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Interactions between the predators and the preys are diverse and complex in ecology. The predators increase the mortality rate of preys by direct predation. In existing literatures, many predator-prey models only involve direct predation for predator-prey interactions [1,2,3,4]. However, besides the population loss caused by direct predation, the prey will modify their behavior, psychology and physiology in response to the predation risk. This is defined as the fear effect by Cannon [5]. Zanette et al. [6] investigated the variation of song sparrows offspring reproduction when the sounds and calls of predators were broadcasted to simulate predation risk. They discovered that the fear effect alone can cause a significant reduction of song sparrows offspring reproduction. Inspired by this experimental result, Wang et al. [7] incorporated the fear effect into the predator-prey model. In their theoretical analysis, linear and Holling type Ⅱ functional response are chosen respectively. According to their results, the structure of the equilibria will not be affected by the fear effects, but the stability of equilibria and Hopf bifurcation are slightly different from models with no fear effects.
Soon afterward, Wang and Zou [8] considered a model with the stage structure of prey (juvenile prey and adult prey) and a maturation time delay. Additionally, Wang and Zou [9] pointed out anti-predation behaviors will not only decrease offspring reproduction of prey but also increase the difficulty of the prey being caught. Based on this assumption, they derived an anti-predation strategic predator-prey model.
Recently, the aforementioned ordinary or delayed differential equation models were extended to reaction diffusion equation [10,11,12]. Wang et al. [11] used several functional responses to study the effect of the degree of prey sensitivity to predation risk on pattern formation. Following this work, Wang et al. [10] introduced spatial memory delay and pregnancy delay into the model. Their numerical simulation presented the effect of some biological important variables, including the level of fear effect, memory-related diffusion, time delay induced by spatial memory and pregnancy on pattern formation. Moreover, Dai and Sun [13] incorporated chemotaxis and fear effect into predator-prey model, and investigated the Turing-Hopf bifurcation by selecting delay and chemotaxis coefficient as two analysis parameters.
Denote by u1(x,t) and u2(x,t), the population of the prey and adult predator at location x and time t, respectively. We suppose juvenile predators are unable to prey on. By choosing simplest linear functional response in the model of Wang and Zou [7], the equation for prey population is given by
∂tu1−d1Δu1=u1(rg(K,u2)−μ1−mu1)−pu1u2, |
where d1 is a diffusion coefficient for prey, μ1 is the mortality rate for prey, m is the intraspecies competition coefficient, p is predation rate, r is reproduction rate for prey and g(K,u2) represents the cost of anti-predator defense induced by fear with K reflecting response level. We assume g(K,u2) satisfies the following conditions.
(H1) g(0,u2)=g(K,0)=1, limu2→∞g(K,u2)=limK→∞g(K,u2)=0, ∂g(K,u2)∂K<0 and ∂g(K,u2)∂u2<0.
Considering the maturation period of the predator, we set b(x,t,a1) be the density of the predator at age a1, location x and time t. Establish the following population model with spatial diffusion and age-structure
(∂a1+∂t)b(x,t,a1)=d(a1)Δb(x,t,a1)−μ(a1)b(x,t,a1),b(x,t,0)=cpu1(x,t)u2(x,t), | (1.1) |
where x∈Ω, a bounded spatial habitat with the smooth boundary ∂Ω, t,a1>0, c is the conversion rate of the prey to predators, τ>0 be the maturation period of predator and age-specific functions
d(a1)={d0, a1≤τ,d2, a1>τ, μ(a1)={γ, a1≤τ,μ2, a1>τ, |
represent the diffusion rate and mortality rate at age a1, respectively. We introduce the total population of the matured predator as u2=∫∞τb(x,t,a1)da1. Thus, (1.1) together with b(x,t,∞)=0 yields
∂tu2=d2Δu2+b(x,t,τ)−μ2u2. | (1.2) |
Let s1=t−a1 and w(x,t,s1)=b(x,t,t−s1). Along the characteristic line, solving (1.1) yields
∂tw(x,t,s1)={d0Δw(x,t,s1)−γw(x,t,s1), x∈Ω, 0≤t−s1≤τ,d2Δw(x,t,s1)−μ2w(x,t,s1), x∈Ω, t−s1>τ,w(x,s1,s1)=b(x,s1,0)=cpu1(x,s1)u2(x,s1), s1≥0; w(x,0,s1)=b(x,0,−s1), s1<0. |
Assume linear operator d0Δ−γ with Neumann boundary conditions yields the C0 semigroups T1(t). Therefore,
b(x,t,τ)=w(x,t,t−τ)={T1(τ)b(⋅,t−τ,0), t>τ,T1(t)b(⋅,0,τ−t), t≤τ. |
In particular, G(x,y,t) denotes the kernel function corresponding to T1(t). Thus
b(x,t,τ)=T1(τ)b(⋅,t−τ,0)=cp∫ΩG(x,y,τ)u1(y,t−τ)u2(y,t−τ)dy, t>τ. |
The above equation together with (1.2) yields a nonlocal diffusive predator-prey model with fear effect and maturation period of predators
∂u1∂t=d1Δu1+u1(x,t)[rg(K,u2(x,t))−μ1−mu1(x,t)]−pu1(x,t)u2(x,t),∂u2∂t=d2Δu2+cp∫ΩG(x,y,τ)u1(y,t−τ)u2(y,t−τ)dy−μ2u2(x,t). | (1.3) |
Since the spatial movement of mature predators is much bigger than that of juvenile predators, we assume that diffusion rate of juvenile predators d0 approaches zero. Hence, the kernel function becomes G=e−γτf(x−y) with a Dirac-delta function f. Thus, the equation of u2(x,t) in (1.3) becomes
∂u2∂t=d2Δu2+cpe−γτ∫Ωf(x−y)u1(y,t−τ)u2(y,t−τ)dy−μ2u2(x,t). |
It follows from the properties of Dirac-delta function that
∫Ωf(x−y)u1(y,t−τ)u2(y,t−τ)dy=limϵ→0∫Bϵ(x)f(x−y)u1(y,t−τ)u2(y,t−τ)dy=u1(x,t−τ)u2(x,t−τ)limϵ→0∫Bϵ(x)f(x−y)dy=u1(x,t−τ)u2(x,t−τ) |
where Bϵ(x) is the open ball of radius ϵ centered at x. Therefore, model (1.3) equipped with nonnegative initial conditions and Neumann boundary conditions is
∂u1(x,t)∂t=d1Δu1(x,t)+u1(x,t)[rg(K,u2(x,t))−μ1−mu1(x,t)]−pu1(x,t)u2(x,t), x∈Ω,t>0,∂u2(x,t)∂t=d2Δu2(x,t)+cpe−γτu1(x,t−τ)u2(x,t−τ)−μ2u2(x,t), x∈Ω,t>0,∂u1(x,t)∂ν=∂u2(x,t)∂ν=0,x∈∂Ω,t>0,u1(x,ϑ)=u10(x,ϑ)≥0,u2(x,ϑ)=u20(x,ϑ)≥0,x∈Ω, ϑ∈[−τ,0]. | (1.4) |
If r≤μ1, then as t→∞, we have (u1,u2)→(0,0) for x∈Ω, namely, two species will extinct. Throughout this paper, suppose r>μ1, which ensures that the prey and predator will persist.
This paper is organized as follows. We present results on well-posedness and uniform persistence of solutions and prove the global asymptotic stability of predator free equilibrium in Section 2. The nonexistence of nonhomogeneous steady state and steady state bifurcation are proven in Section 3. Hopf bifurcation analysis is carried out in Section 4. In Section 5, we conduct numerical exploration to illustrate some theoretical conclusions and further explore the dynamics of the nonlocal model numerically. We sum up our paper in Section 6.
Denote by C:=C([−τ,0],X2) the Banach space of continuous maps from [−τ,0] to X2 equipped with supremum norm, where X=L2(Ω) is the Hilbert space of integrable function with the usual inner product. C+ is the nonnegative cone of C. Let u1(x,t) and u2(x,t) be a pair of continuous function on Ω×[−τ,∞) and (u1t,u2t)∈C as (u1t(ϑ),u2t(ϑ))=(u1(⋅,t+ϑ),u2(⋅,t+ϑ)) for ϑ∈[−τ,0]. By using [14,Corollary 4], we can prove model (1.4) exists a unique solution. Note (1.4) is mixed quasi-monotone [15], together with comparison principle implies that the solution of (1.4) is nonnegative.
Lemma 2.1. For any initial condition (u10(x,ϑ),u20(x,ϑ))∈C+, model (1.4) possesses a unique solution (u1(x,t),u2(x,t)) on the maximal interval of existence [0,tmax). If tmax<∞, then lim supt→t−max(‖u1(⋅,t)‖+‖u2(⋅,t)‖)=∞. Moreover, u1 and u2 are nonnegative for all (x,t)∈¯Ω×[−τ,tmax).
We next prove u1 and u2 are bounded which implies that tmax=∞.
Theorem 2.2. For any initial condition φ=(u10(x,ϑ),u20(x,ϑ))∈C+, model (1.4) possesses a global solution (u1(x,t),u2(x,t)) which is unique and nonnegative for (x,t)∈¯Ω×[0,∞). If u10(x,ϑ)≥0(≢0),u20(x,ϑ)≥0(≢0), then this solution remains positive for all (x,t)∈¯Ω×(0,∞). Moreover, there exists a positive constant ξ independent of φ such that lim supt→∞u1≤ξ, lim supt→∞u2≤ξ for all x∈Ω.
Proof. Consider
∂w1∂t=d1Δw1+rw1−μ1w1−mw21, x∈Ω,t>0,∂w1∂ν=0, x∈∂Ω,t>0, w1(x,0)=supϑ∈[−τ,0]u10(x,ϑ), x∈Ω. | (2.1) |
Clearly, w1(x,t) of (2.1) is a upper solution to (1.4) due to ∂u1/∂t≤d1Δu1+ru1−μ1u1−mu21. By using Lemma 2.2 in [16], (r−μ1)/m of (2.1) is globally asymptotically stable in C(¯Ω,R+). This together with comparison theorem indicates
lim supt→∞u1≤limt→∞w1=r−μ1m uniformly for x∈¯Ω. | (2.2) |
Thus, there exists ˜ξ>0 which is not dependent on initial condition, such that ‖u1‖≤˜ξ for all t>0. T2(t) denotes the C0 semigroups yielded by d2Δ−μ2 with the Neumann boundary condition. Then from (1.4),
u2=T2(t)u20(⋅,0)+cpe−γτ∫t−τ−τT2(t−τ−a)u1(⋅,a)u2(⋅,a)da. |
Let −δ<0 be the principle eigenvalue of d2Δ−μ2 with the Neumann boundary condition. Then, ‖T2(t)‖≤e−δt. The above formula yields
‖u2(⋅,t)‖≤e−δt‖u20(⋅,0)‖+cpe−γτ˜ξ∫t−τ−τe−δ(t−τ−a)‖u2(⋅,a)‖da,≤˜B1+∫t0˜B2‖u2(⋅,a)‖da, |
by choosing constants ˜B1≥‖u2(⋅,0)‖+cpτ˜ξsupa∈[−τ,0]‖u2(⋅,a)‖ and ˜B2≥cp˜ξ. Using Gronwall's inequality yields ‖u2(⋅,t)‖≤˜B1e˜B2t for all 0≤t<tmax. Lemma 2.1 implies tmax=∞. So (u1,u2) is a global solution. Moreover, if u10(x,ϑ)≥0(≢0),u20(x,ϑ)≥0(≢0), then by [17,Theorem 4], this solution is positive for all t>0 and x∈¯Ω.
We next prove (u1,u2) is ultimately bounded by a constant which is not dependent on the initial condition. Due to (2.2), there exist t0>0 and ξ0>0 such that u1(x,t)≤ξ0 for any t>t0 and x∈¯Ω. Let z(x,t)=cu1(x,t−τ)+u2(x,t), μ=min{μ1,μ2} and I1=∫Ωz(x,t)dx. We integrate both sides of (1.4) and add up to obtain
I′1(t)≤∫Ω(c(r−μ1)u1(x,t−τ)−μ2u2(x,t))dx≤crξ0|Ω|−μI1(t), t≥t0+τ. |
Comparison principle implies
lim supt→∞‖u2‖1≤lim supt→∞I1≤crξ0|Ω|/μ. |
Especially, there exist t1>t0 and ξ1>0 such that ‖u2‖1≤ξ1 for all t≥t1.
Now, we define Vl(t)=∫Ω(u2(x,t))ldx with l≥1, and estimate the upper bound of V2(t). For t>t1, the second equation of (1.4) and Young's inequality yield
12V′2(t)≤−d2∫Ω|∇u2|2dx+cpξ0∫Ωu2(x,t−τ)u2(x,t)dx−μ2V2(t)≤−d2‖∇u2‖22+cpξ02V2(t−τ)+cpξ02V2(t). |
The Gagliardo-Nirenberg inequality states:
∀ ϵ>0,∃ ˆc>0, s.t. ‖P‖22≤ϵ‖∇P‖22+ˆcϵ−n/2‖P‖21, ∀ P∈W1,2(Ω). |
We obtain
V′2(t)≤C1+C2V2(t−τ)−(C2+C3)V2(t), |
where C1=ˆcϵ−n/2−12d2ξ21>0, C2=cpξ0>0 and C3=2(d2/ϵ−C2)>0 with small ϵ∈(0,d2/C2). Using comparison principle again yields lim supt→∞V2(t)≤C1/C3, which implies there exist t2>t1 and ξ2>0 such that V2≤ξ2 for t≥t2.
Let Ll=lim supt→∞Vl(t) with l≥1, we want to estimate L2l with the similar method of estimation for L2. Multiply the second equation in (1.4) by 2lv2l−1 and integrate on Ω. Young's inequality implies
V′2l(t)≤−2d2∫Ω|∇ul2|2dx+cpξ0V2l(t−τ)+(2l−1)cpξ0V2l(t). |
Then
V′2l(t)≤2d2ˆcϵ−n/2−1V2l(t)−2d2ϵV2l(t)+cpξ0(V2l(t−τ)+(2l−1)V2l(t)), |
via Gagliardo-Nirenberg inequality. Since Ll=lim supt→∞Vl(t), there exists tl>0 such that Vl≤1+Ll when t>tl. Hence,
V′2l(t)≤2d2ˆcϵ−(n/2+1)(1+Ll)2−2d2ϵV2l(t)+lC4(V2l(t)+V2l(t−τ)) |
with C4=2cpξ0. We choose ϵ−1=(2C4+1)l/(2d2) and C5=2d2ˆc[(2C4+1)/(2d2)]n/2+1. Then for t>tl, we obtain
V′2l(t)≤C5ln/2+1(1+Ll)2+lC4V2l(t−τ)−(lC4+l)V2l(t). |
By comparison principle, the above inequality yields L2l≤C5ln/2(1+Ll)2, with a constant C5 which is not dependent on l and initial conditions. Finally, prove L2s<∞ for all s∈N0. Let B=1+C5 and {bs}∞s=0 be an infinite sequence denoted by bs+1=B(1/2)(s+1)2sn((1/2)(s+2))bs with the first term b0=L1+1. Clearly, L2s≤(bs)2s and
lims→∞lnbs=lnb0+lnB+n2ln2. |
Therefore,
lim sups→∞(L2s)(1/2)s≤lims→∞bs=B(1+L1)2n/2≤B(1+ξ1)2n/2≤ξ:=max{B(ξ1+1)2n/2,(r−μ1)/m}. |
The above inequality leads to lim supt→∞u1≤ξ and lim supt→∞u2≤ξ for all x∈Ω.
In Theorem 2.2, we proved that the solution of model (1.4) is uniformly bounded for any nonnegative initial condition, this implies the boundedness of the population of two species. Clearly model (1.4) exists two constant steady states E0=(0,0) and E1=((r−μ1)/m,0), where E0 is a saddle. Define the basic reproduction ratio [18] by
R0=cpe−γτ(r−μ1)mμ2. | (2.3) |
Thus model (1.4) possesses exactly one positive constant steady state E2=(u∗1,u∗2) if and only if R0>1, which is equivalent to cp(r−μ1)>mμ2 and 0≤τ<τmax:=1γlncp(r−μ1)mμ2. Here,
u∗1=μ2eγτpc, u∗2 satisfies rg(K,u2)−pu2=μ1+mu∗1. |
The linearization of (1.4) at the positive constant steady state (˜u1,˜u2) gives
∂W/∂t=DΔW+L(Wt), | (2.4) |
where domain Y:={(u1,u2)T:u1,u2∈C2(Ω)∩C1(ˉΩ),(u1)ν=(u2)ν=0 on ∂Ω}, W=(u1(x,t),u2(x,t))T, D=diag(d1,d2) and a bounded linear operator L:C→X2 is
L(φ)=Mφ(0)+Mτφ(−τ), for φ∈C, |
with
M=(rg(K,˜u2)−μ1−2m˜u1−p˜u2˜u1(rg′u2(K,˜u2)−p)0−μ2), Mτ=(00cpe−γτ˜u2cpe−γτ˜u1). |
The characteristic equation of (2.4) gives
ρη−DΔη−L(eρ⋅η)=0, for some η∈Y∖{0}, |
or equivalently
det(ρI+σnD−M−e−ρτMτ)=0, for n∈N0. | (2.5) |
Here, σn is the eigenvalue of −Δ in Ω with Neumann boundary condition with respect to eigenfunction ψn for all n∈N0, and
0=σ0<σ1≤σ2≤⋯≤σn≤σn+1≤⋯ and limn→∞σn=∞. | (2.6) |
Theorem 2.3. (i) The trivial constant steady state E0=(0,0) is always unstable.
(ii) If R0>1, then E1=((r−μ1)/m,0) is unstable, and model (1.4) possesses a unique positive constant steady state E2=(u∗1,u∗2).
(iii) If R0≤1, then E1 is globally asymptotically stable in C+.
Proof. (ⅰ) Note, (2.5) at E0 takes form as (ρ+σnd1−r+μ1)(ρ+σnd2+μ2)=0 for all n∈N0. Then r−μ1>0 is a positive real eigenvalue, namely, E0 is always unstable.
(ⅱ) The characteristic equation at E1 gives
(ρ+σnd1+r−μ1)(ρ+σnd2+μ2−cpe−γτr−μ1me−ρτ)=0 for n∈N0. | (2.7) |
Note that one eigenvalue ρ1=−σnd1−r+μ1 remains negative. Hence, we only need to consider the root distribution of the following equation
ρ+σnd2+μ2−cpe−γτr−μ1me−ρτ=0 for n∈N0. | (2.8) |
According to [19,Lemma 2.1], we obtain that (2.8) exists an eigenvalue ρ>0 when R0>1, namely, E1 is unstable when R0>1.
(ⅲ) By using [19,Lemma 2.1] again, any eigenvalue ρ of (2.8) satisfies Re(ρ)<0 when R0<1, namely, E1 is locally asymptotically stable when R0<1. Now, consider R0=1. 0 is an eigenvalue of (2.7) for n=0 and all other eigenvalues satisfy Re(ρ)<0. To prove the stability of E1, we shall calculate the normal forms of (1.4) by the algorithm introduced in [20]. Set
Υ={ρ∈C,ρ is the eigenvalue of equation (2.7) and Reρ=0}. |
Obviously, Υ={0} when R0=1. System (1.4) satisfies the non-resonance condition relative to Υ. Denote ¯u1=(r−μ1)/m, and let w=(w1,w2)T=(¯u1−u1,u2)T and (1.4) can be written as
˙wt=A0wt+F0(wt) on C. |
Here linear operator A0 is given by (A0φ)(ϑ)=(φ(ϑ))′ when ϑ∈[−τ,0) and
(A0φ)(0)=(d1Δ00d2Δ)φ(0)+(−r+μ1(p−rg′u2(K,0))¯u10−μ2)φ(0)+(000μ2)φ(−τ), |
and the nonlinear operator F0 satisfies [F0(φ)](ϑ)=0 for −τ≤ϑ<0. By Taylor expansion, [F0(φ)](0) can be written as
[F0(φ)](0)=(mφ21(0)−r¯u1g″u2(K,0)φ22(0)/2+(rg′u2(K,0)−p)φ1(0)φ2(0)−cpe−γτφ1(−τ)φ2(−τ))+h.o.t. | (2.9) |
Define a bilinear form
⟨β,α⟩=∫Ω[α1(0)β1(0)+α2(0)β2(0)+μ2∫0−τβ2(ϑ+τ)α2(ϑ)dϑ]dx, β∈C([0,τ],X2), α∈C. |
Select α=(1,m/(p−rg′u2(K,0)) and β=(0,1)T to be the right and left eigenfunction of A0 relative to eigenvalue 0, respectively. Decompose wt as wt=hα+δ and ⟨β,δ⟩=0. Notice A0α=0 and ⟨β,A0δ⟩=0. Thus,
⟨β,˙wt⟩=⟨β,A0wt⟩+⟨β,F0(wt)⟩=⟨β,F0(wt)⟩. |
Moreover,
⟨β,˙wt⟩=˙h⟨β,α⟩+⟨β,˙δ⟩=˙h⟨β,α⟩. |
It follows from the above two equations that
˙hm(1+μ2τ)|Ω|p−rg′u2(K,0)=⟨β,F0(hα+δ)⟩=∫ΩβT[F0(hα+δ)](0)dx=∫Ω[F0(hα+δ)]2(0)dx. |
When the initial value is a small perturbation of E1, then δ=O(h2), together with Taylor expansion yields
[F0(hα+δ)]2(0)=−cpe−γτ(hα1(−τ)+δ1(−τ))(hα2(−τ)+δ2(−τ))=−cpe−γτmp−rg′u2(K,0)h2+O(h3). |
Therefore, we obtain the norm form of (1.4) as follows
˙h=−cpe−γτ1+μ2τh2+O(h3). | (2.10) |
Then for any positive initial value, the stability of zero solution of (2.10) implies E1 is locally asymptotically stable when R0=1.
Next, it suffices to show the global attractivity of E1 in C+ when R0≤1. Establish a Lyapunov functional V:C+→R as
V(ϕ1,ϕ2)=∫Ωϕ2(0)2dx+cpe−γτr−μ1m∫Ω∫0−τϕ2(θ)2dθdx for (ϕ1,ϕ2)∈C+. |
Along solutions of (1.4), taking derivative of V(ϕ1,ϕ2) with respect to t yields
dVdt≤−2d2∫Ω|∇u2|2dx+∫Ω2cpe−γτr−μ1mu2(x,t−τ)u2(x,t)−2μ2u22(x,t)+cpe−γτr−μ1m[u22(x,t)−u22(x,t−τ)]dx≤∫Ω2μ2(R0−1)u22(x,t)dx≤0 if R0≤1. |
Note {E1} is the maximal invariant subset of dV/dt=0, together with LaSalle-Lyapunov invariance principle [21,22] implies E1 is globally asymptotically stable if R0≤1.
In Theorem 2.3, E0 is always unstable which suggests that at leat one species will persist eventually. Moreover, if R0≤1, then E1 is globally asymptotically stable in C+, which implies that when the basic reproduction ratio is no more than one, the predator species will extinct and only the prey species can persist eventually. Next, we will prove the solution is uniformly persistent. Θt denotes the solution semiflow of (1.4) mapping C+ to C+; namely, Θtφ:=(u1(⋅,t+⋅),u2(⋅,t+⋅))∈C+. Set ζ+(φ)=∪t≥0{Θtφ} be the positive orbit and ϖ(φ) be the omega limit set of ζ+(φ). Denote
Z:={(φ1,φ2)∈C+:φ1≢0 and φ2≢0}, ∂Z:=C+∖Z={(φ1,φ2)∈C+:φ1≡0 or φ2≡0}, |
Γ∂ as the largest positively invariant set in ∂Z, and Ws((˜u1,˜u2)) as the stable manifold associated with (˜u1,˜u2). We next present persistence result of model (1.4).
Theorem 2.4. Suppose R0>1. Then there exists κ>0 such that lim inft→∞u1(x,t)≥κ and lim inft→∞u2(x,t)≥κ for any initial condition φ∈Z and x∈¯Ω.
Proof. Note Θt is compact, and Theorem 2.2 implies Θt is point dissipative. Then Θt possesses a nonempty global attractor in C+ [23]. Clearly, Γ∂={(φ1,φ2)∈C+:φ2≡0}, and ϖ(φ)={E0,E1} for all φ∈Γ∂. Define a generalized distance function ψ mapping C+ to R+ by
ψ(φ)=minx∈ˉΩ{φ1(x,0),φ2(x,0)}, ∀φ=(φ1,φ2)∈C+. |
Following from strong maximum principle [24], ψ(Θtφ)>0 for all φ∈Z. Due to ψ−1(0,∞)⊂Z, assumption (P) in [25,Section 3] holds. Then verify rest conditions in [25,Theorem 3].
First, prove Ws(E0)∩ψ−1(0,∞)=∅. Otherwise, there exists an initial condition φ∈C+ with ψ(φ)>0, such that (u1,u2)→E0 as t→∞. Thus, for any sufficiently small ε1>0 satisfying rg(K,ε1)−μ1>pε1, there exists t1>0 such that 0<u1,u2<ε1 for all x∈Ω and t>t1. Note that rg(K,0)−μ1>0 and ∂g(K,u2)/∂u2<0 ensure the existence of small ε1>0. Then the first equation in (1.4) and (H1) lead to
∂tu1>d1Δu1+u1[(rg(K,ϵ1)−μ1−pϵ1)−mu1],t>t1. |
Notice
∂tˆu1−d1Δˆu1=ˆu1[(rg(K,ϵ1)−pϵ1−μ1)−mˆu1], x∈Ω,t>t1,∂νˆu1=0, x∈∂Ω,t>t1, |
has a globally asymptotically stable positive steady state (rg(K,ϵ1)−μ1−pϵ1)/m due to [16,Lemma 2.2], together with comparison principle yields limt→∞u1≥limt→∞ˆu1>0. A contradiction is derived, so Ws(E0)∩ψ−1(0,∞)=∅.
Next check Ws(E1)∩ψ−1(0,∞)=∅. If not, there exists φ∈C+ with ψ(φ)>0 such that (u1,u2) converges to E1 as t→∞. According to (2.2), for any small ε2>0 satisfying cpe−γτ((r−μ1)/m−ϵ2)>μ2, there exists t2>0 such that u1>(r−μ1)/m−ε2 for all x∈¯Ω and t>t2−τ. Note that R0>1 ensures the existence of ε2>0. Thus, the second equation of (1.4) yields
∂tu2>d2Δu2+cpe−γτ(r−μ1m−ε2)u2(x,t−τ)−μ2u2(x,t), t>t2. |
In a similar manner, we derive limt→∞u2(x,t)>0 by above inequality, cpe−γτ((r−μ1)/m−ϵ2)>μ2 and comparison principle. A contradiction yields again. Hence, it follows from Theorem 3 in [25] that, for any φ∈C+, there exists κ>0 such that lim inft→∞ψ(Θtφ)≥κ uniformly for any x∈¯Ω.
Theorem 2.4 implies that when the basic reproduction ratio is bigger than one, both the predator species and prey species will persist eventually. We next investigate the stability of E2. The corresponding characteristic equation at E2 gives
ρ2+a1,nρ+a0,n+(b1,nρ+b0,n)e−ρτ=0, n∈N0, | (2.11) |
with
a1,n=σn(d1+d2)+μ2+mu∗1>0, a0,n=(σnd1+au∗1)(σnd2+μ2)>0,b1,n=−μ2<0, b0,n=−μ2(σnd1+mu∗1+u∗2(rg′u2(K,u∗2)−p)). |
Characteristic equation (2.11) with τ=0 is
ρ2+(a1,n+b1,n)ρ+a0,n+b0,n=0, n∈N0. | (2.12) |
We observe that a0,n+b0,n=σnd2(σnd1+mu∗1)−μ2u∗2(rg′u2(K,u∗2)−p)>0, and a1,n+b1,n=σn(d1+d2)+mu∗1>0 for all integer n≥0, which yields that any eigenvalue ρ of (2.12) satisfies Re(ρ)<0. Then, local asymptotic stability of E2 is derived when τ=0 which implies Turing instability can not happen for the non-delay system of (1.4). In addition, a0,n+b0,n>0 for any n∈N0 leads to that (2.11) can not have an eigenvalue 0 for any τ≥0. This suggests we look for the existence of simple ρ=±iδ (δ>0) for some τ>0. Substitute ρ=iδ into (2.11) and then
Gn(δ,τ)=δ4+(a21,n−2a0,n−b21,n)δ2+a20,n−b20,n=0, n∈N0, | (2.13) |
with
a21,n−2a0,n−b21,n=(σnd1+mu∗1)2+(σnd2)2+2μ2σnd2>0,a0,n+b0,n=σnd2(σnd1+mu∗1)−μ2u∗2(rg′u2(K,u∗2)−p)>0,a0,n−b0,n=σ2nd1d2+σn(2μ2d1+mu∗1d2)+μ2(2mu∗1+u∗2(rg′u2(K,u∗2)−p)). |
Thus, a0,n−b0,n≥0 for all n∈N0 is equivalent to
(A0): 2mu∗1≥u∗2(p−rg′u2(K,u∗2)). |
If (A0) holds, then (2.13) admits no positive roots, together with for τ=0, any eigenvalues ρ of (2.11) satisfies Re(ρ)<0, yields the next conclusion.
Theorem 2.5. Suppose R0>1. Then, E2 is locally asymptotically stable provided that (A0) holds.
Now, we consider positive nonhomogeneous steady states. The steady state (u1(x),u2(x)) of (1.4) satisfies the elliptic equation
−d1Δu1=rg(K,u2)u1−μ1u1−mu21−pu1u2, x∈Ω,−d2Δu2=cpe−γτu1u2−μ2u2, x∈Ω,∂νu1=∂νu2=0, x∈∂Ω. | (3.1) |
From Theorem 2.3, all the solutions converge to E1 when R0≤1 and the positive nonhomogeneous steady state may exist only if R0>1. Throughout this section, we assume that R0>1. In what follows, the positive lower and upper bounds independent of steady states for all positive solutions to (3.1) are derived.
Theorem 3.1. Assume that R0>1. Then any nonnegative steady state of (3.1) other than (0,0), and ((r−μ1)/m,0) should be positive. Moreover, there exist constants ¯B,B_>0 which depend on all parameters of (3.1) and Ω, such that B_≤u1(x),u2(x)≤¯B for any positive solution of (3.1) and x∈¯Ω.
Proof. We first show any nonnegative solution (u1,u2) other than E0 and E1, should be u1>0 and u2>0 for all x∈¯Ω. To see this, suppose u2(x0)=0 for some x0∈¯Ω, then u2(x)≡0 via strong maximum principle and
0≤d1∫Ω|∇(u1−r−μ1m)|2dx=∫Ω−mu1(x)(u1(x)−r−μ1m)2dx≤0. |
Thus the above inequality implies u1(x)≡0 or u1(x)≡(r−μ1)/m. Now, we assume u2>0 for all x∈¯Ω. Strong maximum principle yields u1>0 for all x∈¯Ω. Hence, u1>0 and u2>0 for all x∈¯Ω.
We now prove u1 and u2 have a upper bound which is a positive constant. Since −d1Δu1(x)≤(r−μ1−mu1(x))u1(x), we then obtain from Lemma 2.3 in [26] that u1(x)≤(r−μ1)/m for any x∈¯Ω.
By two equations in (3.1), we obtain
−Δ(d1cu1+d2u2)≤rc(r−μ1)/m−min{μ1d1,μ2d2}(d1cu1+d2u2). |
By using [26,Lemma 2.3] again, we conclude
u1(x),u2(x)≤¯B=rc(r−μ1)mmin{cμ1,μ2,μ1d2/d1,μ2d1c/d2}. |
Next, we only need to prove ‖u1(x)‖ and ‖u2(x)‖ have a positive lower bound which is not dependent on the solution. Otherwise, there exists a positive steady states sequence (u1,n(x),u2,n(x)) such that either limn→∞‖u1,n‖∞=0 or limn→∞‖u2,n‖∞=0. Integrating second equation of (3.1) gives
0=∫Ωu2,n(cpe−γτu1,n−μ2)dx. | (3.2) |
If ‖u1,n(x)‖∞→0 as n→∞, then cpe−γτu1,n(x)−μ2<−μ2/2 for sufficiently large n, which yields u2,n(cpe−γτu1,n−μ2)<0, a contradiction derived. Thus, ‖u2,n(x)‖∞→0 as n→∞ holds. We then assume that (u1,n,u2,n)→(u1,∞,0), as n→∞ where u1,∞≥0. Similarly, we obtain that either u1,∞≡0 or u1,∞≡(r−μ1)/m. Obviously, u1,∞≢0 based on the above argument, thus u1,∞≡(r−μ1)/m and limn→∞cpe−γτu1,n(x)−μ2=μ2(R0−1)>0. This again contradicts (3.2). Hence, we have shown ‖u1(x)‖∞ and ‖u2(x)‖∞ have a positive constant lower bound independent on the solution. Therefore, u1(x) and u2(x) have a uniform positive constant lower bound independent on the solution of (3.1) via Harnack's inequality [26,Lemma 2.2]. This ends the proof.
Theorem 3.2. Suppose R0>1. There exists a constant χ>0 depending on r,μ1,p,c,γ,τ,μ2,g and σ1, such that if min{d1,d2}>χ then model (1.4) admits no positive spatially nonhomogeneous steady states, where σ1 is defined in (2.6).
Proof. Denote the averages of the positive solution (u1,u2) of system (3.1) on Ω by
~u1=∫Ωu1(x)dx|Ω| and ~u2=∫Ωu2(x)dx|Ω|. |
By (2.2), we have u1(x)≤¯u1, where ¯u1=(r−μ1)/m, which implies ~u1≤¯u1. Multiplying the first equation by ce−γτ and adding two equations of (3.1) lead to
−(d1ce−γτΔu1+d2Δu2)=ce−γτ(rg(K,u2)u1−μ1u1−mu21)−μ2u2. |
The integration of both sides for the above equation yields
~u2=ce−γτμ2|Ω|∫Ω(rg(K,u2)u1−μ1u1−mu21)dx≤(r−μ1)¯u1ce−γτμ2:=Mv. |
It is readily seen that ∫Ω(u1−~u1)dx=∫Ω(u2−~u2)dx=0. Note u1 and u2 are bounded by two constants ¯u1>0 and ¯u2:=rc¯u1/min{μ1d2/d1,μ2}>0 by Theorem 3.1. Denote Mf=maxu2∈[0,¯u2]|g′u2(K,u2)| and we then obtain
d1∫Ω|∇(u1−~u1)|2dx=∫Ω(u1−~u1)(rg(K,u2)u1−μ1u1−mu21)dx−∫Ωpu1u2(u1−~u1)dx=∫Ω(u1−~u1)(rg(K,u2)u1−μ1u1−mu21−(rg(K,~u2)~u1−μ1~u1−m~u12))dx+∫Ωp(~u1~u2−u1u2)(u1−~u1)dx≤(r−μ1+(rMf+p)¯u12)∫Ω(u1−~u1)2dx+(rMf+p)¯u12∫Ω(u2−~u2)2dx, |
d2∫Ω|∇(u2−~u2)|2dx=∫Ω(cpe−γτu1u2−μ2u2)(u2−~u2)dx=cpe−γτ∫Ω(u1u2−~u1~u2)(u2−~u2)dx−∫Ωμ2(u2−~u2)2dx≤cpe−γτ∫ΩMv2(u1−~u1)2dx+cpe−γτ(¯u1+Mv2)∫Ω(u2−~u2)2dx. |
Set A1=(r−μ1)+((rMf+p)¯u1+cpe−γτMv)/2, and A2=((rMf+p)¯u1+cpe−γτ(2¯u1+Mv))/2. Then, the above inequalities and Poincareˊ inequality yield that
with a positive constant depending on and . Hence, if , then , which implies is a constant solution.
Select as the bifurcation parameter and study nonhomogeneous steady state bifurcating from . Let satisfy , , and . Drop . System (3.1) becomes
for with . Calculating Frchet derivative of gives
Then the characteristic equation follows
(3.3) |
where
Obviously, and for all and . Therefore, (3.3) does not have a simple zero eigenvalue. According to [4], we obtain the nonexistence of steady state bifurcation bifurcating at .
Theorem 3.3. Model (1.4) admits no positive nonhomogeneous steady states bifurcating from .
Next, the stability switches at and existence of periodic solutions of (1.4) bifurcating from are studied. Suppose , namely, and to guarantee the existence of .
Recall the stability of for is proved and is not the root of (2.11) for . So, we only consider eigenvalues cross the imaginary axis to the right which corresponds to the stability changes of . Now, we shall consider the positive root of . Clearly, there exists exactly one positive root of if and only if for . More specifically,
is the sufficient and necessary condition to ensure has exactly one positive zero. For some integer , the assumption is a necessary condition to guarantee exists positive zeros. Set
(4.1) |
Implicit function theorem implies has a unique zero
which is a function for . Hence, is an eigenvalue of (2.11), and satisfies
(4.2) |
for . Let
which is the unique solution of and and satisfies for .
According to [3,27], we arrive at the next properties.
Lemma 4.1. Suppose that and holds.
(i) There exists a nonnegative integer such that for , with , and for , where is defined in (4.1).
(ii) Define
(4.3) |
Then, ; for and , we have , where ; and .
(iii) For each integer and some , has one positive zero if and only if (2.11) has a pair of eigenvalues . Moreover,
(4.4) |
When , cross the imaginary axis from right to left at ; when , cross the imaginary axis from left to right.
If , then in holds for any and ; or only has a zero with even multiplicity in and for any positive integers and . Therefore, is locally asymptotically stable for . The following assumption ensures Hopf bifurcation may occur at .
and has at most two zeros (counting multiplicity) for integer and .
Note is strictly decreasing in due to Lemma 4.1. It then follows from , and , for any integer and , that we can find two positive integers
(4.5) |
and
(4.6) |
Then admits two simple zeros and for and no zeros for . The above analysis, together with Lemma 4.1(ⅲ), yields the next result.
Lemma 4.2. Suppose and and hold. Let , and be defined in (4.3), (4.5) and (4.6).
(i) For integer , there are simple zeros of , , and and for each .
(ii) If there exist exactly two bifurcation values with and , then the double Hopf bifurcation occurs at when .
Collect all values with and in a set. To ensure Hopf bifurcation occurs, remove values which appear more than once. The new set becomes
(4.7) |
Lemma 4.1(ⅱ) implies exists two simple zeros . When with , the Hopf bifurcation occurs at . Moreover, is locally asymptotically stable for and unstable for . Define
(4.8) |
Theorem 4.3. Suppose . Let , , and be defined in (4.1), (4.3), (4.7) and (4.8), respectively.
(i) is locally asymptotically stable for all provided that either or .
(ii) If and hold, then a Hopf bifurcation occurs at when . is locally asymptotically stable for , and unstable for . Further, for , the bifurcating periodic solution is spatially nonhomogeneous; for , the bifurcating periodic solution is spatially homogeneous.
Next investigate the properties of bifurcating periodic solutions by global Hopf bifurcation theorem [28]. Set and write (1.4) as
(4.9) |
where for , and
denotes the semigroup yielded by in , with Neumann boundary condition. Clearly, . The solution of (4.9) can be denoted by
(4.10) |
If is a periodic solution of (4.9), then (4.10) yields
(4.11) |
since as . Hence we only need to consider (4.11). The integral operator of (4.11) is differentiable, completely continuous, and G-equivariant by [24,Chapter 6.5]. The condition ensures (1.4) admits exactly one positive steady state . Using a similar argument as in [29,Section 4.2], – in [24,Chapter 6.5] hold and we shall study the periodic solution.
Lemma 4.4. Suppose , then all nonnegative periodic solutions of (4.9) satisfies for all , where and are defined in Theorems 2.2 and 2.4, respectively.
Lemma 4.4 can be obtained by Theorems 2.2 and 2.4. We further assume
This technical condition is used to exclude the periodic solutions. Note assumption holds when . This, together with that is continuous in , implies there exists such that holds for , that is, holds for the model (1.4) with weak fear effect.
Lemma 4.5. Assume that and holds, then model (1.4) admits no nontrivial periodic solution.
Proof. Otherwise, let be the nontrivial periodic solution, that is, . Thus, we have
(4.12) |
Claim
To see this, establish the Lyapunov functional ,
Along the solution of system (4.12), the time derivative of is
The assumption () ensures for all . The maximal invariant subset of is . Therefore, attracts all positive solution of (4.12) by LaSalle-Lyapunov invariance principle [21,22] which excludes the nonnegative nontrivial periodic solution.
To obtain the nonexistence of periodic solution for model (1.4), we must use the condition which is very restrictive. However, in numerical simulations, Lemma 4.5 remains true even is violated. Thus, we conjecture the nonexistence of -periodic solution for (1.4).
In the beginning of this section, when with , are a pair of eigenvalues of (2.11). Give the next standard notations:
For , is an isolated singular point.
is a closed subset of .
For , is the connected component of in .
For integer , let
We are ready to present a conclusion on the global Hopf branches by a similar manner in [30,Theorem 4.12].
Theorem 4.6. Assume , and – hold. Then we have the following results.
(i) The global Hopf branch is bounded for with and .
(ii) For any , model (1.4) possesses at least one periodic solution.
(iii) For , with and , we have if .
To verify obtained theoretical results, the numerical simulation is presented in this part. We choose the fear effect function as and let
Figure 1 shows the existence and stability of , and , and Hopf bifurcation curve of model (1.4) in plane. Above the line , does not exist, is globally asymptotically stable and is unstable; Below the line , there exist three constant steady states and . In the region which is bounded by and , no Hopf bifurcation occurs and is stable. In the region which is bounded by and , there exist periodic solutions through Hopf bifurcation bifurcating at
Fix , then by simple calculation, we have , , , , , for , and . Thus, all Hopf bifurcation values and are the all zeros of for integer . We summarize the dynamics of model (1.4) as follows.
For , we obtain is globally asymptotically stable and is unstable, see Figure 2(a).
For , we obtain is locally asymptotically stable, and two constant steady state and are unstable, as shown in Figure 2(b).
For , we obtain and are unstable, a periodic solution bifurcates from , as shown in Figure 2(c). Further, a Hopf bifurcation occurs at when .
Next, we explore the global Hopf branches by choosing and
(5.1) |
As shown in Figure 3, we collect all zeros of for nonnegative in set with , namely,
From Theorem 4.3, is locally asymptotically stable when and unstable when , at least one periodic solution emerges for . Moreover, a spatially homogeneous periodic solution bifurcates from , see Figure 4(a); a spatially nonhomogeneous periodic solution bifurcates from , see Figure 4(b).
In model (1.3), the kernel function takes form as by reasonable assumptions and our theoretical results are derived by choosing as Dirac-delta function. Next, we choose
(5.2) |
Here, is the truncated normal distribution. Clearly, . Let and the parameter values chosen according to (5.1). As shown in Figure 5, when , a stable nonhomogeneous periodic solution emerges; when , a homogeneous periodic solution emerges; when , the positive constant steady state is stable. Numerical simulation suggests nonlocal interaction can produce more complex dynamics.
We formulate an age-structured predator-prey model with fear effect. For , the global asymptotic stability for predator-free constant steady state is proved via Lyapunov-LaSalle invariance principle. For , we prove the nonexistence of spatially nonhomogeneous steady states and exclude steady state bifurcation. Finally, we carry out Hopf bifurcation analyses and prove global Hopf branches are bounded.
Our theoretical results are obtained by choosing a special kernel function in model (1.3). However, in numerical results, we explore rich dynamics when the nonlocal interaction is incorporated into the delayed term. The theoretical results concerning the nonlocal model are left as an open problem.
H. Shu was partially supported by the National Natural Science Foundation of China (No. 11971285), the Fundamental Research Funds for the Central Universities (No. GK202201002), the Natural Science Basic Research Program of Shaanxi (No. 2023-JC-JQ-03), and the Youth Innovation Team of Shaanxi Universities. W. Xu was partially supported by a scholarship from the China Scholarship Council while visiting the University of New Brunswick. P. Jiang was partially supported by the National Natural Science Foundation of China (No. 72274119).
The authors declare no conflicts of interest in this paper.
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