλv-(m-1)(β-v-(m-1)u-(m-1)),u_(m)=u_(m-1)+1Mu_(m-1)×(1-u_(m-1)-v-(m-1)u_(m-1)+αv-(m-1)),v_(m)=v_(m-1)+1Mλv_(m-1)(β-v_(m-1)u_(m-1)),
where (u-(0),v-(0))=(u-,v-),  (u_(0),v_(0))=(u_,v_). Then by (58) we can obtain
(61)(u_,v_)≤(u_(m),v_(m))≤(u_(m+1),v_(m+1))≤⋯≤(u-(m+1),v-(m+1))≤(u-(m),v-(m))≤(u-,v-).
Then there exist (u~,v~) and (u~,v~) such that
(62)limm→∞u-(m)=u~,limm→∞v-(m)=v~,limm→∞u~(m)=u~,limm→∞v~(m)=v~,
and (u_,v_)≤(u~,v~)≤(u~,v~)≤(u-,v-). And
(63)1-u~-v~u~+αv~=0,β-v~u~=0,1-u~-v~u~+αv~=0,β-v~u~=0.
According to (63), we have
(64)(1-u~)(u~+αβu~)=βu~,(1-u~)(u~+αβu~)=βu~.
By (64), we can obtain
(65)(u~-u~)(1-αβ+β-u~-u~)=0.
If we assume that u~≠u~, then
(66)1-αβ+β-u~-u~=0.
Substituting (66) into (64) and simplifying, we can obtain
(67)(αβ-1)u~2+u~(1-2αβ+α2β2-αβ2+β)+β(α-1)(1-αβ+β)=0,(αβ-1)u~2+u~(1-2αβ+α2β2-αβ2+β)+β(α-1)(1-αβ+β)=0.
Thus the quadratic equation
(68)(αβ-1)x2+x(1-2αβ+α2β2-αβ2+β)+β(α-1)(1-αβ+β)=0
has two different positive roots u~ and u~.
On the other hand, whenαβ<1 or αβ>1+β,
(69)x1x2=β(α-1)1-αβ+βαβ-1<0,
where x1,x2 are two roots of (68). Thus (68) cannot have two positive roots.
Whenαβ=1, (68) becomes a linear equation and has at most one real root. This contradicts the above. Hence, u~=u~. Similarly, v~=v~. Then from the results in [23, 24], the solution (u(x,t),v(x,t)) of System (5) satisfies
(70)limt→+∞u(x,t)=u*,limt→+∞v(x,t)=v*,00uniformlyinx∈[0,lπ].



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