# Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay

Authors: Yong Ye; Hua Liu; Yu-mei Wei; Ming Ma; Kai Zhang
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

In this paper, a prey-predator model and weak Allee effect in prey growth and its dynamical behaviors are studied in detail. The existence, boundedness, and stability of the equilibria of the model are qualitatively discussed. Bifurcation analysis is also taken into account. After incorporating the searching delay and digestion delay, we establish a delayed predator-prey system with Allee effect. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. Finally, we present some numerical simulations to illustrate our theoretical analysis.


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## Body

## 1. Introduction

Some researchers have conducted extensive research on the dynamics of interacting prey-predator models to understand the long-term behavior of species. A wide variety of nonlinear coupled ordinary differential equation models are proposed and analyzed for the interaction between prey and their predators. The classic predator-prey model is the Lotka-Volterra model, which was independently proposed by Lotka in the United States in 1925 and Volterra in Italy in 1926 [1, 2]. The model was developed on the basis of a single-population growth model and has wide applicability. The mathematical form of the Lotka-Volterra model is(1)dxdt=rx-axydydt=cxy-myIn population dynamics,when the population density is very low, there is a positive correlation between the population unit growth rate and the population density. This phenomenon can be called the Allee effect [3–5], starting with Allee’s research [6]. The Allee effect is classified according to the density-dependent properties at low density. If the population density is low, a strong Allee effect will appear. If the proliferation rate is positive and increases, the Allee effect will be weak. Demographic Allee effects can be either weak or strong [7, 8]. When the density is below the critical threshold, the population affected by the strong Allee effect will have a negative average growth rate. Under deterministic dynamics, we find that populations that do not exceed this threshold will be extinct. Many jobs only consider the strong Allee effect, but in the work of Allee it is clear that the Allee effect also has a weak Allee effect [8–13].Today, it is widely believed that the Allee effect greatly increases the likelihood of local and global extinction and can produce a rich variety of dynamic effects [14–16]. And it is interesting and important to study the impact of Allee effect on the predator-prey models [17–19]. In this paper, we introduced a predator-prey model with weak Allee effect:(2)dxdt=rx1-xKxx+A-axydydt=caxy-myx0≥0y0≥0Here, the weak Allee effect term is P(x)=x/x+A, where A>0 is described as a “weak Allee effect constant” ([12]). x is the prey population and y is the predator population, m is the intrinsic death rate of predators, c is the conversion efficiency from prey to predator, K is the carrying capacity, r is the intrinsic growth rate of prey, and a is the prey capture rate by their predators. It is more realistic to introduce time delay on the basis of traditional predator-prey model because it exists almost everywhere in biological activities and is considered as one of the reasons for the regular change of population density [20–26]. Therefore, in order to make the system established in this paper biologically closer to reality, incorporating the searching delay and digestion delay in the system (2) is interesting. Based on the above considerations, We establish a predator-prey model with time delay and weak Allee effect, as follows:(3)dxtdt=rxt1-xtKxtx+A-axt-τ1yt-τ1dytdt=caxt-τ2yt-τ2-mytx0≥0y0≥0where the time delay τi(i=1,2) is the controlling or perturbed parameters, τ1 is the searching delay, and τ2 is the digestion delay.The latter parts of the paper are described as follows. In Section2, we discuss the boundedness, the stability of the equilibria, and bifurcation of the model (2) in detail. In Section 3, we investigated local stability property of interior equilibrium point of the model (3) with time delay; the Hopf bifurcation around the positive equilibrium point is also studied. In Section 4, we verify the previous theoretical derivation by numerical simulation.


## 2. A Predator-Prey Model with Weak Allee Effect

We easily see that model (2) exhibits three equilibrium points E0=(0,0), E1=(K,0), and E∗=(x∗,y∗). Here x∗=m/ca, y∗=Krx∗-rx∗2/aK(x∗+A). And for the positive equilibrium point(s), we have m/ca<K.


### 2.1. Boundedness

Theorem 1.
For the solutionxt,yt of model,(4)limsupt→∞xt+1cyt≤Km+r24rm.Proof.
We defineχ=x(t)+1/cy(t). Then we can easily see that along the solution of system (2),(5)dχdt=dxdt+1cdydt=rx1-xKxx+A-axy+1ccaxy-1cmy=rx1-xKxx+A-mcy.
Thus, we see that for all larget>0(6)dχdt+mχ=rx1-xKxx+A-mcy+mx+m1cy=rx1-xKxx+A+mx≤rx1-xK+mx=xr+m-rKx≤K4rm+r2. Hence the standard comparison argument shows that(7)limsupt→∞xt+1cyt≤Km+r24rm.


### 2.2. Stability Analysis

Theorem 2.
(1) Trivial equilibrium pointE0 is always a saddle-node point.
(2)E1 is stable for a<m/cK and is a saddle point otherwise.
(3) Coexistence equilibriumE∗ is locally asymptotically stable for A<x∗2/K-2x∗ and is unstable node otherwise.Proof.
Let(8)fx,y=rx1-xKxx+A-axygx,y=caxy-my. So, the Jacobian matrix for the model (2) is given by J=∂f/∂x∂f/∂y∂g/∂x∂g/∂y where(9)∂f∂x=-2rKx3+rKx2-3rAx2+2rAKxKx+A2-ay,∂f∂y=-ax,∂g∂x=cay,∂g∂y=cax-m.
So we get(10)J=-2rx3+rKx2-3rAx2+2rAKxKx+A2-ay-axcaycax-m. First, it can be concluded by calculating the Jacobian matrix of the model (2) at E0 given by(11)J0=000-m. And hence E0 is always a saddle-node point. Then, by evaluating the Jacobian matrix of the model (2) at E1, we find(12)J1=-rKK+A-aK0caK-m. First eigenvalue -rK/K+A is negative; hence E1 is stable if caK-m<0 implying a<m/cK, and E1 is a saddle point when a>m/cK. Finally, the Jacobian matrix for the model (2) evaluated at E∗ is given by(13)J∗=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗-mccay∗0. The characteristic polynomial is(14)Hλ=λ2-Tλ+D where T=-rx∗x∗2+2Ax∗-AK/Kx∗+A2 and D=may∗. Thus, we have the following conclusions. (1) If T<0 and A<x∗2/K-2x∗, then the positive equilibrium is locally asymptotically stable. (2) If T>0 and A>x∗2/K-2x∗, then the positive equilibrium is unstable.Theorem 3.
E 1 = ( K , 0 ) is globally stable when a<m/cK.Proof.
Consider the Lyapunov function:(15)Vx,y=∫Kxu-Kudu+1cy. The derivative of V along the solution of the model is(16)V˙=x-Kxdxdt+1cdydt=x-Kxrx1-xKxx+A-axy+1ccaxy-my=x-Kr1-xKxx+A-ax-Ky+axy-mcy=x-KrKx-rx2Kx+A-ax-Ky+axy-mcy=-rxx-K2Kx+A+aKy-mcy≤aKy-mcy.


### 2.3. Bifurcation Analysis

#### 2.3.1. Transcritical Bifurcation

Theorem 4.
The model enters into transcritical bifurcation aroundE1 at a=a0, where a0=m/cK.Proof.
One of the eigenvalues ofJ1 will be zero if J1=0 which gives a=a0. At this point, the other eigenvalue is -rK/K+A. If V and W denote the eigenvectors corresponding to the eigenvalue 0 of the matrices J1 and J1T, respectively, then we obtain V=-aK+A/r,1T and W=0,1T, where J1T=-rK/K+A0-aK0, V1=-a(K+A)/r, V2=1.(17)WTfax¯,y¯,a0=0,WTDfax¯,y¯,a0V=cK≠0,WTD2fx¯,y¯,a0V,V=WT∂2f1∂x2V12+2∂2f1∂x∂yV1V2+∂2f1∂y2V22∂2f2∂x2V12+2∂2f2∂x∂yV1V2+∂2f2∂y2V22x¯,y¯,a0=-2a2K+Ar≠0.
Therefore, by the Sotomayor theorem, we can find that the model experiences transcritical bifurcation ata=a0 around the axial equilibrium E1.


#### 2.3.2. Hopf Bifurcation

From Theorem2, model (2) undergoes bifurcation if A=x∗2/K-2x∗. The purpose of this section is to show that model (2) undergoes a Hopf bifurcation if A=x∗2/K-2x∗. We analyze the Hopf bifurcation occurring at E∗=(x∗,y∗) by choosing as the bifurcation parameter. Denote(18)A0=x∗2K-2x∗.When A=A0, we have T=-rx∗(x∗2+2Ax∗-AK)/Kx∗+A2=0. Thus, the Jacobian matrix J∗ has a pair of imaginary eigenvalues λ=±imay∗. Let λ=α(A)±β(A)i be the roots of λ2-Tλ+D=0; then(19)α2-β2-αT+D=02αβ-Tβ=0and(20)α=T2β=4D-T22dαdAA=A0=-rx∗32AKx∗+A2<0By the Poincare-Andronov-Hopf Bifurcation Theorem, we know that model (2) undergoes a Hopf bifurcation at E∗=(x∗,y∗) when A=A0. However, the detailed nature of the Hopf Bifurcation needs further analysis of the normal form of the model. Set x=X+x∗ and y=Y+y∗, to (x∗,y∗) as origin of coordinates (X,Y). We have the following model:(21)dXdt=a11X+a12Y+F1X,YdYdt=a21X+a22Y+F2X,Ywhere a11=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗/Kx∗+A2-ay∗, a12=-m/c, a21=cay∗, a22=0, and(22)F1X,Y=A1X2+A2XY+A3Y2+B1X3+B2X2Y+B3XY2+B4Y3+P1X,YF2X,Y=C1X2+C2XY+C3Y2+D1X3+D2X2Y+D3XY2+D4Y3+P2X,YA1=-rx∗3-3rAx∗2-3rA2x∗+rA2KKx∗+A3,A2=-a2,A3=0B1=rx∗3+3rA-rx∗2+3rA2-2rAx∗-rA2+rA2K2Kx∗+A3,B2=0,B3=0,B4=0C1=0,C2=ca2,C3=0D1=0,D2=0,D3=0,D4=0where P1(X,Y) and P2(X,Y) are smooth functions of X and Y at least of order four. Now, using the transformationu=X, v=-1/β(a11X+a12Y), we obtain(23)dudt=-βv+G1u,vdvdt=βu+G2u,vwhere(24)G1u,v=F1u,-1a12a11u+βvG2u,v=-1βa11F1u,-1a12a11u+βv+a12F2u,-1a12a11u+βvso(25)G1u,v=A1u2+A2u-1a12a11u+βv+B1u3G2u,v=-1βa11A1u2+A2u-1a12a11u+βv+B1u3+a12uC2-1a12a11u+βvSet(26)σ=116∂3G1∂u3+∂3G1∂u∂v2+∂3G2∂u2∂v+∂3G2∂v3+116β∂2G1∂u∂v∂2G1∂u2+∂2G1∂v2-∂2G2∂u∂v∂2G2∂u2+∂2G2∂v2-∂2G1∂u2∂2G2∂u2+∂2G1∂v2∂2G2∂v2where(27)∂3G1∂u3=6B1,∂3G1∂u∂v2=0,∂3G2∂u2∂v=0,∂3G2∂v3=0,∂2G1∂u∂v=A2βa12,∂2G2∂u∂v=A2a11a12,∂2G1∂v2=0,∂2G2∂v2=0,∂2G1∂u2=2A1-A2a11a12+6B1u,∂2G2∂u2=2A2a112βa12-A1a11β-6B1a11βu.So(28)σ=3B18+116βA2βa122A1-A2a11a12+6B1u-A2a11a122A2a112βa12-A1a11β-6B1a11βu-2A1-A2a11a12+6B1u2A2a112βa12-A1a11β-6B1a11βuIf σ<0, the equilibrium E∗ is destabilized through a Hopf bifurcation that is supercritical and Hopf bifurcation is subcritical otherwise.


## 2.1. Boundedness

Theorem 1.
For the solutionxt,yt of model,(4)limsupt→∞xt+1cyt≤Km+r24rm.Proof.
We defineχ=x(t)+1/cy(t). Then we can easily see that along the solution of system (2),(5)dχdt=dxdt+1cdydt=rx1-xKxx+A-axy+1ccaxy-1cmy=rx1-xKxx+A-mcy.
Thus, we see that for all larget>0(6)dχdt+mχ=rx1-xKxx+A-mcy+mx+m1cy=rx1-xKxx+A+mx≤rx1-xK+mx=xr+m-rKx≤K4rm+r2. Hence the standard comparison argument shows that(7)limsupt→∞xt+1cyt≤Km+r24rm.


## 2.2. Stability Analysis

Theorem 2.
(1) Trivial equilibrium pointE0 is always a saddle-node point.
(2)E1 is stable for a<m/cK and is a saddle point otherwise.
(3) Coexistence equilibriumE∗ is locally asymptotically stable for A<x∗2/K-2x∗ and is unstable node otherwise.Proof.
Let(8)fx,y=rx1-xKxx+A-axygx,y=caxy-my. So, the Jacobian matrix for the model (2) is given by J=∂f/∂x∂f/∂y∂g/∂x∂g/∂y where(9)∂f∂x=-2rKx3+rKx2-3rAx2+2rAKxKx+A2-ay,∂f∂y=-ax,∂g∂x=cay,∂g∂y=cax-m.
So we get(10)J=-2rx3+rKx2-3rAx2+2rAKxKx+A2-ay-axcaycax-m. First, it can be concluded by calculating the Jacobian matrix of the model (2) at E0 given by(11)J0=000-m. And hence E0 is always a saddle-node point. Then, by evaluating the Jacobian matrix of the model (2) at E1, we find(12)J1=-rKK+A-aK0caK-m. First eigenvalue -rK/K+A is negative; hence E1 is stable if caK-m<0 implying a<m/cK, and E1 is a saddle point when a>m/cK. Finally, the Jacobian matrix for the model (2) evaluated at E∗ is given by(13)J∗=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗-mccay∗0. The characteristic polynomial is(14)Hλ=λ2-Tλ+D where T=-rx∗x∗2+2Ax∗-AK/Kx∗+A2 and D=may∗. Thus, we have the following conclusions. (1) If T<0 and A<x∗2/K-2x∗, then the positive equilibrium is locally asymptotically stable. (2) If T>0 and A>x∗2/K-2x∗, then the positive equilibrium is unstable.Theorem 3.
E 1 = ( K , 0 ) is globally stable when a<m/cK.Proof.
Consider the Lyapunov function:(15)Vx,y=∫Kxu-Kudu+1cy. The derivative of V along the solution of the model is(16)V˙=x-Kxdxdt+1cdydt=x-Kxrx1-xKxx+A-axy+1ccaxy-my=x-Kr1-xKxx+A-ax-Ky+axy-mcy=x-KrKx-rx2Kx+A-ax-Ky+axy-mcy=-rxx-K2Kx+A+aKy-mcy≤aKy-mcy.


## 2.3. Bifurcation Analysis

### 2.3.1. Transcritical Bifurcation

Theorem 4.
The model enters into transcritical bifurcation aroundE1 at a=a0, where a0=m/cK.Proof.
One of the eigenvalues ofJ1 will be zero if J1=0 which gives a=a0. At this point, the other eigenvalue is -rK/K+A. If V and W denote the eigenvectors corresponding to the eigenvalue 0 of the matrices J1 and J1T, respectively, then we obtain V=-aK+A/r,1T and W=0,1T, where J1T=-rK/K+A0-aK0, V1=-a(K+A)/r, V2=1.(17)WTfax¯,y¯,a0=0,WTDfax¯,y¯,a0V=cK≠0,WTD2fx¯,y¯,a0V,V=WT∂2f1∂x2V12+2∂2f1∂x∂yV1V2+∂2f1∂y2V22∂2f2∂x2V12+2∂2f2∂x∂yV1V2+∂2f2∂y2V22x¯,y¯,a0=-2a2K+Ar≠0.
Therefore, by the Sotomayor theorem, we can find that the model experiences transcritical bifurcation ata=a0 around the axial equilibrium E1.


### 2.3.2. Hopf Bifurcation

From Theorem2, model (2) undergoes bifurcation if A=x∗2/K-2x∗. The purpose of this section is to show that model (2) undergoes a Hopf bifurcation if A=x∗2/K-2x∗. We analyze the Hopf bifurcation occurring at E∗=(x∗,y∗) by choosing as the bifurcation parameter. Denote(18)A0=x∗2K-2x∗.When A=A0, we have T=-rx∗(x∗2+2Ax∗-AK)/Kx∗+A2=0. Thus, the Jacobian matrix J∗ has a pair of imaginary eigenvalues λ=±imay∗. Let λ=α(A)±β(A)i be the roots of λ2-Tλ+D=0; then(19)α2-β2-αT+D=02αβ-Tβ=0and(20)α=T2β=4D-T22dαdAA=A0=-rx∗32AKx∗+A2<0By the Poincare-Andronov-Hopf Bifurcation Theorem, we know that model (2) undergoes a Hopf bifurcation at E∗=(x∗,y∗) when A=A0. However, the detailed nature of the Hopf Bifurcation needs further analysis of the normal form of the model. Set x=X+x∗ and y=Y+y∗, to (x∗,y∗) as origin of coordinates (X,Y). We have the following model:(21)dXdt=a11X+a12Y+F1X,YdYdt=a21X+a22Y+F2X,Ywhere a11=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗/Kx∗+A2-ay∗, a12=-m/c, a21=cay∗, a22=0, and(22)F1X,Y=A1X2+A2XY+A3Y2+B1X3+B2X2Y+B3XY2+B4Y3+P1X,YF2X,Y=C1X2+C2XY+C3Y2+D1X3+D2X2Y+D3XY2+D4Y3+P2X,YA1=-rx∗3-3rAx∗2-3rA2x∗+rA2KKx∗+A3,A2=-a2,A3=0B1=rx∗3+3rA-rx∗2+3rA2-2rAx∗-rA2+rA2K2Kx∗+A3,B2=0,B3=0,B4=0C1=0,C2=ca2,C3=0D1=0,D2=0,D3=0,D4=0where P1(X,Y) and P2(X,Y) are smooth functions of X and Y at least of order four. Now, using the transformationu=X, v=-1/β(a11X+a12Y), we obtain(23)dudt=-βv+G1u,vdvdt=βu+G2u,vwhere(24)G1u,v=F1u,-1a12a11u+βvG2u,v=-1βa11F1u,-1a12a11u+βv+a12F2u,-1a12a11u+βvso(25)G1u,v=A1u2+A2u-1a12a11u+βv+B1u3G2u,v=-1βa11A1u2+A2u-1a12a11u+βv+B1u3+a12uC2-1a12a11u+βvSet(26)σ=116∂3G1∂u3+∂3G1∂u∂v2+∂3G2∂u2∂v+∂3G2∂v3+116β∂2G1∂u∂v∂2G1∂u2+∂2G1∂v2-∂2G2∂u∂v∂2G2∂u2+∂2G2∂v2-∂2G1∂u2∂2G2∂u2+∂2G1∂v2∂2G2∂v2where(27)∂3G1∂u3=6B1,∂3G1∂u∂v2=0,∂3G2∂u2∂v=0,∂3G2∂v3=0,∂2G1∂u∂v=A2βa12,∂2G2∂u∂v=A2a11a12,∂2G1∂v2=0,∂2G2∂v2=0,∂2G1∂u2=2A1-A2a11a12+6B1u,∂2G2∂u2=2A2a112βa12-A1a11β-6B1a11βu.So(28)σ=3B18+116βA2βa122A1-A2a11a12+6B1u-A2a11a122A2a112βa12-A1a11β-6B1a11βu-2A1-A2a11a12+6B1u2A2a112βa12-A1a11β-6B1a11βuIf σ<0, the equilibrium E∗ is destabilized through a Hopf bifurcation that is supercritical and Hopf bifurcation is subcritical otherwise.


## 2.3.1. Transcritical Bifurcation

Theorem 4.
The model enters into transcritical bifurcation aroundE1 at a=a0, where a0=m/cK.Proof.
One of the eigenvalues ofJ1 will be zero if J1=0 which gives a=a0. At this point, the other eigenvalue is -rK/K+A. If V and W denote the eigenvectors corresponding to the eigenvalue 0 of the matrices J1 and J1T, respectively, then we obtain V=-aK+A/r,1T and W=0,1T, where J1T=-rK/K+A0-aK0, V1=-a(K+A)/r, V2=1.(17)WTfax¯,y¯,a0=0,WTDfax¯,y¯,a0V=cK≠0,WTD2fx¯,y¯,a0V,V=WT∂2f1∂x2V12+2∂2f1∂x∂yV1V2+∂2f1∂y2V22∂2f2∂x2V12+2∂2f2∂x∂yV1V2+∂2f2∂y2V22x¯,y¯,a0=-2a2K+Ar≠0.
Therefore, by the Sotomayor theorem, we can find that the model experiences transcritical bifurcation ata=a0 around the axial equilibrium E1.


## 2.3.2. Hopf Bifurcation

From Theorem2, model (2) undergoes bifurcation if A=x∗2/K-2x∗. The purpose of this section is to show that model (2) undergoes a Hopf bifurcation if A=x∗2/K-2x∗. We analyze the Hopf bifurcation occurring at E∗=(x∗,y∗) by choosing as the bifurcation parameter. Denote(18)A0=x∗2K-2x∗.When A=A0, we have T=-rx∗(x∗2+2Ax∗-AK)/Kx∗+A2=0. Thus, the Jacobian matrix J∗ has a pair of imaginary eigenvalues λ=±imay∗. Let λ=α(A)±β(A)i be the roots of λ2-Tλ+D=0; then(19)α2-β2-αT+D=02αβ-Tβ=0and(20)α=T2β=4D-T22dαdAA=A0=-rx∗32AKx∗+A2<0By the Poincare-Andronov-Hopf Bifurcation Theorem, we know that model (2) undergoes a Hopf bifurcation at E∗=(x∗,y∗) when A=A0. However, the detailed nature of the Hopf Bifurcation needs further analysis of the normal form of the model. Set x=X+x∗ and y=Y+y∗, to (x∗,y∗) as origin of coordinates (X,Y). We have the following model:(21)dXdt=a11X+a12Y+F1X,YdYdt=a21X+a22Y+F2X,Ywhere a11=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗/Kx∗+A2-ay∗, a12=-m/c, a21=cay∗, a22=0, and(22)F1X,Y=A1X2+A2XY+A3Y2+B1X3+B2X2Y+B3XY2+B4Y3+P1X,YF2X,Y=C1X2+C2XY+C3Y2+D1X3+D2X2Y+D3XY2+D4Y3+P2X,YA1=-rx∗3-3rAx∗2-3rA2x∗+rA2KKx∗+A3,A2=-a2,A3=0B1=rx∗3+3rA-rx∗2+3rA2-2rAx∗-rA2+rA2K2Kx∗+A3,B2=0,B3=0,B4=0C1=0,C2=ca2,C3=0D1=0,D2=0,D3=0,D4=0where P1(X,Y) and P2(X,Y) are smooth functions of X and Y at least of order four. Now, using the transformationu=X, v=-1/β(a11X+a12Y), we obtain(23)dudt=-βv+G1u,vdvdt=βu+G2u,vwhere(24)G1u,v=F1u,-1a12a11u+βvG2u,v=-1βa11F1u,-1a12a11u+βv+a12F2u,-1a12a11u+βvso(25)G1u,v=A1u2+A2u-1a12a11u+βv+B1u3G2u,v=-1βa11A1u2+A2u-1a12a11u+βv+B1u3+a12uC2-1a12a11u+βvSet(26)σ=116∂3G1∂u3+∂3G1∂u∂v2+∂3G2∂u2∂v+∂3G2∂v3+116β∂2G1∂u∂v∂2G1∂u2+∂2G1∂v2-∂2G2∂u∂v∂2G2∂u2+∂2G2∂v2-∂2G1∂u2∂2G2∂u2+∂2G1∂v2∂2G2∂v2where(27)∂3G1∂u3=6B1,∂3G1∂u∂v2=0,∂3G2∂u2∂v=0,∂3G2∂v3=0,∂2G1∂u∂v=A2βa12,∂2G2∂u∂v=A2a11a12,∂2G1∂v2=0,∂2G2∂v2=0,∂2G1∂u2=2A1-A2a11a12+6B1u,∂2G2∂u2=2A2a112βa12-A1a11β-6B1a11βu.So(28)σ=3B18+116βA2βa122A1-A2a11a12+6B1u-A2a11a122A2a112βa12-A1a11β-6B1a11βu-2A1-A2a11a12+6B1u2A2a112βa12-A1a11β-6B1a11βuIf σ<0, the equilibrium E∗ is destabilized through a Hopf bifurcation that is supercritical and Hopf bifurcation is subcritical otherwise.


## 3. Delayed Model with Weak Allee Effect

LetX(t)=x(t)-x∗, Y(t)=y(t)-y∗; then the model (3) can be expressed as in the following matrix form after linearization:(29)ddtXtYt=A1XtYt+A2Xt-τ1Yt-τ1+A3Xt-τ2Yt-τ2,A1=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A200-m,A2=-ay∗-ax∗00,A3=00cay∗cax∗.


### 3.1. Stability Analysis

The characteristic polynomial is(30)Hλ=λ2-Tλ+Dwhere(31)T=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗e-λτ1+cax∗e-λτ2-m,D=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗e-λτ1cax∗e-λτ2-m+ca2x∗y∗e-λτ1e-λτ2.Let(32)G=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2So(33)Hλ=λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2GLet λ(τ)=η(τ)+iω(τ), η(τ0)=0, and ω(τ0)=ω0.Theorem 5.
AssumeA<x∗2/K-2x∗, when τ1>0,τ2=0; we have the following conclusions. (1) When 4FG-2Fm+G2+F2+m2-D2>0 and -2FmG2+F2G2+m2G2-E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) Hopf bifurcation occurs when τ1 passes the critical value(34)τ¯1=1ω0arccos⁡EGm-EFG-ω02mD+ω02FD+ω02GD+ω02Eω02D2+E2.Proof.
The characteristic equation is(35)λ2-G-ay∗e-λτ1+cax∗-mλ-mG+may∗e-λτ1+cax∗G=0. Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(36)-ω02-iω0G-ay∗e-iω0τ1+cax∗-m-mG+may∗e-iω0τ1+cax∗G=0. Then(37)-ω02-iω0G-De-iω0τ1+F-m-mG+Ee-iω0τ1+FG=0. Where D=ay∗, E=may∗, F=cax∗ we know(38)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(39)-ω02-iω0G+iω0Dcos⁡ω0τ1+ω0Dsin⁡ω0τ1-iω0F+iω0m-mG+Ecos⁡ω0τ1-iEsin⁡ω0τ1+FG=0. Separate real and imaginary parts(40)-ω02+ω0Dsin⁡ω0τ1-mG+Ecos⁡ω0τ1+FG=0-ω0G+ω0Dcos⁡ω0τ1-ω0F+ω0m-Esin⁡ω0τ1=0Then(41)ω0Dsin⁡ω0τ1+Ecos⁡ω0τ1=ω02-FG+mGω0Dcos⁡ω0τ1-Esin⁡ω0τ1=ω0G+ω0F-ω0mSo(42)ω04+4FG-2Fm+G2+F2+m2-D2ω02-2FmG2+F2G2+m2G2-E2=0 We assume that(43)W=ω02W2+4FG-2Fm+G2+F2+m2-D2W-2FmG2+F2G2+m2G2-E2=0.If 4FG-2Fm+G2+F2+m2-D2>0 and -2FmG2+F2G2+m2G2-E2>0, then all roots of equation have negative real parts for all τ1>0,τ2=0; that is, the equilibrium E∗=(x∗,y∗) is locally asymptotically stable:(44)W=-M±M2-4-2FmG2+F2G2+m2G2-E22 where(45)M=4FG-2Fm+G2+F2+m2-D2. If -M>0, M2=4(2FmG2+F2G2+m2G2-E2) and -2FmG2+F2G2+m2G2-E2>0, there is a unique positive solution; the equilibrium E∗=(x∗,y∗) is unstable. Also if -M>0, M2>4-2FmG2+F2G2+m2G2-E2, and -2FmG2+F2G2+m2G2-E2>0, then there are two positive solutions. We have(46)-ω02+ω0Dsin⁡ω0τ1-mG+Ecos⁡ω0τ1+FG=0 So(47)sin⁡ω0τ1=ω02+mG-Ecos⁡ω0τ1-FGω0D Then, we get(48)cos⁡ω0τ1=EGm-EFG-ω02mD+ω02FD+ω02GD+ω02Eω02D2+E2. It shows that if -M>0 and -2FmG2+F2G2+m2G2-E2>0,(49)λ2-G-ay∗e-λτ1+cax∗+mλ+mG-may∗e-λτ1+cax∗G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯1)=0
whenτ¯1j±=1/ω0±arccos⁡EGm-EFG-ω02mD+ω02FD+ω02GD+ω02E/ω02D2+E2+2πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(50)dReλdτ1-1≠0 According to(51)λ2-G-ay∗e-λτ1+cax∗-mλ-mG+may∗e-λτ1+cax∗G=0 At this time, τ1=τ¯1, where τ¯1 is the value of τ¯1j at j=0. We get(52)2λdλdτ1-dλdτ1G-De-λτ1+F-m+λDe-λτ1-τ1dλdτ1-λ+Ee-λτ1-τ1dλdτ1-λ=0 Then(53)dλdτ1-1=-τ1λ+DλE+λD+2λ-G+F-mλλ2-λG+F-m-mG+FG.dReλdτ1-1=-D2ω02D2+E2+-G+F-m2+2-mG+FG-ω02G+F-m2ω02+-mG+FG-ω022≠0.Theorem 6.
AssumeA<x∗2/K-2x∗, when τ1=0,τ2>0; we have the following conclusions. (1) If -2EmG-F2G2+m2G2+E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) When -2EmG-F2G2+m2G2+E2<0, if τ2<τ¯2, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable; if τ2>τ¯2, the positive equilibrium is unstable. (3) Hopf bifurcation occurs when τ2 passes the critical value(54)τ¯2=1ω0arccos⁡ω02-m-DF+FGE-mG-ω02F2-E2.Proof.
The characteristic equation is(55)λ2-G-ay∗+cax∗e-λτ2+mλ+mG-may∗+cax∗e-λτ2G=0 Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(56)-ω02-iω0G-ay∗+cax∗e-iω0τ2-m-mG+may∗+cax∗Ge-iω0τ2=0. Then(57)-ω02-iω0G-D+Fe-iω0τ2-m-mG+E+FGe-iω0τ2=0. Where D=ay∗, E=may∗, F=cax∗ we know(58)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(59)-ω02-iω0G+iω0D-ω0Fsin⁡ω0τ2-iω0Fcos⁡ω0τ2+iω0m-mG+E-iFGsin⁡ω0τ2+FGcos⁡ω0τ2=0. Separate real and imaginary parts(60)-ω02-ω0Fsin⁡ω0τ2-mG+E+FGcos⁡ω0τ2=0-ω0G+ω0D-ω0Fcos⁡ω0τ2+ω0m-FGsin⁡ω0τ2=0Then(61)-ω0Fsin⁡ω0τ2+FGcos⁡ω0τ2=ω02-E+mG-ω0Fcos⁡ω0τ2-FGsin⁡ω0τ2=ω0G-ω0D-ω0mSo(62)ω04+-2E-2DG+2Dm+G2-F2+m2+D2ω02-2EmG-F2G2+m2G2+E2=0 We assume that(63)W=ω02W2+-2E-2DG+2Dm+G2-F2+m2+D2W-2EmG-F2G2+m2G2+E2=0We can easily find(64)-2E-2DG+2Dm+G2-F2+m2+D2>0. If -2EmG-F2G2+m2G2+E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable,-2EmG-F2G2+m2G2+E2<0, and the positive equilibrium is unstable. We have(65)-ω0Fsin⁡ω0τ2+FGcos⁡ω0τ2=ω02-E+mG So(66)sin⁡ω0τ1=ω02-E+mG-FGcos⁡ω0τ2-ω0F Then, we get(67)cos⁡ω0τ2=ω02-m-DF-FG-E+mG-ω02F2-E2. It shows that if-2EmG-F2G2+m2G2+E2<0,(68)λ2-G-ay∗+cax∗e-λτ2-mλ-mG+may∗+cax∗e-λτ2G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯2)=0
whenτ¯2j±=1/ω0±arccos⁡ω02(-m-D)F+FG(E-mG)/-ω02F2-E2+2πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(69)dReλdτ2-1≠0 According to(70)λ2-G-ay∗+cax∗e-λτ2-mλ-mG+may∗+cax∗e-λτ2G=0 At this time,τ2=τ¯2, where τ¯2 is the value of τ¯2j at j=0. We get(71)2λdλdτ2-dλdτ2G-D+Fe-λτ2-m-λFe-λτ2-τ2dλdτ2-λ+FGe-λτ2-τ2dλdτ2-λ=0 Then(72)dλdτ2-1=-τ2λ-FλG-λ+2λ-G-D-mλλ2-λG-D-m-mG+EdReλdτ2-1=-Fω02+G2+2-mG+E-ω02-G-D-m2ω02G-D-m2+-mG+E-ω022≠0.Theorem 7.
AssumeA<x∗2/K-2x∗, when τ1≠0,τ2≠0; we have the following conclusions. (1) When and G2-F2+m2-D2+2DF>0 and -2EFG-F2G2+m2G2-E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) Hopf bifurcation occurs when τ1 passes the critical value(73)τ¯=1ω0arctan⁡ω02+mGω0D-ω0F-ω0G-m-ω02+mGFG+E+ω0G-m.Proof.
The characteristic equation is(74)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0. Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(75)-ω02-iω0G-De-iω0τ1+Fe-iω0τ2+m-mG+Ee-iω0τ1+FGe-iω0τ2=0. Where D=ay∗, E=may∗, F=cax∗ we know(76)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(77)-ω02-iω0G+iω0Dcos⁡ω0τ1+ω0Dsin⁡ω0τ1-ω0Fsin⁡ω0τ2-iω0Fcos⁡ω0τ2+iω0m-mG+Ecos⁡ω0τ1-iEsin⁡ω0τ1-iFGsin⁡ω0τ2+FGcos⁡ω0τ2=0 Separate real and imaginary parts(78)-ω02+ω0Dsin⁡ω0τ1-ω0Fsin⁡ω0τ2-mG+Ecos⁡ω0τ1+FGcos⁡ω0τ2=0-ω0G+ω0Dcos⁡ω0τ1-ω0Fcos⁡ω0τ2+ω0m-FGsin⁡ω0τ2-Esin⁡ω0τ1=0Let τ1=τ2=τ>0. Then(79)ω0Dsin⁡ω0τ-ω0Fsin⁡ω0τ+Ecos⁡ω0τ+FGcos⁡ω0τ=ω02+mGω0Dcos⁡ω0τ-ω0Fcos⁡ω0τ-FGsin⁡ω0τ-Esin⁡ω0τ=ω0G-ω0mSo(80)ω04+G2-F2+m2-D2+2DFω02-2EFH-F2G2+m2G2-E2=0 We assume that(81)W=ω02W2+G2-F2+m2-D2+2DFW-2EFH-F2G2+m2G2-E2=0If G2-F2+m2-D2+2DF>0 and -2EFH-F2G2+m2G2-E2>0, then all roots of equation have negative real parts for all τ1>0,τ2>0; that is, the equilibrium E∗=(x∗,y∗) is locally asymptotically stable:(82)W=-N±N2-4-2EFH-F2G2+m2G2-E22 where(83)N=G2-F2+m2-D2+2DF If -N>0, N2=4-2EFH-F2G2+m2G2-E2, and -2EFH-F2G2+m2G2-E2>0, there is a unique positive solution; the equilibrium E∗=(x∗,y∗) is unstable. Also if -N>0, N2>4-2EFH-F2G2+m2G2-E2, and -2EFH-F2G2+m2G2-E2>0, then there are two positive solutions. We have(84)ω0Dsin⁡ω0τ-ω0Fsin⁡ω0τ+Ecos⁡ω0τ+FGcos⁡ω0τ=ω02+mGω0Dcos⁡ω0τ-ω0Fcos⁡ω0τ-FGsin⁡ω0τ-Esin⁡ω0τ=ω0G-ω0mSo(85)sin⁡ω0τ=ω02+mGω0D-ω0F-ω0G-mω0D-ω0F2-FG+E2cos⁡ω0τ=-ω02+mGFG+E+ω0G-mω0D-ω0F2-FG+E2Then, we get(86)tan⁡ω0τ=ω02+mGω0D-ω0F-ω0G-m-ω02-mGFG+E+ω0G-m. It shows that if -2EFH-F2G2+m2G2-E2<0,(87)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯1)=0
whenτ¯j±=1/ω0±arctan⁡(ω02+mG)(ω0±D-ω0±F)-ω0±(G-m)/-(ω02+mG)(FG+E)+ω0±(G-m)+πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(88)dReλdτ-1≠0 According to(89)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0 At this time,τ=τ¯, where τ¯ is the value of τ¯j at j=0. We get(90)2λdλdτ-dλdτG-De-λτ+Fe-λτ-m+λDe-λτ-τdλdτ-λλ-λFe-λτ-τdλdτ-λ+Ee-λτ-τdλdτ-λ+FGe-λτ-τdλdτ-λ=0 Then(91)dλdτ-1=-τλ-F-DλλD-λF+E+FG+2λ-G-mλλ2-λG-m-mGdReλdτ-1=-F-D2ω02F-D2+FG+E2+2-mG-ω02-G-m2ω02G-m2+-mG-ω022≠0.


## 3.1. Stability Analysis

The characteristic polynomial is(30)Hλ=λ2-Tλ+Dwhere(31)T=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗e-λτ1+cax∗e-λτ2-m,D=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2-ay∗e-λτ1cax∗e-λτ2-m+ca2x∗y∗e-λτ1e-λτ2.Let(32)G=-2rx∗3+rKx∗2-3rAx∗2+2rAKx∗Kx∗+A2So(33)Hλ=λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2GLet λ(τ)=η(τ)+iω(τ), η(τ0)=0, and ω(τ0)=ω0.Theorem 5.
AssumeA<x∗2/K-2x∗, when τ1>0,τ2=0; we have the following conclusions. (1) When 4FG-2Fm+G2+F2+m2-D2>0 and -2FmG2+F2G2+m2G2-E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) Hopf bifurcation occurs when τ1 passes the critical value(34)τ¯1=1ω0arccos⁡EGm-EFG-ω02mD+ω02FD+ω02GD+ω02Eω02D2+E2.Proof.
The characteristic equation is(35)λ2-G-ay∗e-λτ1+cax∗-mλ-mG+may∗e-λτ1+cax∗G=0. Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(36)-ω02-iω0G-ay∗e-iω0τ1+cax∗-m-mG+may∗e-iω0τ1+cax∗G=0. Then(37)-ω02-iω0G-De-iω0τ1+F-m-mG+Ee-iω0τ1+FG=0. Where D=ay∗, E=may∗, F=cax∗ we know(38)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(39)-ω02-iω0G+iω0Dcos⁡ω0τ1+ω0Dsin⁡ω0τ1-iω0F+iω0m-mG+Ecos⁡ω0τ1-iEsin⁡ω0τ1+FG=0. Separate real and imaginary parts(40)-ω02+ω0Dsin⁡ω0τ1-mG+Ecos⁡ω0τ1+FG=0-ω0G+ω0Dcos⁡ω0τ1-ω0F+ω0m-Esin⁡ω0τ1=0Then(41)ω0Dsin⁡ω0τ1+Ecos⁡ω0τ1=ω02-FG+mGω0Dcos⁡ω0τ1-Esin⁡ω0τ1=ω0G+ω0F-ω0mSo(42)ω04+4FG-2Fm+G2+F2+m2-D2ω02-2FmG2+F2G2+m2G2-E2=0 We assume that(43)W=ω02W2+4FG-2Fm+G2+F2+m2-D2W-2FmG2+F2G2+m2G2-E2=0.If 4FG-2Fm+G2+F2+m2-D2>0 and -2FmG2+F2G2+m2G2-E2>0, then all roots of equation have negative real parts for all τ1>0,τ2=0; that is, the equilibrium E∗=(x∗,y∗) is locally asymptotically stable:(44)W=-M±M2-4-2FmG2+F2G2+m2G2-E22 where(45)M=4FG-2Fm+G2+F2+m2-D2. If -M>0, M2=4(2FmG2+F2G2+m2G2-E2) and -2FmG2+F2G2+m2G2-E2>0, there is a unique positive solution; the equilibrium E∗=(x∗,y∗) is unstable. Also if -M>0, M2>4-2FmG2+F2G2+m2G2-E2, and -2FmG2+F2G2+m2G2-E2>0, then there are two positive solutions. We have(46)-ω02+ω0Dsin⁡ω0τ1-mG+Ecos⁡ω0τ1+FG=0 So(47)sin⁡ω0τ1=ω02+mG-Ecos⁡ω0τ1-FGω0D Then, we get(48)cos⁡ω0τ1=EGm-EFG-ω02mD+ω02FD+ω02GD+ω02Eω02D2+E2. It shows that if -M>0 and -2FmG2+F2G2+m2G2-E2>0,(49)λ2-G-ay∗e-λτ1+cax∗+mλ+mG-may∗e-λτ1+cax∗G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯1)=0
whenτ¯1j±=1/ω0±arccos⁡EGm-EFG-ω02mD+ω02FD+ω02GD+ω02E/ω02D2+E2+2πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(50)dReλdτ1-1≠0 According to(51)λ2-G-ay∗e-λτ1+cax∗-mλ-mG+may∗e-λτ1+cax∗G=0 At this time, τ1=τ¯1, where τ¯1 is the value of τ¯1j at j=0. We get(52)2λdλdτ1-dλdτ1G-De-λτ1+F-m+λDe-λτ1-τ1dλdτ1-λ+Ee-λτ1-τ1dλdτ1-λ=0 Then(53)dλdτ1-1=-τ1λ+DλE+λD+2λ-G+F-mλλ2-λG+F-m-mG+FG.dReλdτ1-1=-D2ω02D2+E2+-G+F-m2+2-mG+FG-ω02G+F-m2ω02+-mG+FG-ω022≠0.Theorem 6.
AssumeA<x∗2/K-2x∗, when τ1=0,τ2>0; we have the following conclusions. (1) If -2EmG-F2G2+m2G2+E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) When -2EmG-F2G2+m2G2+E2<0, if τ2<τ¯2, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable; if τ2>τ¯2, the positive equilibrium is unstable. (3) Hopf bifurcation occurs when τ2 passes the critical value(54)τ¯2=1ω0arccos⁡ω02-m-DF+FGE-mG-ω02F2-E2.Proof.
The characteristic equation is(55)λ2-G-ay∗+cax∗e-λτ2+mλ+mG-may∗+cax∗e-λτ2G=0 Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(56)-ω02-iω0G-ay∗+cax∗e-iω0τ2-m-mG+may∗+cax∗Ge-iω0τ2=0. Then(57)-ω02-iω0G-D+Fe-iω0τ2-m-mG+E+FGe-iω0τ2=0. Where D=ay∗, E=may∗, F=cax∗ we know(58)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(59)-ω02-iω0G+iω0D-ω0Fsin⁡ω0τ2-iω0Fcos⁡ω0τ2+iω0m-mG+E-iFGsin⁡ω0τ2+FGcos⁡ω0τ2=0. Separate real and imaginary parts(60)-ω02-ω0Fsin⁡ω0τ2-mG+E+FGcos⁡ω0τ2=0-ω0G+ω0D-ω0Fcos⁡ω0τ2+ω0m-FGsin⁡ω0τ2=0Then(61)-ω0Fsin⁡ω0τ2+FGcos⁡ω0τ2=ω02-E+mG-ω0Fcos⁡ω0τ2-FGsin⁡ω0τ2=ω0G-ω0D-ω0mSo(62)ω04+-2E-2DG+2Dm+G2-F2+m2+D2ω02-2EmG-F2G2+m2G2+E2=0 We assume that(63)W=ω02W2+-2E-2DG+2Dm+G2-F2+m2+D2W-2EmG-F2G2+m2G2+E2=0We can easily find(64)-2E-2DG+2Dm+G2-F2+m2+D2>0. If -2EmG-F2G2+m2G2+E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable,-2EmG-F2G2+m2G2+E2<0, and the positive equilibrium is unstable. We have(65)-ω0Fsin⁡ω0τ2+FGcos⁡ω0τ2=ω02-E+mG So(66)sin⁡ω0τ1=ω02-E+mG-FGcos⁡ω0τ2-ω0F Then, we get(67)cos⁡ω0τ2=ω02-m-DF-FG-E+mG-ω02F2-E2. It shows that if-2EmG-F2G2+m2G2+E2<0,(68)λ2-G-ay∗+cax∗e-λτ2-mλ-mG+may∗+cax∗e-λτ2G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯2)=0
whenτ¯2j±=1/ω0±arccos⁡ω02(-m-D)F+FG(E-mG)/-ω02F2-E2+2πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(69)dReλdτ2-1≠0 According to(70)λ2-G-ay∗+cax∗e-λτ2-mλ-mG+may∗+cax∗e-λτ2G=0 At this time,τ2=τ¯2, where τ¯2 is the value of τ¯2j at j=0. We get(71)2λdλdτ2-dλdτ2G-D+Fe-λτ2-m-λFe-λτ2-τ2dλdτ2-λ+FGe-λτ2-τ2dλdτ2-λ=0 Then(72)dλdτ2-1=-τ2λ-FλG-λ+2λ-G-D-mλλ2-λG-D-m-mG+EdReλdτ2-1=-Fω02+G2+2-mG+E-ω02-G-D-m2ω02G-D-m2+-mG+E-ω022≠0.Theorem 7.
AssumeA<x∗2/K-2x∗, when τ1≠0,τ2≠0; we have the following conclusions. (1) When and G2-F2+m2-D2+2DF>0 and -2EFG-F2G2+m2G2-E2>0, the positive equilibrium E∗=(x∗,y∗) is locally asymptotically stable. (2) Hopf bifurcation occurs when τ1 passes the critical value(73)τ¯=1ω0arctan⁡ω02+mGω0D-ω0F-ω0G-m-ω02+mGFG+E+ω0G-m.Proof.
The characteristic equation is(74)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0. Next we suppose that λ(τ0)=iω0 is a solution of H(λ) for some τ>0; then we have(75)-ω02-iω0G-De-iω0τ1+Fe-iω0τ2+m-mG+Ee-iω0τ1+FGe-iω0τ2=0. Where D=ay∗, E=may∗, F=cax∗ we know(76)e-iω0τ=cos⁡ω0τ-isin⁡ω0τ. So we get(77)-ω02-iω0G+iω0Dcos⁡ω0τ1+ω0Dsin⁡ω0τ1-ω0Fsin⁡ω0τ2-iω0Fcos⁡ω0τ2+iω0m-mG+Ecos⁡ω0τ1-iEsin⁡ω0τ1-iFGsin⁡ω0τ2+FGcos⁡ω0τ2=0 Separate real and imaginary parts(78)-ω02+ω0Dsin⁡ω0τ1-ω0Fsin⁡ω0τ2-mG+Ecos⁡ω0τ1+FGcos⁡ω0τ2=0-ω0G+ω0Dcos⁡ω0τ1-ω0Fcos⁡ω0τ2+ω0m-FGsin⁡ω0τ2-Esin⁡ω0τ1=0Let τ1=τ2=τ>0. Then(79)ω0Dsin⁡ω0τ-ω0Fsin⁡ω0τ+Ecos⁡ω0τ+FGcos⁡ω0τ=ω02+mGω0Dcos⁡ω0τ-ω0Fcos⁡ω0τ-FGsin⁡ω0τ-Esin⁡ω0τ=ω0G-ω0mSo(80)ω04+G2-F2+m2-D2+2DFω02-2EFH-F2G2+m2G2-E2=0 We assume that(81)W=ω02W2+G2-F2+m2-D2+2DFW-2EFH-F2G2+m2G2-E2=0If G2-F2+m2-D2+2DF>0 and -2EFH-F2G2+m2G2-E2>0, then all roots of equation have negative real parts for all τ1>0,τ2>0; that is, the equilibrium E∗=(x∗,y∗) is locally asymptotically stable:(82)W=-N±N2-4-2EFH-F2G2+m2G2-E22 where(83)N=G2-F2+m2-D2+2DF If -N>0, N2=4-2EFH-F2G2+m2G2-E2, and -2EFH-F2G2+m2G2-E2>0, there is a unique positive solution; the equilibrium E∗=(x∗,y∗) is unstable. Also if -N>0, N2>4-2EFH-F2G2+m2G2-E2, and -2EFH-F2G2+m2G2-E2>0, then there are two positive solutions. We have(84)ω0Dsin⁡ω0τ-ω0Fsin⁡ω0τ+Ecos⁡ω0τ+FGcos⁡ω0τ=ω02+mGω0Dcos⁡ω0τ-ω0Fcos⁡ω0τ-FGsin⁡ω0τ-Esin⁡ω0τ=ω0G-ω0mSo(85)sin⁡ω0τ=ω02+mGω0D-ω0F-ω0G-mω0D-ω0F2-FG+E2cos⁡ω0τ=-ω02+mGFG+E+ω0G-mω0D-ω0F2-FG+E2Then, we get(86)tan⁡ω0τ=ω02+mGω0D-ω0F-ω0G-m-ω02-mGFG+E+ω0G-m. It shows that if -2EFH-F2G2+m2G2-E2<0,(87)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0 has a pair of imaginary eigenvalues λ=±iω0, η(τ¯1)=0
whenτ¯j±=1/ω0±arctan⁡(ω02+mG)(ω0±D-ω0±F)-ω0±(G-m)/-(ω02+mG)(FG+E)+ω0±(G-m)+πj/ω0±,j=0,1,2,⋯.
Next verify the cross-sectional conditions:(88)dReλdτ-1≠0 According to(89)λ2-G-ay∗e-λτ1+cax∗e-λτ2-mλ-mG+may∗e-λτ1+cax∗e-λτ2G=0 At this time,τ=τ¯, where τ¯ is the value of τ¯j at j=0. We get(90)2λdλdτ-dλdτG-De-λτ+Fe-λτ-m+λDe-λτ-τdλdτ-λλ-λFe-λτ-τdλdτ-λ+Ee-λτ-τdλdτ-λ+FGe-λτ-τdλdτ-λ=0 Then(91)dλdτ-1=-τλ-F-DλλD-λF+E+FG+2λ-G-mλλ2-λG-m-mGdReλdτ-1=-F-D2ω02F-D2+FG+E2+2-mG-ω02-G-m2ω02G-m2+-mG-ω022≠0.


## 4. Numerical Simulations

In this section, we present some numerical simulations to illustrate our theoretical analysis.First, we theoretically analyze a predator-prey model with Allee effect in the article and obtain the stability conditions ofE1=(K,0) and E∗=(x∗,y∗). Secondly, we carry out numerical simulation and select appropriate the parameters, which draw the stable positions of the equilibrium points E1=(K,0) and E∗=(x∗,y∗), shown in Figures 1 and 2. In the analysis later in this chapter, we use A as the bifurcation parameter to obtain the critical value of the Hopf bifurcation generated by the model (2). By comparing Figures 3 and 4, we find that as the parameter A changes the equilibrium point changes from a steady state to a limit cycle. To further verify our point of view, we have made a bifurcation diagram as shown in Figure 5. We found that the bifurcation parameter produced a bifurcation at about 100, which coincided with our previous guess. By the same analysis method, we give a set of timing diagrams for comparison as shown in Figures 6 and 7; we found that with the increase of the parameter A the model (2) is shown in Figure 6 and the population gradually becomes stable with the increase of time, while Figure 7 is a periodic change.Figure 1
The phase portrait of model (2) with weak Allee effect. The parameters are taken as A=0.01, r=2.65, K=900, a=0.002, c=0.215, and m=1.06; E1=(K,0) is locally asymptotically stable.Figure 2
The phase portrait of model (2) with weak Allee effect. The parameters are taken as A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) is locally asymptotically stable.Figure 3
The phase portrait of model (2) with weak Allee effect. The parameters are taken as x(0)=260, y(0)=70, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) is locally asymptotically stable, with no limit cycle.Figure 4
The phase portrait of model (2) with weak Allee effect. The parameters are taken as x(0)=260, y(0)=70, A=150, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) becomes unstable; a limit cycle is formed.Figure 5
Bifurcation diagram with respect to the parameterA; other parameter values are r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 6
A Time series diagram for prey and predator. The figure depicts local stability of the interior equilibrium for the model (2), where the parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 7
A Time series diagram for prey and predator. Existence of periodic solution around the interior equilibriumE∗=(x∗,y∗) for the model (2), where the parameter values are x(0)=230, y(0)=95, A=150, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Next, we performed a numerical simulation of the model (3), mainly to study the effect of the time-delay parameter τ on the stability of the coexistence equilibrium point. Here, we compare three sets of timing diagrams, which are the effect of τ1 on the stability of the coexistence equilibrium point, the influence of τ2 on the stability of the coexistence equilibrium point, and the influence of τ1=τ2=τ on the stability of the coexistence equilibrium point. First, by comparing Figures 8 and 9, we find that when τ1=0.1, the population gradually becomes stable with the increase of time. When τ1=0.25, the population gradually shows periodicity with the increase of time. By comparing Figures 10 and 11, we find that when τ2=0.1, the population gradually becomes stable with the increase of time. When τ2=1.5, the population gradually shows periodicity with the increase of time; by comparing Figures 12 and 13, we found that when τ=0.1, the population gradually becomes stable with the increase of time. When τ=0.19, the population gradually shows periodicity with the increase of time. At the same time, we selected the appropriate parameters and gave three sets of phase diagrams for comparison. By comparing Figures 14 and 15, we find that when τ1 increases from 0.1 to 0.25, the coexistence equilibrium point changes from a steady state to a limit cycle; by comparing Figures 16 and 17, we find that τ2 is increased from 0.1 to 1.5. At this time, the coexistence equilibrium point changes from a steady state to a limit cycle; by comparing Figures 18 and 19, we find that when τ increases from 0.1 to 0.19, the coexistence equilibrium point changes from a steady state to a limit cycle. From this, we can conclude that the stability of the equilibrium point of the model (3) changes with the increase of the time lag when the time-delay parameter is introduced, and the generation of the limit cycle (periodic solution) is accompanied by this change.Figure 8
A Time series diagram for prey and predator. The figure depicts local stability of the interior equilibrium for the delayed model (3) with the time delays τ1=0.1 and τ2=0, where the other parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 9
A Time series diagram for prey and predator. Existence of periodic solution around the interior equilibriumE∗=(x∗,y∗) for delayed model (3) with the time delays τ1=0.25 and τ2=0, where the parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 10
A Time series diagram for prey and predator. The figure depicts local stability of the interior equilibrium for the delayed model (3) with the time delays τ1=0 and τ2=0.1, where the other parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 11
A Time series diagram for prey and predator. Existence of periodic solution around the interior equilibriumE∗=(x∗,y∗) for delayed model (3) with the time delays τ1=0 and τ2=1.5, where the parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 12
A Time series diagram for prey and predator. The figure depicts local stability of the interior equilibrium for the delayed model (3) with the time delay τ1=τ2=0.1, where the other parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 13
A Time series diagram for prey and predator. Existence of periodic solution around the interior equilibriumE∗=(x∗,y∗) for delayed model (3) with the time delay τ1=τ2=0.19, where the parameter values are x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06.Figure 14
The phase portrait of delayed model (3) with the time delays τ1=0.1 and τ2=0. The parameters are taken as x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) is locally asymptotically stable, with no limit cycle.Figure 15
The phase portrait of delayed model (3) with the time delays τ1=0.25 and τ2=0. The parameters are taken as x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) becomes unstable; a limit cycle is formed.Figure 16
The phase portrait of delayed model (3) with the time delays τ1=0 and τ2=0.1. The parameters are taken as x0=230,y0=95,A=0.01,r=2.65,K=900,a=0.02,c=0.215,andm=1.06; E∗=(x∗,y∗) is locally asymptotically stable, with no limit cycle.Figure 17
The phase portrait of delayed model (3) with the time delays τ1=0 and τ2=1.5. The parameters are taken as x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) becomes unstable; a limit cycle is formed.Figure 18
The phase portrait of delayed model (3) with the time delay τ1=τ2=0.1. The parameters are taken as x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) is locally asymptotically stable, with no limit cycle.Figure 19
The phase portrait of delayed model (3) with the time delay τ1=τ2=0.19. The parameters are taken as x(0)=230, y(0)=95, A=0.01, r=2.65, K=900, a=0.02, c=0.215, and m=1.06; E∗=(x∗,y∗) becomes unstable; a limit cycle is formed.


## 5. Conclusions

In this paper, we establish a predator-prey model with a weak Allee effect and demonstrate and analyze the boundedness and stability of the model. We also prove that the model (2) experiences transcritical bifurcation around the axial equilibrium and the model (2) undergoes a Hopf bifurcation at E∗=(x∗,y∗) when A=A0; we also analyzed the direction and stability of Hopf bifurcation. Immediately after we introduced the searching delay and digestion delay in the model (2), a new model was obtained, and the model (3) was analyzed for stability changes caused by time lag. It is concluded that the stability of the coexistence equilibrium point of the model (3) changes as the time lag increases. Finally, we verify our theoretical derivation by numerical simulation. First, we select the appropriate parameters to satisfy the stable conditions that we deduced in the article and obtain the stable phase diagrams of the equilibrium points E1=(K,0) and E∗=(x∗,y∗). Second, we try to change the value of parameter A, the timing diagram, phase diagram, and bifurcation diagram corresponding to each parameter drawn. Further verification of our conclusion is as A increases, the model (2) will produce bifurcation. We also carry out a numerical model of the model (3) and discuss our numerical results in three groups: at the beginning, let τ1 change, τ2=0; we observe the timing diagram corresponding to the model (3) as τ1 increases. In the change of the phase diagram, we find that with the increase of τ1 the coexistence equilibrium point of the model (3) begins to stabilize and becomes unstable and is also accompanied by the generation of the limit cycle (periodic solution); we also change the τ2, τ1=0; we observe the increase of τ2, corresponding to the change of the phase diagram of the time series of the model (3); we find that with the increase of τ2 the coexistence equilibrium point of the model (3) begins to stabilize and becomes unstable. At the same time, it is accompanied by the generation of the limit cycle (periodic solution); the same we let τ1=τ2=τ, so that τ increases; we also found similar changes. In this process, we also found an interesting phenomenon. When τ1 takes a small value, the stability of the coexistence equilibrium point of the model changes. However, τ2 requires a larger value than τ1. When τ1=τ2=τ, τ only needs to take a small value, and the stability of the coexistence equilibrium point of the model changes. From a biological point of view, we find that the introduction of weak Allee effect will change the stability of the model; that is to say, the stable state between populations will be broken. Similarly, delays can destroy the stability of the original predator-prey model. Moreover, the introduction of delay and weak Allee effect makes the model closer to reality and makes us more accurately understand the dynamic changes of interspecific relationships. Therefore, we can find that Allee effect and delay play an important role in describing population dynamics.



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