# Hopf Bifurcation Analysis of a Delayed Diffusive Predator-Prey Model with Predator Interference or Foraging Facilitation

Authors: Wenlong Wang; Zijun Liu; Ruizhi Yang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

A delayed diffusive predator-prey model with predator interference or foraging facilitation is studied. We are interested in the existence of Turing instability, local stability, and Hopf bifurcation. We analyze the direction and stability of bifurcating periodic solutions by the normal form method. Our results suggest that diffusion can induce Turing instability and time delay can induce oscillation of prey and predator’s densities. In addition, the interference parameterλ0 has stabilizing and destabilizing effect on the positive equilibrium.


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## Body

## 1. Introduction

The predator-prey model is one of the most important models in population dynamics. The existence of equilibrium points, periodic solutions, and bifurcation of the model are important problems in population dynamics. Many biologists and mathematicians have studied the predator-prey models [1–3]. In [4], Zhang and Zhu considered the following model:(1)∂u∂t=d1Δu+ru1−uK−λvu1+hλvuv,∂v∂t=d2Δv+zvλvu1+hλvu−m,x,t∈Ω×0,+∞,ux,0=φx,vx,0=ψx,x∈Ω,∂ux,t∂ν=∂vx,t∂ν,x∈∂Ω,where ux,t and vx,t represent the densities of the prey and predator population, respectively, d1 and d2 represent the diffusion coefficient of prey and predator, respectively, h is the handling time, r is the average rate of growth inherent in the prey population, the environmental capacity of the prey is denoted by K, z represents the efficiency with which the prey is transformed into a predator’s newborn, zm is the death rate of predators, and λv is the encounter rate. If λ′v<0>0, it represents predator interference (foraging facilitation). The common forms of λv are as follows:(2)λv=λ0b+vω,Berec5,L−λ0b+vω,Pribylovaetal.6,7,where b, λ0, and ω>0 are some parameters. A detailed explanation can be found in [5–7]. Zhang [4] mainly studied the nonconstant positive steady states and Hopf bifurcation.Time delays exist widely in predator-prey models such as maturation time, capturing time, and gestation time and can induce periodic oscillations [8–16]. Motivated by this, we want to study the effect of time delay τ>0 in the resource limitation of the prey on model (1). We study the following model:(3)∂u∂t=d1Δu+ru1−ut−τK−λvu1+hλvuv,x∈0,lπ,t>0,∂v∂t=d2Δv+zvλvu1+hλvu−m,x∈0,lπ,t>0,uxt,0=vxt,0=0,uxt,lπ=vxt,lπ=0,t>0,u0,x=u0x≥0,v0,x=v0x≥0,x∈0,lπ.The aim of this paper is to study the local stability and Hopf bifurcation induced by time delayτ.The article is arranged as follows. In Section2, we study Turing instability, local stability and the existence of Hopf bifurcation. In Section 3, we study the property of Hopf bifurcation. In Section 4, we give some numerical simulations. Finally, a summarization is given in Section 5.


## 2. Stability Analysis

In [4], Zhang and Zhu have studied the existence of positive equilibrium. In the following, we just suggest that system (3) has positive equilibrium and denoted as u∗,v∗. We use the similar analysis process in our previous work [17] to analyze the stability of u∗,v∗. It is easy to obtain that the characteristic equation is(4)Δnλ,τ=λ2+Anλ+Bn+Cn+λru∗Ke−λτ=0,where(5)An=d1+d2n2l2−A−zD,Bn=d1d2n4l4−d1zD+d2An2l2+zAD+BC,Cn=d2n2l2−zDru∗K, (6)A≔−u∗2v∗guu∗,v∗−u∗v∗gu∗,v∗,B≔u∗2v∗gvu∗,v∗+u∗2gu∗,v∗,C≔u∗v∗guu∗,v∗+v∗gu∗,v∗,D≔u∗v∗gvu∗,v∗,gu,v=λv1+hλvu.Make the following hypothesis:(7)HBC+DA−ru∗K>0,A−ru∗K+zD<0.


### 2.1.τ=0

The characteristic equation ofτ=0 is(8)λ2−trnλ+Δn=0,n∈N∗,where(9)trn=A−ru∗K+zD−d1+d2n2l2,Δn=d1d2n4l4−d2A−ru∗K+d1zDn2l2+zBC+DA−ru∗K.The eigenvalues are given by(10)λinr=trn±trn2−4Δn2,n∈N∗.Denote(11)p±=d2A−ru∗/K+d1zD±d2A−ru∗/K+d1zD2−4zd1d2DA−ru∗/K+BC2d1d2, (12)Γ0=d2A−ru∗K+d1zD2−4d1d2zBC+DA−ru∗K.By analyzing the characteristic root, we can easily get the following theorem.Theorem 2.1.
Ifd1>0 and d2>0, suppose H hold. Then, we have the following results:(1)
Ifd2A−ru∗/K+d1zD≤0, then u∗,v∗ is locally asymptotically stable(2)
Ifd2A−ru∗/K+d1zD>0 and Γ0<0, then u∗,v∗ is locally asymptotically stable(3)
Ifd2A−ru∗/K+d1zD>0 and Γ0>0, but n2/l2∉p−,p+ (for all n∈N), then u∗,v∗ is locally asymptotically stable(4)
Ifd2A−ru∗/K+d1zD>0 and Γ0>0 and k2/l2∈p−,p+ (for some k∈N), then u∗,v∗ is Turing unstable
Choose(13)d1=0.008,d2=0.2,l=4,r=2,K=8,z=1,m=1,h=0.25,λ0=5,b=0.5,ω=1.5,r=2,λv=λ0b+vw.By direct calculation,u∗,v∗≈0.29,0.56 is a unique positive equilibrium. d2A−ru∗/K+d1zD≈0.08>0, Γ0≈0.01, p−≈−57.14, and p+≈9.05. Then, u∗,v∗ is Turing unstable (shown in Figure 1).Figure 1
The numerical simulations of system (3) with τ=0.


### 2.2.τ>0

AssumeH and one of conditions 1−3 in Theorem 2.1 hold. Suppose iω is a root of (4); then,(14)−ω2+iωAn+Bn+Cn+iωru∗Kcosωτ−isinωτ=0.Then, we have(15)−ω2+Bn+Cncosωτ+ωru∗Ksinωτ=0,ωAn−Cnsinωτ+ωru∗Kcosωτ=0,which leads to(16)ω4+ω2An2−2Bn−ru∗K2+Bn2−Cn2=0.Letp=ω2; then, (16) is changed to(17)p2+pAn2−2Bn−ru∗K2+Bn2−Cn2=0,and its roots are given by(18)pn±=12−An2−2Bn−ru∗K2±An2−2Bn−ru∗K22−4Bn2−Cn2.IfH and one of conditions 1−3 in Theorem 2.1 hold, we have(19)An2−2Bn−ru∗K2=d12+d22n4l4−2d1A+d2zDn2l2+D2z2−2BCz−ru∗K2+A2,Bn−Cn=d1d2n4l4−d2A+ru∗K+zDd1n2l2+zBC+DA+ru∗K,Bn+Cn=Δn>0.DefineD=k∈ℕ|Eq.2.8have positive roots withn=k. For n∈D, if pn+>0, then (4) has a pair of purely imaginary roots ±iωnj,+ at τnj,+,j∈ℕ0. And if pn−>0, then (4) has a pair of purely imaginary roots ±iωn− at τnj,−,j∈ℕ0, where ωn±=pn±,(20)τnj,±=τn0,±+2jπωn±,j=0,1,2,…, (21)τn0,±=1ωn±arccosCn−Anru∗/Kωn±2−BnCnCn2+ru∗/K2ωn2,From (21), we have τn0<τnjj∈ℕ0. For n∈D, denote(22)τ∗=minτk0,±orτn0,+|k∈D.Lemma 2.1.
SupposeH and one of conditions 1−3 in Theorem 2.1 hold. Then,(23)αn′τnj=dλdτ|τ=τnj,+>0,αn′τnj=dλdτ|τ=τnj,−<0,forτ∈Dj∈ℕ0.Proof.
By (4), we have(24)dλdτ−1=2λ+An+e−λτru∗/KCn+λru∗/Ke−λτ−τλ.
Then,(25)Redλdττ=τnj,±−1=Re2λ+An+e−λτru∗/KCn+λru∗/Ke−λτ−τλτ=τnj,±=1Λω22ω2+An2−2Bn−ru∗K2τ=τnj,±=±1Λω2An2−2Bn−ru∗K22−4Bn2−Cn2τ=τnj,±,where Λ=ω4D2+Cn2ω2>0. Therefore, the conclusion holds.Theorem 2.2.
For system (3), assume H and one of conditions 1−3 in Theorem 2.1 hold.(1)
Pu∗,v∗ is locally asymptotically stable for τ≥0 when D=∅(2)
Pu∗,v∗ is locally asymptotically stable for τ∈0,τ∗ and unstable for τ∈τ∗,τ∗+ϵ with some ϵ>0 when D≠∅(3)
Hopf bifurcation occurs whenτ=τnj,+orτ=τnj,−, for j∈ℕ0 and n∈D


## 2.1.τ=0

The characteristic equation ofτ=0 is(8)λ2−trnλ+Δn=0,n∈N∗,where(9)trn=A−ru∗K+zD−d1+d2n2l2,Δn=d1d2n4l4−d2A−ru∗K+d1zDn2l2+zBC+DA−ru∗K.The eigenvalues are given by(10)λinr=trn±trn2−4Δn2,n∈N∗.Denote(11)p±=d2A−ru∗/K+d1zD±d2A−ru∗/K+d1zD2−4zd1d2DA−ru∗/K+BC2d1d2, (12)Γ0=d2A−ru∗K+d1zD2−4d1d2zBC+DA−ru∗K.By analyzing the characteristic root, we can easily get the following theorem.Theorem 2.1.
Ifd1>0 and d2>0, suppose H hold. Then, we have the following results:(1)
Ifd2A−ru∗/K+d1zD≤0, then u∗,v∗ is locally asymptotically stable(2)
Ifd2A−ru∗/K+d1zD>0 and Γ0<0, then u∗,v∗ is locally asymptotically stable(3)
Ifd2A−ru∗/K+d1zD>0 and Γ0>0, but n2/l2∉p−,p+ (for all n∈N), then u∗,v∗ is locally asymptotically stable(4)
Ifd2A−ru∗/K+d1zD>0 and Γ0>0 and k2/l2∈p−,p+ (for some k∈N), then u∗,v∗ is Turing unstable
Choose(13)d1=0.008,d2=0.2,l=4,r=2,K=8,z=1,m=1,h=0.25,λ0=5,b=0.5,ω=1.5,r=2,λv=λ0b+vw.By direct calculation,u∗,v∗≈0.29,0.56 is a unique positive equilibrium. d2A−ru∗/K+d1zD≈0.08>0, Γ0≈0.01, p−≈−57.14, and p+≈9.05. Then, u∗,v∗ is Turing unstable (shown in Figure 1).Figure 1
The numerical simulations of system (3) with τ=0.


## 2.2.τ>0

AssumeH and one of conditions 1−3 in Theorem 2.1 hold. Suppose iω is a root of (4); then,(14)−ω2+iωAn+Bn+Cn+iωru∗Kcosωτ−isinωτ=0.Then, we have(15)−ω2+Bn+Cncosωτ+ωru∗Ksinωτ=0,ωAn−Cnsinωτ+ωru∗Kcosωτ=0,which leads to(16)ω4+ω2An2−2Bn−ru∗K2+Bn2−Cn2=0.Letp=ω2; then, (16) is changed to(17)p2+pAn2−2Bn−ru∗K2+Bn2−Cn2=0,and its roots are given by(18)pn±=12−An2−2Bn−ru∗K2±An2−2Bn−ru∗K22−4Bn2−Cn2.IfH and one of conditions 1−3 in Theorem 2.1 hold, we have(19)An2−2Bn−ru∗K2=d12+d22n4l4−2d1A+d2zDn2l2+D2z2−2BCz−ru∗K2+A2,Bn−Cn=d1d2n4l4−d2A+ru∗K+zDd1n2l2+zBC+DA+ru∗K,Bn+Cn=Δn>0.DefineD=k∈ℕ|Eq.2.8have positive roots withn=k. For n∈D, if pn+>0, then (4) has a pair of purely imaginary roots ±iωnj,+ at τnj,+,j∈ℕ0. And if pn−>0, then (4) has a pair of purely imaginary roots ±iωn− at τnj,−,j∈ℕ0, where ωn±=pn±,(20)τnj,±=τn0,±+2jπωn±,j=0,1,2,…, (21)τn0,±=1ωn±arccosCn−Anru∗/Kωn±2−BnCnCn2+ru∗/K2ωn2,From (21), we have τn0<τnjj∈ℕ0. For n∈D, denote(22)τ∗=minτk0,±orτn0,+|k∈D.Lemma 2.1.
SupposeH and one of conditions 1−3 in Theorem 2.1 hold. Then,(23)αn′τnj=dλdτ|τ=τnj,+>0,αn′τnj=dλdτ|τ=τnj,−<0,forτ∈Dj∈ℕ0.Proof.
By (4), we have(24)dλdτ−1=2λ+An+e−λτru∗/KCn+λru∗/Ke−λτ−τλ.
Then,(25)Redλdττ=τnj,±−1=Re2λ+An+e−λτru∗/KCn+λru∗/Ke−λτ−τλτ=τnj,±=1Λω22ω2+An2−2Bn−ru∗K2τ=τnj,±=±1Λω2An2−2Bn−ru∗K22−4Bn2−Cn2τ=τnj,±,where Λ=ω4D2+Cn2ω2>0. Therefore, the conclusion holds.Theorem 2.2.
For system (3), assume H and one of conditions 1−3 in Theorem 2.1 hold.(1)
Pu∗,v∗ is locally asymptotically stable for τ≥0 when D=∅(2)
Pu∗,v∗ is locally asymptotically stable for τ∈0,τ∗ and unstable for τ∈τ∗,τ∗+ϵ with some ϵ>0 when D≠∅(3)
Hopf bifurcation occurs whenτ=τnj,+orτ=τnj,−, for j∈ℕ0 and n∈D


## 3. Property of Hopf Bifurcation

Similar to our previous work [17], by the standardized method, in [18–20], we give some parameters to determine the property of Hopf bifurcation. For a critical value τnj,+orτnj,−, we denote it τ˜. Let u˜x,t=ux,τt−u∗ and v˜x,t=vx,τt−v∗. Then, system (3) is (after drop tilde)(26)∂u∂t=τd1Δu+fu,v,ut−1,∂v∂t=τd2Δv+hu,v,where f˜u,v,ut−1=ru+u∗1−ut−1+u∗/K−λv+v∗u+u∗v+v∗/1+hλv+v∗u+u∗, and h˜u,v=zv+v∗λv+v∗u+u∗/1+hλv+v∗u+u∗−m, for x∈0,lπ, and t>0. Let(27)τ=τ˜+μ,u1t=u⋅,t,u2t=v⋅,t,U=u1,u2T.Then, (25) can be rewritten in an abstract form in the phase space C1≔C−1,0,X:(28)dUtdt=τ˜DΔUt+Lτ˜Ut+FUt,μ,where Lμϕ and Fϕ,μ are defined by(29)Lμϕ=μAϕ10−Bϕ20−ru∗Kϕ1−1zCϕ10+zDϕ20,Fϕ,μ=μDΔϕ+Lμϕϕ,μ,with(30)fϕ,μ=τ˜+μF1ϕ,μ,F2ϕ,μT,F1ϕ,μ=rϕ10+u∗1−ϕ1−1+u∗K−λϕ20+v∗ϕ20+v∗ϕ10+u∗1+hλϕ20+v∗ϕ10+u∗−Aϕ10+Bϕ20+u∗ϕ1−1,F2ϕ,μ=zϕ20+v∗λϕ20+v∗ϕ10+v∗ϕ10+u∗1+hλϕ20+v∗ϕ10+u∗−mϕ20+v∗−zCϕ10−zDϕ20,respectively, for ϕ=ϕ1,ϕ2T∈C1. Consider the linear equation:(31)dUtdt=τ˜DΔUt+Lτ˜Ut.According to the results in Section2, we know that Λn:=iωnτ˜,−iωnτ˜ are characteristic values of system (31) and the linear functional differential equation:(32)dztdt=−τ˜Dn2l2zt+Lτ˜zt.There exists a2×2 matrix function ηnσ,τ˜−1≤σ≤0, such that(33)−τ˜Dn2l2ϕ0+Lτ˜ϕ=∫−10dηnσ,τϕσ,for ϕ∈C−1,0,ℝ2. In fact, we can choose(34)ηnσ,τ=τE,σ=0,0,σ∈−1,0,−τF,σ=−1,where(35)E=A−d1n2l2−BzCzD−d2n2l2,F=−u∗000.Define the following linear paring:(36)ψ,ϕ=ψ0ϕ0−∫−10∫ξ=0σψξ−σdηnσ,τ˜ϕξdξ=ψ0ϕ0+τ˜∫−10ψξ+1Fϕξdξ.for ϕ∈C−1,0,ℝ2, ψ∈C−1,0,ℝ2. Choose p1θ=1,ξTeiωnτ˜σσ∈−1,0,p2σ=p1σ¯, is a basis of Aτ˜ with Λn and q1r=1,ηe−iωnτ˜rr∈0,1,q2r=q1r¯ is a basis of A∗ with Λn, where(37)ξ=zCiωn+d2n2/l2−zD,η=Biωn−d2n2/l2+zD.LetΦ=Φ1,Φ2 and Ψ∗=Ψ1∗,Ψ2∗T with Φ1σ=p1σ+p2σ/2, Φ2σ=p1σ−p2σ/2i, for θ∈−1,0, and Ψ1∗r=q1r+q2r/2, Ψ2∗r=q1r−q2r/2i, for r∈0,1. Then, we can compute by (36):(38)D1∗:=Ψ1∗,Φ1,D2∗:=Ψ1∗,Φ2,D3∗:=Ψ2∗,Φ1,D4∗:=Ψ2∗,Φ2.DefineΨ∗,Φ=Ψj∗,Φk=D1∗D2∗D3∗D4∗ and(39)Ψ=Ψ1,Ψ2T=Ψ∗,Φ−1Ψ∗.Then,Ψ,Φ=I2. In addition, define fn≔βn1,βn2, where(40)βn1=cosnlx0,βn2=0cosnlx.We also define(41)c⋅fn=c1βn1+c2βn2,forc=c1,c2T∈C1.Thus, the center subspace of linear (31) is given by PCNC1⊕PSC1 and PSC1 denotes the complement subspace of PCNC1 in C1:(42)u,v≔1lπ∫0lπu1v1¯dx+1lπ∫0lπu2v2¯dx.for u=u1,u2, v=v1,v2, u,v∈X, and ϕ,f∗=ϕ,f∗1,ϕ,f∗2T. Let Aτ˜ denote the infinitesimal generator of an analytic semigroup induced by the linear system (31), and (26) can be rewritten as the following abstract form:(43)dUtdt=Aτ˜Ut+RUt,μ,where(44)RUt,μ=0,θ∈−1,0,FUt,μ,θ=0.By the decomposition ofC1, the solution above can be written as(45)Ut=Φx1x2fn+hx1,x2,μ,where(46)x1x2=Ψ,Ut,fn, (47)hx1,x2,μ∈PSCC1,h0,0,0=0,Dh0,0,0=0.In particular, the solution of (28) on the center manifold is given by(48)Ut=Φx1tx2tfn+hx1,x2,0.Letz=x1−ix2, and notice that p1=Φ1+iΦ2. Then, we have(49)Φx1x2fn=Φ1,Φ2z+z¯2iz−z¯2fn=12p1z+p¯1z¯fn, (50)hx1,x2,0=hz+z¯2,iz−z¯2,0.Hence, (48) is(51)Ut=12p1z+p¯1z¯fn+hz+z¯2,iz−z¯2,0=12p1z+p¯1z¯fn+Wz,z¯,where(52)Wz,z¯=hz+z¯2,iz−z¯2,0.From [18], z satisfies(53)z˙=iωnτ˜z+gz,z¯,where(54)gz,z¯=Ψ10−iΨ20FUt,0,fn.Let(55)Wz,z¯=W20z22+W11zz¯+W02z¯22+…,(56)gz,z¯=g20z22+g11zz¯+g02z¯22+…By (51) and (55), we have(57)ut0=12z+z¯cosnxl+W2010z22+W1110zz¯+W0210z¯22+⋯,vt0=12ξ+ξ¯z¯cosnxl+W2020z22+W1120zz¯+W0220z¯22+⋯,ut−1=12ze−iωnτ˜+z¯eiωnτ˜cosnxl+W201−1z22+W111−1zz¯+W021−1z¯22+⋯,and(58)F¯1Ut,0=1τ˜F1=−rKut0ut−1+12fuuut20+fuvut0vt0+12fvvvt20+16fuuuut30+12fuuvut20vt0+12fuvvut0vt20+16fvvvvt30+O4,F¯2Ut,0=1τ˜F2=12huuut20+huvut0vt0+12hvvvt20+16huuuut30+12huuvut20vt0+12huvvut0vt20+16hvvvvt30+O4,with(59)fuu=f˜uu0,0,0,fuv=f˜uv0,0,0,fvv=f˜vv0,0,0,fuuu=f˜uuu0,0,0,fuuv=f˜uuv0,0,0,fuvv=f˜uvv0,0,0,fvvv=f˜vvv0,0,0,huu=h˜uu0,0,huv=h˜uv0,0,hvv=h˜vv0,0,huuu=h˜uuu0,0,huuv=h˜uuv0,0,huvv=h˜uvv0,0,hvvv=h˜vvv0,0.Hence,(60)F¯1Ut,0=cos2nxlz22χ20+zz¯χ11+z¯22χ¯20+z2z¯2ς11cosnxl+ς12cos3nxl+⋯,F¯2Ut,0=cos2nxlz22ς20+zz¯ς11+z¯22ς¯20+z2z¯2ς21cosnxl+ς22cos3nxl+⋯,FUt,0,fn=τ˜F¯1Ut,0fn1+F¯2Ut,0fn2=z22τ˜χ20ς20Γ+zz¯τ˜χ11ς11Γ+z¯22τ˜χ¯20ς¯20Γ+z2z¯2τ˜κ1κ2+⋯,with(61)Γ=1lπ∫0lπcos3nxldx,κ1=ς11lπ∫0lπcos2nxldx+ς12lπ∫0lπcos4nxldx,κ1=ς21lπ∫0lπcos2nxldx+ς22lπ∫0lπcos4nxldx,and(62)χ20=14e−iτ˜ωn−2rK+eiτ˜ωnfuu+2ξfuv+fvvξ,χ11=−14e−iτωn1+e2iτωnrK−eiτωnfuu+fvvξ¯ξ+fuvξ¯+ξ,ς11=fuu+fuvξ−re−iτ˜KW1110+fuv+fvvξW1120−rW111−1K−rW201−12K+fuuK−eiτωnr+fuvKξ¯2KW2010+12fuv+fvvξ¯W2020,ς12=W1110fuu−e−iτωnrK+fuvξ+W1120fuv+fvvξ−rW111−12K+W2010fuuK−eiτωnr+fuvKξ¯2K+12W2020fuv+fvvξ¯−rW201−12K,ϱ20=14huu+ξ2huv+hvvξ,ϱ11=14huu+hvvξ¯ξ+huvξ¯+ξ,ς21=W1110huu+huvξ+W1120huv+hvvξ+12W2010huu+huvξ¯+12W2020huv+hvvξ¯,ς22=18huuu+huuvξ¯+2ξ+ξ2huvvξ¯+huvvξ+hvvvξ¯ξ.Denote(63)Ψ10−iΨ20:=γ1γ2.Notice that(64)1lπ∫0lπcos3nxldx=0,n=1,2,3,…,and we have(65)Ψ10−iΨ20FUt,0,fn=z22γ1χ20+γ2ς20Γτ˜+zz¯γ1χ11+γ2ς11Γτ˜+z¯22γ1χ¯20+γ2ς¯20Γτ˜+z2z¯2τ˜γ1κ1+γ2κ2+⋯Then, by (54), (56), and (65), we have g20=g11=g02=0, for n=1,2,3,⋯. If n=0, we have the following quantities:(66)g20=γ1τ˜χ20+γ2τ˜ς20,g11=γ1τ˜χ11+γ2τ˜ς11,g02=γ1τ˜χ¯20+γ2τ˜ς¯20.And forn∈ℕ0, g21=τ˜γ1κ1+γ2κ2.From [18], we have(67)W˙z,z¯=W20zz˙+W11z˙z¯+W11zz¯˙+W02z¯z¯˙+⋯,Aτ˜Wz,z¯=Aτ˜W20z22+Aτ˜W11zz¯+Aτ˜W02z¯22+⋯,and Wz,z¯ satisfies(68)W˙z,z¯=Aτ˜W+Hz,z¯,where(69)Hz,z¯=H20z22+W11zz¯+H02z¯22+⋯=X∗FUt,0−ΦΨ,X∗FUt,0,fn⋅fn.Hence, we have(70)2iωnτ˜−Aτ˜W20=H20,−Aτ˜W11=H11,−2iωnτ˜−Aτ˜W02=H02,that is,(71)W20=2iωnτ˜−Aτ˜−1H20,W11=−Aτ˜−1H11,W02=−2iωnτ˜−Aτ˜−1H02.By (65), we have that, for θ∈−1,0,(72)Hz,z¯=−Φ0Ψ0<FUt,0,fn>⋅fn=−p1θ+p2θ2,p1θ−p2θ2iΦ1θΦ20FUt,0,fn⋅fn=−12p1θΦ10−iΦ20+p2θΦ10+iΦ20FUt,0,fn⋅fn=−12p1θg20+p2θg¯02z22+p1θg11+p2θg¯11zz¯+p1θg02+p2θg¯20z¯22+⋯.Therefore, by (69), for θ∈−1,0,(73)H20θ=0,n∈ℕ,−12p1θg20+p2θg¯02⋅f∗,n=0,H11θ=0,n∈ℕ,−12p1θg11+p2θg¯11⋅f∗,n=0,H02θ=0,n∈ℕ,−12p1θg02+p2θg¯20⋅f∗,n=0, (74)Hz,z¯0=FUt,0−ΦΨ,FUt,0,fn⋅fn,where(75)H200=τ˜χ20ς20cos2nxl,n∈ℕ,τ˜χ20ς20−12p10g20+p20g¯02⋅f∗,n=0.H110=τ˜χ11ς11cos2nxl,n∈ℕ,τ˜χ11ς11−12p10g11+p20g¯11⋅f∗,n=0.By the definition ofAτ˜ and (70), we have(76)W˙20=Aτ˜W20=2iωnτ˜W20+12p1θg20+p2θg¯02⋅fn,−1≤θ<0.That is,(77)W20θ=i2iωnτ˜g20p1θ+g¯023p2θ⋅fn+E1e2iωnτ˜θ,where(78)E1=W200,n∈ℕ,W200−i2iωnτ˜g20p1θ+g¯023p2θ⋅f∗,n=0.By (70), we have(79)−g20p10+g¯023p20⋅f∗+2iωnτ˜E1−Aτ˜i2ωnτ˜g20p10+g¯023p20⋅f∗−Aτ˜E1−Lτ˜i2ωnτ˜g20p10+g¯023p20⋅fn+E1e2iωnτ˜θ=τ˜χ20ς20−12p10g20+p20g¯02⋅f∗.As(80)Aτ˜p10+Lτ˜p1⋅f=iω∗p10⋅f∗, (81)Aτ˜p20+Lτ˜p2⋅f∗=−iω∗p20⋅f∗,we have(82)2iωnE1−Aτ˜E1−Lτ˜E1e2iωn=τ˜χ20ς20cos2nxl,n∈ℕ0.That is,(83)E1=τ˜Eχ20ς20cos2nxl,where(84)E=2iωnτ˜+d1n2l2−A+ru∗Ke−2iτ˜ωnB−zCe−2iωnτ˜−zD+2iωnτ˜+d2n2l2−1.Similarly, from (72), we have(85)−W˙11=i2ωnτ˜p1θg11+p2θg¯11⋅fn,−1≤θ<0.That is,(86)W11θ=i2iωnτ˜p1θg¯11−p1θg11+E2.Similarly, we have(87)E2=τ˜E∗χ11ς11cos2nxl,where(88)E∗=d1n2l2−A+ru∗KB−zC−zD+d2n2l2−1.Thus, we have(89)c10=i2ωnτ˜g20g11−2g112−g0223+12g21,μ2=−Rec10Reλ′τnj,T2=−1ωnτ˜Imc10+μ2Imλ′τnj,β2=2Rec10.By [18], we have the following theorem.Theorem 3.1.
For any critical valueτnj, we have the Hopf bifurcation is forward μ2>0 or backward μ2<0. The bifurcating periodic solutions are orbitally asymptotically stable β2<0 or unstable β2>0. The period increases T2>0 or decreases T2<0.


## 4. Numerical Simulations

Choose parameters(90)d1=0.008,d2=1,l=30,r=1,K=4,z=2,m=2,h=0.25,b=1,w=3,λv=λ0b+vw.We give the bifurcation diagram of system (3) with parameter λ0 in Figure 2. It shows that the interference parameter λ0 has stabilizing and destabilizing effect on the positive equilibrium under parameters (90).Figure 2
Bifurcation diagram of system (3) with parameter λ0.If we chooseλ0=6.3, then u∗,v∗≈2.13,0.50 is a unique positive equilibrium. Then, τ∗≈2.07. Then, Pu∗,v∗ is locally asymptotically stable for τ∈0,τ∗0 (Figure 3). By computation, we have(91)μ2≈21.68>0,β2≈−6.29<0,T2≈9.34>0.Figure 3
Numerical simulations of system (3) with τ=2 and the initial condition at 2.1+0.0001sinx,0.5−0.0001cosx.Hence, the stable bifurcating period solution exists forτ>τ∗ (Figure 4).Figure 4
Numerical simulations of system (3) with τ=2.54 and the initial condition at 2.1+0.0001sinx,0.5−0.0001cosx.


## 5. Conclusion

In the paper, we consider a delayed diffusive predator-prey model with predator interference or foraging facilitation. The Turing instability and Hopf bifurcation are analyzed. Compared with model (1), time delay can affect the coexistence mode of the prey and the predator. When the delay larger than the critical value, the prey and predator will coexist in the form of periodic oscillation. Time delay may also cause spatially inhomogeneous periodic solutions, but unfortunately, it does not appear in our numerical simulation. In addition, we observed that the interference parameter λ0 has stabilizing and destabilizing effect on the positive equilibrium.



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*Source: 5278036-2022-02-22.xml*