# Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

Authors: Lv-Zhou Zheng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.


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## Body

## 1. Introduction

In general, delay differential equations display more complicated dynamics than ordinary differential equations as a time delay could bring a switch in the stability of equilibria and induce various oscillations and periodic solutions. Neural Networks with time delay or distributed delays have been investigated by some researchers [1–4]. Different from [1–4], distributed delays are introduced into a Lotka-Volterra predator-prey system. Lotka-Volterra predator-prey system is always the hot question in mathematical ecology, and many scholars have made deep and broad study in this area [5–10]. May [5] first proposed a model describing a predator-prey system with delay:
(1)x˙(t)=x(t)[r1-a11x(t-τ)-a12y(t)],y˙(t)=y(t)[-r2-a21x(t)-a22y(t)],
where x(t), y(t) denote the densities of prey and predator at time t, respectively, τ>0 denotes growth time of prey, r1>0, r2>0 denote the growth rates of prey and predator at time t, respectively, aii>0(i=1,2) denote the inner impact factors in two populations, respectively, and aij>0(i≠j) denote the mutual impact factors in two populations, respectively.Recently, Song and Wei [9] developed a Logistic predator-prey model with discrete and distributed delays:
(2)x˙(t)=rx(t)[1-a1x(t-τ)-a2∫-∞tf(t-s)x(s)ds],
where r,τ,a1, and a2 are positive constants and f(t) is the kernel function satisfying f(t)≥0(t≥0) and ∫0∞f(t)dt=1.In nature, living space and resources are limited. Therefore it is advisable to study a predator-prey model with competition term. The competition term describes the competitive interactions between the two species. Associated with these works in [5, 9, 10], in this paper, a predator-prey model with distributed delays and competition term is considered:
(3)x--˙(t)=x--(t)[r-r1x--(t-τ)∫-∞tf(t-s)-r2∫-∞tf(t-s)x--(s)dshhhhhhh-a1y--(t-τ)∫-∞tf(t-s)],y--˙(t)=y--(t)[-r3+a2x--(t-τ)-r4y--(t-τ)]-cy--(t)y--2(t-τ).Obviously, system (3) improves and contains most of existing models. For (3), we define the kernel function
(4)f(t)=σe-σt,σ>0.Denote(5)u--(t)=∫-∞tσe-σ(t-s)x--(s)ds.
Then we have
(6)x--˙(t)=x--(t)[r-r1x--(t-τ)-r2u--(t)-a1y--(t-τ)]u--˙(t)=σx--(t)-σu--(t)y--˙(t)=y--(t)[-r3+a2x--(t-τ)-r4y--(t-τ)]hhhhh-cy--(t)y--2(t-τ).Letx--(t)=(r/r1)x-(t), u--(t)=(r/r1)u-(t), and y--(t)=(r3/r4)y-(t); by (6),
(7)x-˙(t)=rx-(t)[1-x-(t-τ)-r2r1u-(t)-a1r3rr4y-(t-τ)]u-˙(t)=σx-(t)-σu-(t)y-˙(t)=r3y-(t)[-y-(t-τ)-cr3r42y-2(t-τ)-1+a2rr1r3x-(t-τ)cr3r42hhhhhhhhhhhh-y-(t-τ)-cr3r42y-2(t-τ)].Denotea=r2/r1, b=a1r3/rr4, c=a2r/r1r3, and d=cr3/r42; from (7), it follows that
(8)x-˙(t)=rx-(t)[1-x-(t-τ)-au-(t)-by-(t-τ)]u-˙(t)=σx-(t)-σu-(t)y-˙(t)=r3y-(t)hhhhh×[dy-2-1+cx-(t-τ)-y-(t-τ)-dy-2(t-τ)].


## 2. Main Results

To ensure system (8) with positive equilibrium points, let us suppose system (8) with conditionsG1:
Δ-(1+a)d>0,(2+b)(1+a)d-Δ>0,where Δ=(1+a+bc)2-4(1+a)d(1+a-c).Clearly,z*=(x*,u*,y*) is a positive equilibrium point of system (8) if and only if z*=(x*,u*,y*) satisfies
(9)0=1-x*-au*-by*0=bx*-bu*0=-1+cx*-y*-dy*2.Obviously, under the assumptionG1, system (8) has unique positive equilibrium points z*=(x*,u*,y*), where
(10)x*=(2+b)(1+a)d-bΔ2(1+a)2du*=x*y*=Δ-(1+a+bc)2(1+a)d.Letx-(t)=x(t)-x*, u-(t)=u(t)-u*, and y-(t)=y(t)-y*; then
(11)x˙(t)=r[x(t)+x*][-x(t-τ)-au(t)-by(t-τ)]u˙(t)=bx(t)-bu(t)y˙(t)=r3[y(t)+y*]hhhhl×[cx(t-τ)-(1+2dy*)y(t)-dy2(t)].Linearizing system (11) at (0,0,0),
(12)x˙(t)=rx*[-x(t-τ)-au(t)-by(t-τ)]u˙(t)=bx(t)-bu(t)y˙(t)=r3y*[cx(t-τ)-(1+2dy*)y(t)],
then the characteristic equation of (12) is
(13)λ3+d1λ2+d2λ-d3+(d4λ2+d5λ-d6)e-λτ+(d7λ+d8)e-2λτ=0,
where
(14)d1=b-r3y*(1+2dy*),d2=rabx*-rr3x*y*(1+2dy*),d3=rr3abx*y*(1+2dy*),d4=rx*,d5=rbx*-rr3x*y*(1+2dy*),d6=rr3bx*y*(1+2dy*),d7=rr3bcx*y*,d8=rr3b2cx*y*.By (13), we can get
(15)(λ3+d1λ2+d2λ-d3)eλτ+(d4λ2+d5λ-d6)+(d7λ+d8)e-λτ=0.Clearly,iω(ω>0) is a solution of (15) if and only if ω satisfies
(16)(-iω3-d1ω2+d2iω-d3)eiω+(-d4ω2+d5iω-d6)+(d7iω+d8)e-iω=0.
Substituting eiω=cos⁡ωτ+isinωτ, e-iω=cos⁡ωτ-isinωτ into (16), then
(17)(-iω3-d1ω2+d2iω-d3)(cos⁡ωτ+isinωτ)+(-d4ω2+d5iω-d6)+(d7iω+d8)(cos⁡ωτ-isinωτ)=0;
that is,
(18)[(ω3-d2ω)-d7ω]cos⁡ωτ+[d8+(d1ω+d3)ω3]sinωτ=d5ω,[(ω3-d2ω)+d7ω]sinωτ+[d8-(d1ω+d3)]cos⁡ωτ=d4ω2+d6;
thus
(19)sinωτ=(d4ω5+[d6-d4(d2+d7)]ω3+d1d5ω2hh+[d5(d3-d8)-d6(d2+d7)]ωω5)×(ω6-2d2ω4+(d12+d22-d72)ω2hhhhl+2d1d3ω+d32-d82)-1,cos⁡ωτ=(d5ω4-d1d4ω3+(d5d7-d2d5-d4d8-d3d4)ω2hh-d1d6ω-(d3+d8)d6d5ω4)×(ω6-2d2ω4+(d12+d22-d72)ω2hhhhl+2d1d3ω+d32-d82)-1.Let(20)e1=-2d2,e2=d12+d22-d72,e3=2d1d3,e4=d32-d82,e5=d4,e6=d6-d4(d2+d7),e7=d1d5,e8=[d5(d3-d8)-d6(d2+d7)],e9=d5,e10=-d1d4,e11=d5d7-d2d5-d4d8-d3d4,e12=-d1d6,e13=-(d3+d8),
so
(21)sinωτ=e5ω5+e6ω3+e7ω2+e8ωω6+e1ω4+e2ω2+e3ω+e4,cos⁡ωτ=e9ω4+e10ω3+e11ω2+e12ω+e13ω6+e1ω4+e2ω2+e3ω+e4.
Combined with sin2ωτ+cos⁡2⁡ωτ=1, we get
(22)ω12+f1ω10+f2ω8+f3ω7+f4ω6+f5ω5+f6ω4+f7ω3+f8ω2+f9ω+f10=0,
where
(23)f1=2e1-e52,f2=2e2+e12-2e5e6-e92,f3=2(e3-e5e7-e9e10),f4=2e4+2e1e2-e62-e102-2e5e8-2e9e11,f5=2(e1e3-e6e7-e9e12-e10e11),f6=e22+2e1e4-e72-e112-2e6e8-2e9e13-2e10e12,f7=2(e2e3-e7e8-e10e13-e11e12),f8=e32+2e2e4-e82-e122-2e11e13,f9=2(e3e4-e12e13),f10=e42-e132.Let(24)G(ω)=ω12+f1ω10+f2ω8+f3ω7+f4ω6+f5ω5+f6ω4+f7ω3+f8ω2+f9ω+f10.AssumeG2:
system (22) has at least one positive real solution.On the other hand,(25)cos⁡ωkτ=e9ωk4+e10ωk3+e11ωk2+e12ωk+e13ωk6+e1ωk4+e2ωk2+e3ωk+e4;
thus
(26)τk(j)=1ωk×{arccose9ωk4+e10ωk3+e11ωk2+e12ωk+e13ωk6+e1ωk4+e2ωk2+e3ωk+e4+2jπ},
where k=1,…,12, j=0,1,…,±iωk. Define
(27)τ0=τk0(0)=min⁡k∈{1,…,12}⁡{τk(0)},ω0=ωk0.Whenτ=0, we can obtain
(28)λ3+(d1+d4)λ2+(d2+d5+d7)λ+d8-d3-d6=0.
By Hurwitz criterion, it follows that the solutions of (28) are all of negative real parts when τ=0, so the positive equilibrium point Z* is locally asymptotically stable.In the following, we assumeG3:
Re(dλ/dτ)|τ=τ0≠0.From (13),
(29)(3λ2+2d1λ+d2)eλτ+(λ3+d1λ2+d2λ-d3)eλτ×(λ+τdλdτ)+(2d4λ+d5)dλdτ+d7e-λτdλdτ-(d7λ+d8)e-λτ(λ+τdλdτ)=0;
therefore
(30)dλdτ=(-λ(λ3+d1λ2+d2λ-d3)eλτ+λ(d7λ+d8)e-λτ)×((3λ2+2d1λ+d2)eλτ+(λ3+d1λ2+d2λ-d3)τeλτhhhh+2d4λ+d5+d7e-λτ-(d7λ+d8)e-λτ)-1,
and then
(31)(dλdτ)-1=(3λ2+2d1λ+d2)eλτ+2d4λ+d5+d7e-λτ-λ(λ3+d1λ2+d2λ-d3)eλτ+λ(d7λ+d8)e-λτ-τλ=(3λ2+2d1λ+d2)eλτ+2d4λ+d5+d7e-λτd4λ3+d5λ2-d6λ+2λ(d7λ+d8)e-λτ-τλ=(-3ω2+2d1iω+d2)eλτ+2d4iω+d5+d7e-λτ-d4iω3-d5ω2-d6iω+2iω(d7iω+d8)e-λτ-τiω.
Substituting eiω=cos⁡ωτ+isinωτ, e-iω=cos⁡ωτ-isinωτ into (31), we get
(32)(dλdτ)-1=((-3ω2+2d1iω+d2)(cos⁡ωτ+isinωτ)hhh+2d4iω+d5+d7(cos⁡ωτ-isinωτ)ω2)×(-d4iω3-d5ω2-d6iωhhhh+2iω(d7iω+d8)(cos⁡ωτ-isinωτ)-d4iω3-d5ω2-d6iω)-1-τiω=((-3ω2+d2+d7)cos⁡ωτ-2d1ωsinωτ+d5)×((-d5ω2-2d7ω2cos⁡ωτ+2d8ωsinωτ)hhhhh+i(-d4ω3-d6ω+2d8ωcos⁡ωτ+2d7ω2sinωτ))-1+(i[2d1ωcos⁡ωτ+(-3ω2+d2-d7)sinωτ+2d4ω])×((-d5ω2-2d7ω2cos⁡ωτ+2d8ωsinωτ)hhhhh+i(-d4ω3-d6ω+2d8ωcos⁡ωτ+2d7ω2sinωτ))-1-τiω.Let(33)Q=(-d5ω2-2d7ω2cos⁡ωτ+2d8ωsinωτ)2+(-d4ω3-d6ω+2d8ωcos⁡ωτ+2d7ω2sinωτ)2>0QRe(dλdτ)-1=[(-3ω2+d2+d7)cos⁡ωτ-2d1ωsinωτ+d5]×[(-d5ω2-2d7ω2cos⁡ωτ+2d8ωsinωτ)]+[2d1ωcos⁡ωτ+(-3ω2+d2-d7)sinωτ+2d4ω]×[-d4ω3-d6ω+2d8ωcos⁡ωτ+2d7ω2sinωτ].
Combined with
(34)sign⁡{Re(dλdτ)|τ=τ0}=sign⁡{Re(dλdτ)|τ=τ0-1},
then we can derive the following theorem.Theorem 1.
If assumptionsG2, G3 hold, then (i)
whenτ∈[0,τ0), the positive equilibrium point z*(x*,u*,y*) of system (11) is asymptotically stable;(ii)
whenτ=τ0, the region near the positive equilibrium point z*(x*,u*,y*) of system (11) can appear Hopf bifurcation.


## 3. Conclusion

As we know, time delay has a strong impact on the dynamic evolution of a population, which could bring a switch in the stability of equilibria and induce various oscillations and periodic solutions. However, these effects would induce ecosystem unbalance and even biological disaster. In order to maintain the sustainable development of the biological resources, we investigate a predator-prey model with delays. In this paper, on the basis of Lotka-Volterra predator-prey system, a general class of predator-prey models with distributed delays and competition term is formulated and studied. Some sufficient conditions in terms of algebraic criterion are obtained, in order to achieve stability and Hopf bifurcation. These theoretical analyses can characterize the fundamental properties of ecological system and provide convenience for applications. By using the same method, similar results may be obtained for the predator-prey model with functional response which describes prey eaten per predator per unit of time.In the future, we will investigate the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions based on the normal form approach theory and center manifold theory introduced by Hassard et al. [11].



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