# Bogdanov-Takens Bifurcation of a Delayed Ratio-Dependent Holling-Tanner Predator Prey System

Authors: Xia Liu; Yanwei Liu; Jinling Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A delayed predator prey system with refuge and constant rate harvesting is discussed by applying the normal form theory of retarded functional differential equations introduced by Faria and Magalhães. The analysis results show that under some conditions the system has a Bogdanov-Takens singularity. A versal unfolding of the system at this singularity is obtained. Our main results illustrate that the delay has an important effect on the dynamical behaviors of the system.


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## Body

## 1. Introduction

It is well known that the multiple bifurcations will occur when a predator prey system (ODE) with more interior positive equilibria, such as Bogdanov-Takens bifurcation, Hopf bifurcation, and backward bifurcation; see [1–5] for example. However, when the predator prey systems with delay and Bogdanov-Takens bifurcation are researched relative few (see [6–8] and the reference therein) using similar methods as [6–8], the authors of [9–11] consider the Bogdanov-Takens bifurcation of some delayed single inertial neuron or oscillator models.Motivated by the works of [5, 6], we mainly consider the Bogdanov-Takens bifurcation of the following system:
(1)x˙=rx(1-xK)-αy(x-m-)Ay+x-m--h-,y˙=sy(1-by(t-τ-)x(t-τ-)-m-),
where x and y stand for prey and predator population (or densities) at time t, respectively. The predator growth is of logistic type with growth rate r and carrying capacity K in the absence of predation; α and A stand for the predator capturing rate and half saturation constant, respectively; s is the intrinsic growth rate of predator; however, carrying capacity x/b (b is the conversion rate of prey into predators) is the function on the population size of prey. The parameters α, A, m-, h-, s, b, and τ- are all positive constants. m- is a constant number of prey using refuges; h- is the rate of prey harvesting.For simplicity, we first rescale system (1). Let X=x-m-, Y=y; system (1) can be written as (still denoting X, Y as x, y)
(2)x˙=r(x+m-)(1-x+m-K)-αxyAy+x-h-,y˙=sy(1-by(t-τ-)x(t-τ-)).Next, lett-=rt, X(t-)=x(t)/K, and Y(t-)=αy(t)/rK; then system (2) takes the form (still denoting X, Y, t- as x, y, t)
(3)x˙=(x+m)(1-x-m)-xyay+x-h,y˙=δy(β-y(t-τ)x(t-τ)),
where m=m-/K, a=Ar/α, δ=sh-/α, β=α/br, h=h-/r, and τ=rτ-.Whenτ=0, we have known that for some parameter values system (3) exhibits Bogdanov-Takens bifurcation (see [5]). Summarizing the methods used by [6] and the formulae in [12], the sufficient conditions which depend on delay to guarantee that system (3) has a Bogdanov-Takens singularity will be given. Therefore, the delay has effect on the occurrence of Bogdanov-Takens bifurcation.In the next section we will compute the normal form and give the versal unfolding of system (3) at the degenerate equilibrium.


## 2. Bogdanov-Takens Bifurcation

System (3) can also be written as
(4)x˙=τ((x+m)(1-x-m)-xyay+x-h),y˙=τδy(β-y(t-1)x(t-1)).Let(5)0<m<12(1-βaβ+1),h=h~=14(βaβ+1-1)2+mβaβ+1.ThenP*=(x*,y*) is an interior positive equilibrium of systems (3) and (4), where x*=-(1/2)((β/(aβ+1))+2m-1), y*=βx*.In order to discuss the properties of system (4) in the neighborhood of the equilibrium P*=(x*,y*), let x-=x-x*, y-=y-y*; then P* is translated to (0,0), and system (4) becomes (still denoting x-, y- as x, y)
(6)x˙=τβ(aβ+1)2x-τ(aβ+1)2y+g1x2x˙=+g2xy+g3y2+h.o.t,y˙=τδβ2x(t-1)-τδβy(t-1)+g4x2(t-1)y˙=+g5x(t-1)y(t-1)+g6x(t-1)yy˙=+g7y(t-1)y+h.o.t,
where h.o.t denotes the higher order terms and
(7)g1=-(2aδβ2τβ+(2m-1)(aβ+1)+τ),g2=4aδβτβ+(2m-1)(aβ+1),g3=-2aδτβ+(2m-1)(aβ+1),g4=2τδβ2(aβ+1)β+(2m-1)(aβ+1),g5=-2τδβ(aβ+1)β+(2m-1)(aβ+1),g6=-2τδβ(aβ+1)β+(2m-1)(aβ+1),g7=2τδ(aβ+1)β+(2m-1)(aβ+1).The characteristic equation of the linearized part of system (6) is
(8)F(λ)=λ2+[τδβe-λ-τβ(aβ+1)2]λ.
Clearly, if
(9)δ=1(aβ+1)2,τ≠1δβ,that is,τ≠(aβ+1)2β;
then λ=0 is double zero eigenvalue; if δ=1/(aβ+1)2 and τ=1/δβ, that is, τ=(aβ+1)2/β, then λ=0 is triple zero eigenvalue. We will mainly discuss the first case in this paper.According to the normal form theory developed by Faria and Magalhães [13], first, rewrite system (4) as X˙(t)=L(Xt), where X(t)=(x1(t),x2(t)), L(ϕ)=L(ϕ1(-1)ϕ2(0)), and ϕ=(ϕ1,ϕ2). Take A0 as the infinitesimal generator of system. Let Λ={0}, and denote by P the invariant space of A0 associated with the eigenvalue λ=0; using the formal adjoint theory of RFDE in [13], the phase space C1 can be decomposed by Λ as C1=P⊕Q. Define Φ and Ψ as the bases for P and P*, the space associated with the eigenvalue λ=0 of the adjoint equation, respectively, and to be normalized such that (Ψ,Φ)=I, Φ˙=ΦJ, and Ψ˙=-JΨ, where Φ and Ψ are 2×2 matrices.Next we will find theΦ(θ) and Ψ(s) based on the techniques developed by [14].Lemma 1 (see Xu and Huang [14]).
The bases ofP and its dual space P* have the following representations:
(10)P=spanΦ,Φ(θ)=(φ1(θ),φ2(θ)),-1≤θ≤0,P*=spanΨ,Ψ(s)=col(ψ1(s),ψ2(s)),0≤s≤1,
where φ1(θ)=φ10∈Rn∖{0}, φ2(θ)=φ20+φ10θ, and φ20∈Rn and ψ2(s)=ψ20∈Rn*∖{0}, and ψ1(s)=ψ10-sψ20, ψ10∈Rn*, which satisfy(1)
(A+B)φ10=0,(2)
(A+B)φ20=(B+I)φ10,(3)
ψ20(A+B)=0,(4)
ψ10(A+B)=ψ20(B+I),(5)
ψ20φ20-(1/2)ψ20Bφ10+ψ20Bφ20=1,(6)
ψ10φ20-(1/2)ψ10Bφ10+ψ10Bφ20+(1/6)ψ20Bφ10-(1/2)ψ20Bφ20=0.By (6), we have A=(τδβ-τδ00) and B=(00τδβ2-τδβ); using Lemma 1, we obtain
(11)Φ(θ)=(11τδβ+θββθ),-1≤θ≤0,Ψ(s)=(m1+sβnm2-sn-βnn),0≤s≤1,
where m1=τδβ(τδβ-2)/2(τδβ-1)2=τβ[τβ-2(aβ+1)2]/2[τβ-(aβ+1)2]2, m2=(1+(τδβ-1)2)/2β(τδβ-1)2=(1/2β)(1+(aβ+1)4/(τβ-(aβ+1))2), and n=τδ/(τδβ-1)=τ/(τβ-(aβ+1)2).The matrixJ satisfying Φ˙=ΦJ is given by J=(0100). System (6) in the center manifold is equivalent to the system z˙=Jz+Ψ(0)F(Φz), where(12)F(ϕ)=(g1ϕ12(0)+g2ϕ1(0)ϕ2(0)+g3ϕ22(0)+h.o.tg4ϕ12(-1)+g5ϕ1(-1)ϕ2(-1)+g6ϕ1(-1)ϕ2(0)+g7ϕ2(-1)ϕ2(0)+h.o.t).It is easy to obtain
(13)Φ(θ)z=(z1+(1τδβ+θ)z2β(z1+x2θ)),ϕ1(0)=z1+1τδβz2,ϕ1(-1)=z1+(1τδβ-1)z2,ϕ2(0)=βz1,ϕ2(-1)=β(z1-z2).
Hence (6) becomes
(14)z˙1=z2-τm1z12-2m1δβz1z2+l1z22+h.o.t,z˙2=nτβz12+2nδz1z2+l2z22+h.o.t,
where l1=m1g1/τ2δ2β2+2m2(δ+μ1)(aβ+1)(1-τδβ)/τδ2(β+(aβ+1)(2m-1)), l2=-(ng1/βτ2δ2)+2n(δ+μ1)(aβ+1)(1-τδβ)/τδ2(β+(aβ+1)(2m-1)).After a series of transformations we obtain(15)z˙1=z2+h.o.t,z˙2=d1z12+d2z1z2+h.o.t,
where d1=βnτ=βτ2/(τβ-(aβ+1)2), d2=-τm1+n/δ=-(τ2β2+2(aβ+1)2[(aβ+1)2-2τβ])/(τβ-(aβ+1)2)2; if τ≠(2±2)/δβ, then d2≠0.Hence, we have the following theorem.Theorem 2.
Let (5), (9), and τ≠(2±2)/δβ hold. Then the equilibrium P* of system (4) is a Bogdanov-Takens singularity.In the following, we will do versal unfolding of system (4) at P*:
(16)x˙=τ((x+m)(1-x-m)-xyay+x-h),y˙=τ(δ+μ1)y(β+μ2-y(t-1)x(t-1)).
When μ1=μ2=0, system (16) has a Bogdanov-Takens singularity P* and a two-dimensional center manifold exists.The Taylor expansion of system (16) at P* takes the form
(17)x˙=τδβx-τδy+g1x2+g2xy+g3y2+h.o.t,y˙=w0μ2+τ(δ+μ1)β2x(t-1)-τ(δ+μ1)βy(t-1)y˙=+τδμ2y+w4x2(t-1)+w5x(t-1)y(t-1)y˙=+w6x(t-1)y+w7y(t-1)y+h.o.t,
where
(18)w0=-τδβ[β+(2m-1)(aβ+1)]2(aβ+1),w4=2τ(δ+μ1)β2(aβ+1)β+(2m-1)(aβ+1),w5=-2τ(δ+μ1)β(aβ+1)β+(2m-1)(aβ+1),w6=-2τ(δ+μ1)(aβ+1)β+(2m-1)(aβ+1),w7=2τ(δ+μ1)(aβ+1)β+(2m-1)(aβ+1).We decompose the enlarged phase spaceBC of system as BC=P⊕Kerπ. Then y in system can be decomposed as y=Φz+u with z∈R2 and u∈Q′. Hence, system is decomposed as
(19)z˙=B1+B2z+Ψ(0)G(Φz+u),u˙=AQ′u+(I-π)X0u˙=×[B0+B2(Φ0z+u(0))+G(Φz+u)],
where(20)X0(θ)={I,θ=00,-1≤θ<0,B0=(0w0μ2),B1=Ψ(0)B0=(w0m2μ2w0nμ2),B2=Ψ(0)(τδβϕ1(0)-τδϕ2(0)τ(δ+μ1)β2ϕ1(-1)-τ(δ+μ1)βϕ2(-1)+τδμ2ϕ2(0)),G(ϕ)=(g1ϕ12(0)+g2ϕ1(0)ϕ2(0)+g3ϕ22(0)+h.o.tw4ϕ12(-1)+w5ϕ1(-1)ϕ2(-1)+w6ϕ1(-1)ϕ2(0)+w7ϕ2(-1)ϕ2(0)+h.o.t).To compute the normal form of system at P*, consider z˙=B1+B2z+Ψ(0)G(Φz); together with (13) we obtain
(21)z˙1=w0m2μ2+z2+m2τδβμ2z1+m2βμ1δz2z˙1=-τm1z12-2m1δβz1z2+l1z22+h.o.t,z˙2=w0nμ2+nτδβμ2z1+nβμ1δz2z˙2=+nτβz12+2nδz1z2+l2z22+h.o.t.Following the normal form formula in Kuznetsov [12], system (21) can be reduced to
(22)z˙1=z2+h.o.t,z˙2=γ1+γ2z1+d1z12+d2z1z2+h.o.t,
where
(23)γ1=-τδβ[β+(2m-1)(aβ+1)]2(aβ+1)nμ2γ1=τ2β[β+(2m-1)(aβ+1)]2(aβ+1)3[(aβ+1)2-τβ]μ2,γ2=β2n2τn-τm1δμ1γ2=+(nτδβ-2w0nδβ-w0n(l2δβ-2+2m2β)δβaaaaaaaa+(-2w0m22β-2w0m2l2δβ)nτ(m22βm2τδ2β2+2l1w0nδβ+2w0m2+++++++aaaa-2w0m22β-2w0m2l2δβ))+++++×(-τδ+m2τδβ+n)-12w0nδβ-w0n(l2δβ-2+2m2β)δβ)μ2.Hence system (4) exist the following bifurcation curves in a small neighborhood of the origin in the (μ1,μ2) plane.Theorem 3.
Let (5), (9), and τ≠(2±2)/δβ hold. Then system (4) admits the following bifurcations:(i)
a saddle-node bifurcation curveSN={(μ1,μ2):γ1=(1/4d1)γ22};(ii)
a Hopf bifurcation curveH={(μ1,μ2):μ2=0,γ2<0};(iii)
a homoclinic bifurcation curveHL={(μ1,μ2):γ1=-(6/25d1)γ22,γ2<0}.



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