# Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay

Authors: Hua Liu; Yong Ye; Yumei Wei; Weiyuan Ma; Ming Ma; Kai Zhang
Journal: Complexity
 (2019)

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## Abstract

In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.


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## Body

## 1. Introduction

Since the Allee effect was proposed by Allee [1] in 1931, the predator-prey model with Allee effect has been studied extensively [2–27]. From the ordinary differential equation predator-prey model with Allee effect to the partial differential equation model, many researchers have achieved rich results [4, 7, 9–16, 28]. Cai et al. [6] established a Leslie–Gower predator-prey model with additive Allee effect on prey, and they found Allee effect can increase the risk of ecological extinction. Sen et al. [5] established a two-prey one-predator model with Allee effect, and the effects of Allee effect on the dynamics of predator population are discussed. Of course, the research on reaction-diffusion predator-prey model with Allee effect is also very rich. For example, Wang et al. [7] established a reaction-diffusion predator-prey model and found the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes. They also studied Allee effect induced instability in a reaction-diffusion predator-prey model [4]. Petrovskii et al. found that the deterministic system with Allee effect can induce patch invasion [23]. Sun et al. found that predator mortality plays an important role in the pattern formation of populations [13]. It is now believed that the spatial composition of population interactions has been identified as an important factor in how ecological communities operate and form. Pattern formation in the predator-prey model is an appropriate tool for understanding the basic mechanism of spatiotemporal population dynamics. We find that there are few studies on delays in reaction-diffusion predator-prey model with Allee effect. So next we discuss the effects of Allee effect and delay on pattern formation. First, we consider a predator-prey model with hyperbolic mortality established by Zhang et al. [10], the model is obtained as follows:(1)∂U∂T−d1ΔU=aU1−UK−bUVc+U,x∈Ω,T>0,∂V∂T−d2ΔV=mUVc+U−hV,x∈Ω,T>0,∂U∂n=∂V∂n=0,x∈∂Ω,T>0,Ux,0=U0x≥0,Vx,0=V0x≥0,x∈Ω,where U and V are the population densities of prey and predator, respectively; a is the birth rate, K is the carrying capacity and b is the maximum uptake rate of the prey; c is the prey density at which the predator has the maximum kill rate; m is the birth rate of predator; function hV reflects the predator death rate; the habitat Ω⊂Rn is a bounded domain with smooth boundary ∂Ω; n is the outward unit normal vector on ∂Ω; d1 and d2 are the diffusion coefficients, respectively; and Δ is the Laplacian operator. The homogeneous Neumann boundary condition implies that the system above is self-contained and there is no host across the boundary. After nondimensionalization,(2)U⟶Ku,V⟶acbv,Kc⟶β,am⟶α,T⟶tm.Then, considering that the predator-prey model with Allee effect is more realistic, people begin to introduce delay into the predator-prey model and discuss the effects of Allee effect and delay on the dynamics of the model [2, 17–22]. We try to introduce weak Allee effect and searching delay into model (1), and then we get(3)∂u∂t−d1Δu=αu1−uuu+A−αut−τvt−τ1+βut−τ,x∈Ω,t>0,∂v∂t−d2Δv=βuv1+βu−hv,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,θ=φx,θ≥0,vx,θ=ψx,θ≥0,x,θ∈Ω×−τ,0,where hv=γv2/e+ηv. For hyperbolic mortality, γ is the death rate of the predator, e and η are coefficients of light attenuation by water and self-shading in the context of plankton mortality, and τ is the searching delay. The weak Allee effect term is u/u+A, where A>0 is described as a weak Allee effect constant.


## 2. Turing Instability

First, we consider the model withτ=0:(4)∂u∂t−d1Δu=αu1−uuu+A−αuv1+βu,x∈Ω,t>0,∂v∂t−d2Δv=βuv1+βu−γv2e+ηv,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,0=u0x≥0,vx,0=v0x≥0,x∈Ω.Obviously, ifd1=d2=0, without diffusion in model (4), then we can obtain the following ordinary differential equations:(5)dudt=αu1−uuu+A−αuv1+βu=fu,v,dvdt=βuv1+βu−γv2e+ηv=gu,v.We mainly focus on the stability of the positive equilibrium of model (4). Clearly, the positive equilibrium E∗=u∗,v∗ of the ordinary differential equation (ODE) or the partial differential equation (PDE) model (4) satisfies fu∗,v∗=0 and gu∗,v∗=0:(6)dudt=αu1−uuu+A−αuv1+βu=0,dvdt=βuv1+βu−γv2e+ηv=0.For simplicity of discussion, in this paper, we shall concentrate the case ofη=γ and e=1. We easily see that model (4) exhibits a positive equilibrium point E∗=u∗,v∗ when β>γ/1−γ, 0<γ<1, and A<γ/β. When β>γ/1−γ, 0<γ<1, and γ/β<A<βγ−γ−β2+4βγ2/4β2γ, model (4) exhibits two positive equilibrium points E1∗=u1∗,v1∗ and E2∗=u2∗,v2∗. In this work, we mainly focus on a positive equilibrium point, where(7)u∗=βγ−γ−β2+γ−βAβγ2βγ+βγ−γ−β2βγ,v∗=βγu∗.We calculate the Jacobian matrix of model (5) at E∗, which is given by J∗=a10a01b10b01, where(8)a10=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2,a01=−αu∗1+βu∗,b10=β2u∗γ1+βu∗2,b01=−βu∗1+βu∗2.We can easily know that the characteristic polynomial is(9)Hλ=λ2−Tλ+D,where(10)T=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2−βu∗1+βu∗2,D=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2−βu∗1+βu∗2+αβ2u∗2γ1+βu∗3.Thus, we have the following conclusions:(a)
IfT<0 and D>0, then the positive equilibrium is locally asymptotically stable(b)
IfT>0, then the positive equilibrium is unstableNext, let us consider the PDE model (4); we choose the perturbation function consisting of the following two-dimensional Fourier modes:(11)uv=eλkt+ikxx,J=a10−k2d1a01b10b01−k2d2,so, we can find(12)Tk=a10+b01−k2d1+d2,Dk=a10−k2d1b01−k2d2−a01b10.We know, whenE∗ is stable,(13)T=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2−βu∗1+βu∗2<0.We can easily find thatTk=a11+a22−k2d1+d2<0. So, if model (3) changes from stable to unstable, it needs to be(14)Dk=a10−k2d1b01−k2d2−a01b10<0,that is,(15)2Dd1d2<a10+b01d1d2,d1<d2.


## 3. Delay-Induced Instability

Finally, we consider the PDE model (4) with delay (searching delay), and we get model (3). Considering τ and spatial diffusion, if τ is small enough, the following changes are made [29]:(16)ux,y,t−τ=ux,y,t−τ∂u∂t,vx,y,t−τ=vx,y,t−τ∂v∂t,we substitute (16) into model (3) to get(17)∂u∂t−d1Δu=αu1−uuu+A−αux,y,t−τ∂u/∂tvx,y,t−τ∂v/∂t1+βux,y,t−τ∂u/∂t,x∈Ω,t>0,∂v∂t−d2Δv=βuv1+βu−γv2e+ηv,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,θ=φx,θ≥0,vx,θ=ψx,θ≥0,x,θ∈Ω×−τ,0.Expanding in Taylor series and neglecting the higher-order nonlinearities, we find(18)∂u∂t−d1Δu=αu1−uuu+A−αuv1+βu−τfut−τut−τ,vt−τ∂u∂t,−τfvt−τut−τ,vt−τ∂v∂t,x∈Ω,t>0,∂v∂t−d2Δv=βuv1+βu−γv2e+ηv,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,θ=φx,θ≥0,vx,θ=ψx,θ≥0,x,θ∈Ω×−τ,0,where(19)fut−τ,vt−τ=αu1−uuu+A−αut−τvt−τ1+βut−τ.We can see that iffu∗,v∗=0 and gu∗,v∗=0 are satisfied at equilibrium point E∗=u∗,v∗, then we can get the model:(20)∂u∂t−d1Δu=fuu,vu−u∗+fvu,vv−v∗−τfut−τut−τ,vt−τ∂u∂t,−τfvt−τut−τ,vt−τ∂v∂t,x∈Ω,t>0,∂v∂t−d2Δv=guu,vu−u∗+gvu,vv−v∗,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,θ=φx,θ≥0,vx,θ=ψx,θ≥0,x,θ∈Ω×−τ,0,where(21)fu,v=αu1−uuu+A−αuv1+βu,gu,v=βuv1+βu−γv2e+ηv.We consider that the stable equilibrium pointE∗=u∗,v∗ is subjected to a small perturbation of u¯∗andv¯∗. Let u=u∗+u¯∗andv=v∗+v¯∗, so we get(22)∂u¯∗∂t−d1Δu¯∗=A10u¯∗+A01v¯∗+ταβu∗γ1+βu∗2∂u¯∗∂t+ταu∗1+βu∗∂v¯∗∂t,x∈Ω,t>0,∂v¯∗∂t−d2Δv¯∗=B10u¯∗+B01v¯∗,x∈Ω,t>0,∂u∂n=∂v∂n=0,x∈∂Ω,t>0,ux,θ=φx,θ≥0,vx,θ=ψx,θ≥0,x,θ∈Ω×−τ,0.Assuming that the solution of the system has the following form,(23)u¯∗x,t=u¯∗0eλtcoskxx,v¯∗x,t=v¯∗0eλtcoskxx.So, we get(24)Jkτ=a10+τb10N−k2d11−Ma01+τb10N−τd2Nk21−Mb10b01−k2d2,where(25)M=αβu∗γ1+βu∗2,N=αu∗1+βu∗.We can easily find(26)Tkτ=a10+τb10N−k2d11−M+b01−k2d2,Dkτ=a10+τb10N−k2d11−Mb01−k2d2−b10a01+τb01N−τd2Nk21−M.Whenk=0, model (3) undergoes Hopf bifurcation at Tkτ=0, so the critical value for undergoing Hopf bifurcation can be obtained:(27)τH=−a10+b01+b01Mb10N.We know, whenE∗ is stable,(28)a10b01−a01b10>0,Dkτ=a10+τb10N−k2d11−Mb01−k2d2−b10a01+τb01N−τd2Nk21−M>0.So, we just need to judge(29)Tkτ=a10+τb10N−k2d11−M+b01−k2d2>0.It is easy to know when(30)τ>−a10+k2d1+−b01+k2d21−Mb10N,Tkτ=a10+τb10N−k2d11−M+b01−k2d2>0.So, the instability condition caused by delay is as follows:(31)T=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2−βu∗1+βu∗2<0,D=−2αu∗3−3αAu∗2+αu∗2+2αAu∗u∗+A2−αβu∗γ1+βu∗2−βu∗1+βu∗2+αβ2u∗2γ1+βu∗3>0,τ>−a10+k2d1+−b01+k2d21−Mb10N.


## 4. Amplitude Equations and Pattern Selection

We rewrite the transformed form of system (3) at the positive spatially homogeneous steady state E∗=u∗,v∗ as follows and denote by U,VT the perturbation solution U−u∗,V−v∗T of the system:(32)∂X∂t=LX+H,where X=U,VT.Then, let the linear operatorL be defined as follows:(33)L=Jτ+DaΔ=a10+τa01N1−Mb10+τb01N1−Mb10b01+−k2d11−M−τd2Nk21−M0−k2d2,and H be given by(34)H=A20U2+A11UV+A02V2+A30U3+A21U2V+A12UV2+A03V3+oε3B20U2+b11UV+B02V2+B30U3+B21U2V+B12UV2+B03V3+oε3,where(35)A20=11−M−αu∗3−3αAu∗3+3αAu∗2−αAu∗+αA2u∗+A3+αβ2u∗γ1+βu∗3−2τNβ2v∗1−M1+βu∗3,A11=−α21+βu∗1−M+τNβ1+βu∗21−M,A02=0,B20=−2β2v∗1+βu∗3,B11=β1+βu∗2,B02=−2γ1+γv∗2,A30=11−M−9αA2u∗2−6αAu∗2+6αA2u∗+2αAu∗−4αA23u∗+A4−αβ3u∗γ1+βu∗4+6τNβ3v∗1−M1+βu∗4,A21=−2τNβ21−M1+βu∗3,A12=0,A03=0,B30=6β3v∗1+βu∗4,B21=−2β21+βu∗3,B12=0,B03=−2γ1+γv∗3.Next, near the Turing bifurcation threshold, we expand the control parameter τ as(36)τT−τ=ετ1+ε2τ2+ε3τ3+oε3,where ε≪1. Similarly, expand the solution X, linear operator L, and the nonlinear term H into Taylor series at ε=0:(37)X=εU1V1+ε2U2V2+ε3U3V3+oε3,(38)H=ε2h2+ε3h3+oε3,(39)L=LT+τT−τM,where(40)h2=h21h22=A20TU12+A11TU1V1+A02TV12B20TU12+B11TU1V1+B02TV12,h3=h31h32=A30TU3+A21TU2V+A12TUV2+A03TV3+2A20TU1U2+A02TV1V2+A11TU1V2+V1U2B30TU3+B21TU2V+B12TUV2+B03TV3+2B20TU1U2+B02TV1V2+B11TU1V2+V1U2−α1A20′U12+A11′U1V1+A02′V12α1B20′U12+B11′U1V1+B02′V12.are terms corresponding to the second and third orders in the expansion of the nonlinear term and for the linear operator(41)L=LT+τT−τM.We have,(42)LT=a10+τa01N−k2d11−Mb10+τb01N−τd2Nk21−Mb10b01−k2d2τ=τT,(43)M=m11m12m21m22,where m11=−a21N/1−M, m12=−a22N+d2Nk2/1−M, m21=0, and m22=0 at U=u∗.Finally, we introduce multiple time scales:(44)∂∂t=ε∂∂T1+ε2∂∂T2+oε2.Then, substituting equations (33)–(44) into equation (32) and expanding it with respect to different orders of εi,i=1,2,3,(45)ε:LTu1v1=0,ε2:LTu2v2=∂∂T1u1v1−α1Mu1v1−h2,ε3:LTu3v3=∂∂T1u2v2+∂∂T2u1v1−α1Mu2v2−α2Mu1v1−h3.In what follows, we seek the amplitude equations by solving system (45). Since LT has an eigenvector associated with the zero eigenvalue,(46)f,1T,f=−a01−τb01N+τd2Nk2a10+τb10N−k2d1.The general solution of the first system of (45) can be written as(47)U1V1=f1∑j=13Wjeikj⋅r+c.c.,where Wj is the amplitude of the mode eikj⋅r. Notice that the second system of (45) is nonhomogeneous, and LT∗, the adjoint operator of LT, has zero eigenvectors in the form of(48)1geikj⋅r+c.c.,j=1,2,3,with g=−b011−M/a10+τb10N−k2d1. Let(49)FUFV=∂∂T1U1V1−β1m11U1+m12V1m21U1+m22V1−α2Mh21h22.Then, in view of the Fredholm solvability conditions,(50)1,gFUjFVj=0,where FUj and FVj are the coefficients of eikj⋅r in FU and FV, respectively. It follows after some routine calculation that, for jl=1,2,3 and jl≠lm, if l≠m,(51)f+g∂Wj1∂T1=α1h3Wj1−2h1+gh2W¯j2W¯j3,where(52)h1=−f2A20T+fA11T+A02T,h2=−f2B20T+fB11T+B02T,h3=fm11+m12+gfm21+m22.Notice the forms ofU1 and V1 given by (47). We have a particular solution for the second system of (45) as follows:(53)U2V2=U¯0V¯0+∑j=13U¯jV¯jeikj⋅r+∑j=13U¯jjV¯jjei2kj⋅r+U¯12V¯12eik1−k2⋅r+U¯23V¯23eik2−k3⋅r+U¯31V¯31eik3−k1⋅r+c.c.,with the coefficients being given below at αT=α:(54)U¯0V¯0=2B01h1−A01h2Δ02A10h2−B10h1Δ0∑j=13Wj2,≡zU0zV0∑j=13Wj2,U¯j=fV¯j,XjjYjj≡zU1zV1Wj2,=1A10−4d1kc2B01−4d2kc2−A01B10×B01−4d2kc2h1−A01h2A10−4d1kc2h2−B10h1Wj2,XjkYjk≡zU2zV2WjW¯k,=1A10−3d1kc2B01−3d2kc2−A01B10×B01−3d2kc2h1−A01h2A10−3d1kc2h2−B10h1WjW¯k.Again, apply the Fredholm solvability condition to the third system of (45). We have, for j=1,(55)f+g∂Vj∂T1+∂Wj∂T2=hα1Vj+α2Wj+h4W¯lW¯m+HV¯lW¯m+V¯mW¯l−G1W12+G2W22+W32Wj,with(56)h4=−2α1A20′f2+A11′f+A02′+gB20′f2+A11′f+B02′,H=−2h1+gh2,G1=−3A30f3+2A11fzV0+A11fzV1+4A20fzU0+2A20fzU1+3A21f2+4A02zV0+2A02zV1+2A11zU0+A11zU1+3A12f+3A03−gB30f3+2B11fzV0+B11fzV1+4B20fzU0+2B20fzU1+3B21f2+4B02zV0+2B02zV1+2B11zU0+B11zU1+3B12f+3B03,G2=−6A30f3+2A11fzV0+A11fzV2+4A20fzU0+2A20fzU2+6A21f2+4A02zV0+2A02zV2+2A11zU0+A11zU2+6A12f+6A03−g6B30f3+2B11fzV0+B11fzV2+4B20fzU0+2B20fzU2+6B21f2+4B02zV0+2B02zV2+2B11zU0+B11zU2+6B12f+6B03.The combination of equations (51) and (55) gives the amplitude equation (57) for the amplitude(57)τ0∂Aj∂t=μAj+hA¯lA¯m−g1A12+g2A22+A32Aj,where(58)τ0=f+gτTfm11+m12+gfm21+m22,μ=τT−ττT,h=HτTfm11+m12+gfm21+m22,gi=GiτTfm11+m12+gfm21+m22.Please notice that system (57) is in complex form. Following to reference [30], for the purpose of convenience of discussion, we convert it into the real form by Aj=ρjexpiφj with ρj as the real amplitudes and φj as phase angles:(59)τ0∂φ∂t=−hρ12ρ22+ρ12ρ32+ρ22ρ32ρ1ρ2ρ3sinφ,τ0∂ρ1∂t=−μρ1+hρ2ρ3cosφ−g1ρ13−g2ρ32+ρ22ρ1,τ0∂ρ2∂t=−μρ2+hρ1ρ3cosφ−g1ρ23−g2ρ32+ρ12ρ2,τ0∂ρ3∂t=−μρ3+hρ2ρ1cosφ−g1ρ33−g2ρ12+ρ22ρ3,where φ=φ1+φ2+φ3. Since we are only interested in the stable steady states and notice the fact that hρi≠0, from the first equation of (59), we have φ=0 or π. Also, noticing the fact that τ0>0, it implies that when h>0, the state corresponding to φ=0 is stable, but the one corresponding to φ=π when h<0. Then, system of amplitude equation (59) becomes(60)τ0∂ρ1∂t=μρ1+hρ2ρ3−g1ρ13−g2ρ32+ρ22ρ1,τ0∂ρ2∂t=μρ2+hρ1ρ3−g1ρ23−g2ρ32+ρ12ρ2,τ0∂ρ3∂t=μρ3+hρ2ρ1−g1ρ33−g2ρ12+ρ22ρ3.Please notice that generally the amplitude equations are valid only when the control parameter is in the Turing space. It is easy to see that the above system of ordinary differential equation (60) has five equilibria, which corresponds five kinds of steady states [10, 30, 31]. Noticing the symmetry of the system, we have the following:(1)
System (60) always has an equilibrium E0=0,0,0, which is stable for μ<μ2=0 and unstable for μ>μ2(2)
System (60) has an equilibrium Es=μ/g1,0,0 corresponding to stripe patterns, which is stable for μ>μ3=h2g1/g2−g12 and unstable for μ>μ3(3)
System (60) has an equilibrium Eh=ρ1,ρ2,ρ3 corresponding to hexagon patterns, with φ=0 or φ=π, and ρ1+=h+h2+4g1+2g2μ/2g1+2g2 is stable for μ<μ4=h22g1+g2/g2−g12 and ρ1−=h−h2+4g1+2g2μ/2g1+2g2 is unstable, where ρ1=ρ2=ρ3=h±h2+4g1+2g2μ/2g1+2g2(4)
System (60) has an equilibrium Em=ρ1,ρ2,ρ3 corresponding to mixed patterns, with g1>g2, μ>g1ρ12 which is unstable, where ρ1=h/g2−g1,ρ2=ρ3=μ−g1ρ12/g2+g1


## 5. Numerical Simulations

In this section, we will further study the dynamic behavior of the coexistence equilibrium of the delayed reaction-diffusion model (3) using numerical simulation in two-dimensional space. In this paper, a two-dimensional delay reaction-diffusion model is treated by the finite difference method in the discrete region of 100×100. The spatial distance between two lattices is defined as the step size ΔxandΔy, using the standard five-point approximation for the 2D Laplacian with the zero-flux boundary conditions, and the time step is expressed as Δt. Take a fixed time step Δt=0.01. What needs to be further explained is that the concentrations Si,jn+1,Ii,jn+1 at the moment n+1Δt at the mesh position i,j are given by(61)Si,jn+1=Si,jn+Δtd1ΔhSi,jn+ΔtfSi,jn,Ii,jn,Ii,jn+1=Ii,jn+Δtd2ΔhIi,jn+ΔtgSi,jn,Ii,jn,with the diffusion term (Laplacian) are defined by(62)ΔhSi,jn=Si+1,jn+Si,j+1n+Si−1,jn+Si,j−1n−4Si,jnh2,ΔhIi,jn=Ii+1,jn+Ii,j+1n+Ii−1,jn+Ii,j−1n−4Ii,jnh2.Other parameters are fixed as(63)α=0.65,β=6,γ=0.5,e=1,η=0.5,d1=0.001,d2=0.1.First, we discuss the effect of weak Allee effect on Turing pattern information. We try to take the Allee effect constant toA=0, A=0.02, and A=0.1. Here, we first discuss the situation without delay. When A=0, Zhang et al. [10] give the condition of Turing instability. Here, we just give the pattern formations. By comparing Figures 1 and 2 (A=0 and A=0.02), we find that the initial state is the coexistence of stripes and spots, and the stripes are very long. With the increase of Allee parameters, the length of stripes decreases and some stripes even form a circle. Then, we continue to increase the value of the weak Allee parameter like Figure 3 (A=0.1), and we find that pattern formations have changed again. As time goes on, we find that pattern formations show a cycle when t=100 and when t=500 and we find that the cycle diffuses outward (indicating that pattern formations are not stable) to form a butterfly-like shape; and finally, when we increase to t=2000, we find that the pattern formations are not stable. It was found that pattern formations became stripes and spots again. After we tried to add more time, we found that the pattern formation did not change again.Figure 1
Mixture patterns obtained with model (1) for A=0 and τ=0. Time: (a) t=100, (b) t=500, (c) t=2000, and (d) t=5000.
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(d)Figure 2
Mixture patterns obtained with model (1) for A=0.02 and τ=0. Time: (a) t=100, (b) t=500, (c) t=2000, and (d) t=5000.
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(d)Figure 3
Mixture patterns obtained with model (1) for A=0.1 and τ=0. Time: (a) t=100, (b) t=500, (c) t=2000, and (d) t=5000.
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(b)
(c)
(d)Next, let us discuss the pattern formation change of the model with time delay and without Allee effect. We change the delay toτ=0.25. By comparing with Figure 1, we find that the stripes and spots of the original pattern formations change to stripes like Figure 4, and the pattern formations will not change as time goes on.Figure 4
Stripes pattern obtained with model (1) for A=0 and τ=0.25. Time: (a) t=100, (b) t=500, (c) t=1000, and (d) t=2000.
(a)
(b)
(c)
(d)Finally, we discuss the pattern formations of models with Allee effect and time delay (A=0.02 and τ=0.02). We find that pattern formations are spots when t=100; as time goes on t=500, we find that pattern formations change again, similar to Figure 3, but the final pattern formations change differently. We can see that pattern formations change into strips surrounded by spots; we increase the time again (t=2000) to find that the pattern will spread outward in this form, forming the phenomenon of strip pattern surrounded by spots pattern; we further increase the time (t=5000) to find that the pattern such as in Figure 5 tends to stabilize and does not change again.Figure 5
Mixture patterns obtained with model (1) for A=0.02 and τ=0.02. Time: (a) t=100, (b) t=500, (c) t=2000, and (d) t=5000.
(a)
(b)
(c)
(d)


## 6. Conclusion

This paper is based on a model that considers a predator-prey model with nonlinear mortality and Holling II functional response. The weak Allee effect is introduced and the effect of the Allee effect on pattern formations is considered. Furthermore, we consider a class of reaction-diffusion predator-prey models with searching delay and weak Allee effect, considering the effects of delay on pattern formations. We give the stability and Turing instability of the positive equilibrium pointE∗. As a result of diffusion, model (3) and model (4) exhibits stationary Turing pattern. Furthermore, through numerical simulation, comparing Figures 1 and 2, we find that the Allee effect will reduce the length of the strip pattern in Figure 1, and there will be some “cycle” pattern as shown in Figure 2. From an ecological point of view, we know that the Allee effect increases the risk of population extinction, while the effect of the longer stripe pattern in Figure 1 increases the likelihood of predation. However, the shorter stripes and spots in Figure 2 reduce the likelihood of predation. As the Allee effect parameter continues to increase, we find that the pattern has changed again. The type of the pattern is similar to that of Figure 1, but the density and size of the pattern will change slightly as shown in Figure 3. We believe that in order to avoid predator hunting, predators are concentrated in a certain area rather than scattered throughout the habitat, which further reduces the contact area between predator and prey. Over time, prey needs to migrate to new habitats. The aggregation pattern diffuses slowly, the predator follows the pursuit, and the aggregation point enlarges gradually. It is worth noting when the Allee effect parameter is A=0.1, there are two positive equilibrium points in model (4). Next, we consider the effect of delay on pattern formations. By comparing Figure 1 with Figure 4, we find when the delay is τ=0.25, the pattern changes from the state where the starting spots pattern and the strip pattern coexist to the case where only the strip pattern exists. Finally, we try to consider the Allee effect and delay to observe the changes in pattern formations, where A=0.02 and τ=0.02. We find that when both are present, the spots pattern is surrounded by strip patterns as shown in Figure 5. This reminds us of animals in the natural world at the lower end of the food chain, often with a means of protection. Juvenile animals are surrounded by adult animals to reduce the probability of their juvenile animals being preyed. This may be an interesting finding or not. So, we find that Allee effect and delay play an important role in spatial invasion of populations.



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