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Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

  • Zaowang Xiao , Zhong Li , Zhenliang Zhu and Fengde Chen EMAIL logo
Published/Copyright: March 26, 2019
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Open Mathematics
From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

Keywords: Hopf bifurcation; Predator-prey; Beddington-DeAngelis; Stage structure; Global stability
MSC 2010: 34D23; 92B05; 34D40

1 Introduction

Predator-prey model is one of the basic models between different species in real world. As we all know, there are always many species going through two stages, immature and mature, which reflect the different characteristics of species at each stage. Therefore, to exhibit the real world phenomenon, stage structure population models are more reasonable than other models. In recent years, numerous papers have considered the predator-prey system with stage structure (see [1, 2, 3, 4, 5]).

On the other hand, in general, the consumption of prey by predator throughout its past history governs the present birth rate of the predator, in other words, time delay due to gestation is a common example. Obviously, delay differential equations exhibit much more complicated dynamics than ordinary differential equations (see [6, 7, 8, 9, 10, 11, 12, 13, 14]). For example, Wang and Chen [15] considered the following predator prey system with stage structure for the predator population:

  x˙(t)=x(t)(r−ax(t−τ1)−μy2(t)),y˙1(t)=ημx(t−τ2)y2(t−τ2)−βy1(t)−dy1(t),y˙2(t)=βy1(t)−ey2(t). (1.1)

The authors studied the asymptotic behavior of system (1.1). When a time delay due to gestation of the predators and a time delay from a crowding effect of the prey are incorporated, we establish conditions for the permanence of the populations and sufficient conditions for the existence of globally stable positive equilibrium of system (1.1).

Beddington [16] and DeAngelis et al. [17] established a famous B-D functional response that is a predator dependent functional response. In many cases, predators need to search for food and share or compete for food. Therefore, the stage-structured predator-prey models incorporating Beddington-DeAngelis functional response better reflect the ecology. Chen et al. [18] discussed the stability of the boundary solution of a nonautonomous predator-prey model with the Beddington-DeAngelis functional response, which reflects the dynamics of interacting predators and prey in a fluctuating environment. Xia et al. [19] considered stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting. Chen et al. [20] discussed the extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances. Khajanchi [21] investigated the dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model:

 x˙(t)=x(t)(r−rkx(t))−μx(t)y2(t)1+bx(t)+cy2(t),y˙1(t)=ημx(t)y2(t)1+bx(t)+cy2(t)−βy1(t)−dy1(t),y˙2(t)=βy1(t)−ey2(t). (1.2)

By analyzing the above system, conditions for positivity, boundedness, uniform persistence, existence of positive equilibria with their local stability have been established. Also, the author showed the existence of Hopf bifurcation when the conversion parameter k1 passes the critical value. Finally, the conditions for the occurrence of global stability for the unique interior equilibrium point were derived.

In the real world, refuge is a strategy to reduce the risk of predation. It is clear that the existence of refuge can have a significant impact on the coexistence of predator species and prey species. In recent years, many papers [22, 23, 24, 25, 26, 27] have proposed and analyzed predator-prey models incorporating prey refuges. Recently, Wei and Fu [28] discussed the Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refuge

x˙1(t)=ax2(t)−rx1(t)−bx1(t),x˙2(t)=bx1(t)−cx2(t)−αx22(t)−β(1−m)x2(t)y(t)a1+b1(1−m)x2(t)+c1y(t), y˙(t)=dβ(1−m)x2(t−τ)y(t−τ)a1+b1(1−m)x2(t−τ)+c1y(t−τ)−γy(t). (1.3)

By using the characteristic equations, the local stability of each feasible equilibrium of model (1.3) was discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium was established.

Motivated by the works [21] and [28], a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge is investigated in this paper. The proposed model is as follows:

 x˙(t)=x(t)(r−ax(t))−μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t),y˙1(t)=ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−βy1(t)−dy1(t),y˙2(t)=βy1(t)−ey2(t), (1.4)

where x(t), y1(t) and y2(t) denote the densities of prey species, immature predator species and mature predator species at time t, respectively; r is the intrinsic growth rate of prey species; a is the intraspecific competition rate of prey species; d and e are the death rates of the immature and mature predator species respectively; μ(1 – m) is the capturing rate of the mature predator; η is the conversion rate of nutrients into the production of predator species; τ denotes the time delay due to the gestation of the mature predator; m ∈ [0, 1) is refuge rate to prey; the predator species consumes the prey species with Beddington-DeAngelis functional response incorporating prey refuge μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t), and ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ) denotes the growth rate of predator which are pregnant at time t – τ.

The initial conditions for system (1.4) take the form

x(θ)=ϕ(θ),   y1(θ)=ψ1(θ),   y2(θ)=ψ2(θ),ϕ(θ)≥0,       ψ1(θ)≥0,        ψ2(θ)≥0,       θ∈[−τ,0),ϕ(0)>0,       ψ1(0)>0,        ψ2(0)>0, (1.5)

where (ϕ(θ), ψ1(θ), ψ2(θ)) ∈ C([–τ, 0], R+3 ), which is the Banach space of continuous functions mapping the interval [–τ, 0] into R+3 , where R+3 = {(x1, x2, x3) : xi ≥ 0, i = 1, 2, 3}.

The rest of this paper is organized as follows. The boundedness and local stability of the equilibrium and the existence of Hopf bifurcation at positive equilibrium of system (1.4) are derived in the next section. In Section 3, we study the permanence of system (1.4). In Section 4, the global stability of system (1.4) are investigated. In Section 5, the influence of refuge rate on the densities to predator species and prey species is discussed. We end this paper with some examples and a briefly discussion.

2 Boundedness, Local stability and Hopf bifurcation

In this section, we study the boundedness and local stability of the equilibrium as well as the existence of Hopf bifurcation at positive equilibrium of system (1.4). It is obvious that solutions of model (1.4) with initial conditions (1.5) are positive for all t ≥ 0. The result is a direct consequence of Nagumo’s theorem [29].

2.1 Boundedness

Theorem 2.1

Every solution of system (1.4) with initial conditions (1.5) is bounded for all t ≥ 0 and all of these solutions are ultimately bounded.

Proof

Let V(t) = ηx(t – τ) + y1(t) + y2(t), and calculating the derivative of V(t) with respect to t along the positive solution of system (1.4), we have

V˙(t)=ηx˙(t)+y˙1(t)+y˙2(t)=η[x(t−τ)(r−ax(t−τ))−μ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)]+ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−βy1(t)−dy1(t)+βy1(t)−ey2(t)=ηx(t−τ)[r−ax(t−τ)]−dy1(t)−ey2(t). (2.1)

For a small positive constant s ≤ min{d, e},

V˙(t)+sV(t)=(s−d)y1(t)+(s−e)y2(t)+ηx(t−τ)[s+r−ax(t−τ)] ≤ηx(t−τ)[s+r−ax(t−τ)]. (2.2)

Hence there exists a positive constant M=η(s+r)24a such that

V˙(t)+sV(t)≤M, (2.3)

that is

V(t)≤(V(0)−Ms)e−st+Ms. (2.4)

Thus V(t) is ultimately bounded, that is, each solution z(t) = (x(t), y1(t), y2(t)) of system (1.4) is ultimately bounded. The proof is complete.

2.2 Equilibria

Obviously, system (1.4) always has a trivial equilibrium E0(0, 0, 0) and a predator-extinction equilibrium E1(r/a, 0, 0). Further, if the following holds:

μη>be(1+dβ) , 0≤m<1−ae(1+dβ)r[μη−be(1+dβ)], (H1)

then model (1.4) has a unique coexistence equilibrium E∗(x∗,y1∗,y2∗), where

x∗=−K+Δ2,   y1∗=eβy2∗,   y2∗=(1−m)[μη−be(1+dβ)]x∗−e(1+dβ)ce(1+dβ) (2.5)

with

K=1a((1−m)[μη−be(1+dβ)]ηc−r),    Δ=K2+4eacη(1+dβ).

Let E͠ = (x̃, ỹ1, ỹ2) be any arbitrary equilibrium, then the variational matrix of system (1.4) at E͠ is given by

J=r−2ax~−μ(1−m)y~2(1+cy~2)[1+b(1−m)x~+cy~2]20−μ(1−m)x~[1+b(1−m)x~][1+b(1−m)x~+cy~2]2ημ(1−m)y~2(1+cy~2)e−λτ[1+b(1−m)x~+cy~2]2−(β+d)ημ(1−m)x~[1+b(1−m)x~]e−λτ[1+b(1−m)x~+cy~2]20β−e

and the characteristic equation becomes

(λ+β+d)(λ+e)(λ−r+2ax~+μ(1−m)y~2(1+cy~2)[1+b(1−m)x~+cy~2]2)+μ(1−m)y~2(1+cy~2)[1+b(1−m)x~+cy~2]2×βημ(1−m)x~[1+b(1−m)x~][1+b(1−m)x~+cy~2]2e−λτ−(λ−r+2ax~+μ(1−m)y~2(1+cy~2)[1+b(1−m)x~+cy~2]2)×βημ(1−m)x~[1+b(1−m)x~][1+b(1−m)x~+cy~2]2e−λτ=0. (2.6)

2.3 E0 = (0, 0, 0)

First, we analyze the stability of equilibrium E0.

Theorem 2.2

The trivial equilibrium E0(0, 0, 0) of system (1.4) is unstable.

Proof

The characteristic equation (2.6) takes the form at the trivial equilibrium E0(0, 0, 0)

(λ−r)(λ+β+d)(λ+e)=0. (2.7)

It is readily seen that Eq. (2.7) has a positive root, thus the equilibrium E0 is always unstable. The proof is complete.

2.4 E1 = (r/a, 0, 0)

After that, we consider the stability of equilibrium E1.

Theorem 2.3

If the following holds:

μη>be(1+dβ) , 1−ae(1+dβ)r[μη−be(1+dβ)]<m<1, (H2)

then the predator-extinction equilibrium E1(r/a, 0, 0) of system (1.4) is locally asymptotically stable; if (H1) holds, then E1 is unstable.

Proof

The characteristic equation (2.6) at predator-extinction equilibrium E1 becomes

(λ+r)[(λ+β+d)(λ+e)−βημr(1−m)a+br(1−m)e−λτ]=0. (2.8)

Clearly, the equation λ + r = 0 has one negative real root, which implies that all other roots of Eq. (2.8) are determined by

λ2+h1λ+h2+h3e−λτ=0, (2.9)

where h1 = e + β + d > 0, h2 = e(β + d), h3 = −βημr(1−m)a+br(1−m).

When τ = 0, Eq. (2.9) turns to

λ2+h1λ+h2+h3=0. (2.10)

According to (H2), we have h2 + h3 > 0. By the Routh-Hurwitz criterion, the boundary equilibrium E1 is locally asymptotically stable. If (H1) holds, then Eq. (2.10) has at least a positive real root, thus the predator-extinction equilibrium E1 is unstable.

For τ > 0, we investigate the existence of purely imaginary roots of (2.9). If iω1(ω1 > 0) is a solution of (2.9) if and only if ω1 satisfies

−ω12+h1ω1i+h2+h3(cos⁡(τω1)−isin⁡(τω1))=0.

Separating the real and imaginary parts, we obtain

h1ω1=h3sin⁡(τω1),ω12−h2=h3cos⁡(τω1), (2.11)

which implies

ω14+(h12−2h2)ω12+h22−h32=0. (2.12)

Note that

h12−2h2=(e+β+d)2−2e(β+d)=e2+(β+d)2>0,h2−h3=e(β+d)+βημr(1−m)a+br(1−m)>0,

and h2 + h3 > 0, then h22−h32>0. Hence (2.9) has no positive real roots. By Theorem 3.4.1 in [30], if (H2) holds, then all the roots of (2.9) have negative real parts for all τ ≥ 0, this implies that the boundary equilibrium E1 is locally asymptotically stable for all τ ≥ 0. The proof is complete.

2.5 E∗=(x∗,y1∗,y2∗)

Further, we analyze the stability of equilibrium E*.

For the positive equilibrium E∗=(x∗,y1∗,y2∗), the characteristic equation (2.6) reduces to

λ3+p2λ2+p1λ+p0+(q1λ+q0)e−λτ=0, (2.13)

where

p2=β+d+e+A+2ax∗−r,p1=Q+(β+d+e)(A+2ax∗−r),p0=Q(A+2ax∗−r),q1=−B,q0=−B(2ax∗−r),

with

Q=e(β+d),A=μ(1−m)y2∗(1+cy2∗)[1+b(1−m)x∗+cy2∗]2,B=βημ(1−m)x∗[1+b(1−m)x∗][1+b(1−m)x∗+cy2∗]2=Q[e(1+dβ)+be(1+dβ)(1−m)x∗]μη(1−m)x∗.

When τ = 0, Eq. (2.13) turns to

λ3+p2λ2+(p1+q1)λ+p0+q0=0. (2.14)

From (H1), we derive that B=Q[e(1+dβ)+be(1+dβ)(1−m)x∗]μη(1−m)x∗<Q, thus

p2=β+d+e+(A+2ax∗−r),p1+q1=Q−B+(β+d+e)(A+2ax∗−r),p0+q0=QA+(Q−B)2ax∗−(Q−B)r,p2(p1+q1)−(p0+q0)=(β+d+e+A+2ax∗−r)(Q−B)+(A+2ax∗−r) ×[(β+d)(β+d+A+2ax∗−r)+e(e+A+2ax∗−r)] +QA+(Q+B)2ax∗−(Q+B)r.

Denote

r1=QAQ+B+2ax∗,   r2=A+2ax∗,   r3=QAQ−B+2ax∗,r4=r1+(β+d+e+r2−r)(Q−B)Q+B+(r2−r)[(β+d)(β+d+r2−r)+e(e+r2−r)]Q+B.

We see that if r < min{r2, r4} and (H1) hold, then p2 > 0, p1 + q1 > 0, p0 + q0 > 0 and p2(p1 + q1) – (p0 + q0) > 0. Therefore, by the Routh-Hurwitz criterion, the positive equilibrium E* is locally asymptotically stable.

For τ > 0, if iω(ω > 0) is a solution of (2.13) if and only if ω satisfies

−ω3i−p2ω2+p1ωi+p0+(q1ωi+q0)(cos⁡(τω)−isin⁡(τω))=0.

Separating the real and imaginary parts, we have

ω3−p1ω=q1ωcos⁡(τω)−q0sin⁡(τω),p2ω2−p0=q1ωsin⁡(τω)+q0cos⁡(τω), (2.15)

which implies

ω6+(p22−2p1)ω4+(p12−2p0p2−q12)ω2+p02−q02=0, (2.16)

where

p22−2p1=(β+d+e+A+2ax∗−r)2−2Q−2(β+d+e)(A+2ax∗−r) =(β+d)2+e2+(A+2ax∗−r)2>0,p12−2p0p2−q12=[Q+(β+d+e)(A+2ax∗−r)]2−2Q(A+2ax∗−r)×(β+d+e+A+2ax∗−r)−B2>(A+2ax∗−r)2[(β+d)2+e2]>0,p02−q02=[QA+(Q−B)2ax∗−(Q−B)r]×[QA+(Q+B)2ax∗−(Q+B)r].

Obviously, if r < r1 and (H1) hold, then p02−q02>0, this implies that (2.16) has no positive real roots. Therefore, by Theorem 3.4.1 in [30], if r < r1 and (H1) are satisfied, then all the roots of (2.16) have negative real parts for all τ ≥ 0. Hence the positive equilibrium E∗=(x∗,y1∗,y2∗) is locally asymptotically stable for all τ ≥ 0.

If r1 < r < min{r2, r4} holds, which implies p02−q02<0, then there exists a unique positive root ω0 satisfying (2.16). From (2.15), we have

cos⁡(τω0)=q1ω04+(p2q0−p1q1)ω02−p0q0q12ω02+q02. (2.17)

Denote

τ0n=1ω0arccosq1ω04+(p2q0−p1q1)ω02−p0q0q12ω02+q02+2nπω0,   n=0,1,2,⋯. (2.18)

By Theorem 3.4.1 in Kuang [30], we see that if p02−q02<0 hold, then E* remains stable for τ < τ0 := τ00.

We now claim that

d(Reλ)dττ=τ0>0.

This shows that there exists at least one eigenvalue with a positive real part for τ > τ0. Moreover, the conditions for the existence of a Hopf bifurcation [31] are then satisfied yielding a periodic solution. To this end, differentiating Eq. (2.13) with respect to τ, it follows that

dλdτ−1=−3λ2+2p2λ+p1λ(λ3+p2λ2+p1λ+p0)+q1λ(q1λ+q0)−τλ.

Hence, a direct calculation shows that

sgnd(Reλ)dτλ=iω0=sgnRedλdτ−1λ=iω0=sgn−(p1−3ω02)(ω02−p1)+2p2(p0−p2ω02)(ω03−p1ω0)2+(p0−p2ω02)2−q12q12ω02+q02.

We derive from (2.15) that

(ω03−p1ω0)2+(p0−p2ω02)2=q12ω02+q02.

Therefore, we yield

sgnd(Reλ)dτλ=iω0=sgn3ω04+2(p22−2p1)ω02+p12−2p0p2−q12q12ω02+q02>0.

Thus, the transversal condition holds and a Hopf bifurcation occurs at ω = ω0, τ = τ0. Now, let us summarize our results as follows:

Theorem 2.4

  1. If (H1) and r < r1 hold, then the positive equilibrium E* of system (1.4) is locally asymptotically stable for all τ ≥ 0.

  2. If (H1) and r1 < r < min{r2, r4} hold, then there exists a τ0 > 0 such that E* is locally asymptotically stable when τ ∈ [0, τ0). Furthermore, system (1.4) undergoes a Hopf bifurcation at E* when τ = τ0.

Remark

  1. Let τ = 0 and m = 0 in (1.4), then Theorem 2.3 of this paper is equivalent to Theorem 3 of [21]. Obviously, we generalize the conclusion of boundary equilibrium in [21], and show that prey refuge affect the stability of the boundary equilibrium. Further, the global stability of the boundary equilibrium will be studied in Section 4.

  2. Notice that the conditions of positive equilibrium E* is locally asymptotically stable in [21] is very complicated. Let τ = 0 and m = 0 in model (1.4), compare Theorem 2.4 of this paper with Lemma 4 of [21], we find that the conditions of our positive equilibrium locally stable are more extensive and concise than that of [21].

3 Permanence

Lemma 3.1

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Assume that (H1) holds, then

lim supt→+∞ x(t)≤L1,  lim supt→+∞ y1(t)≤L2,  lim supt→+∞ y2(t)≤L3, (3.1)

where

L1=ra,L2=βημ(1−m)L1−e(β+d)[1+b(1−m)L1]cβ(β+d),L3=βeL2.

Proof

It follows from (H1) that L2=βημ(1−m)L1−e(β+d)[1+b(1−m)L1]cβ(β+d)>0. Hence, there exists an enough small positive constant ε such that

L2ε=defβημ(1−m)(L1+ε)−e(β+d)[1+b(1−m)(L1+ε)]cβ(β+d)>0,L3ε=defβeL2ε>0.

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). From the first equation of system (1.4), it follows that

x˙(t)=x(t)(r−ax(t))−μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t)≤x(t)(r−ax(t)). (3.2)

Applying Lemma 2.3 in [32] to (3.2), it immediately follows that

lim supt→+∞ x(t)≤ra=defL1. (3.3)

Then for above ε > 0 sufficiently small there exists a T1 > 0 such that if t > T1, x(t) ≤ L1 + ε. We derive from the second and the third equations of system (1.4) that for t > T1 + τ,

y˙1(t)≤ημ(1−m)(L1+ε)y2(t−τ)1+b(1−m)(L1+ε)+cy2(t−τ)−βy1(t)−dy1(t),y˙2(t)=βy1(t)−ey2(t). (3.4)

Consider the following auxiliary equations:

u˙1(t)=ημ(1−m)(L1+ε)u2(t−τ)1+b(1−m)(L1+ε)+cu2(t−τ)−βu1(t)−du1(t),u˙2(t)=βu1(t)−eu2(t). (3.5)

Using a similar argument as that in the proof of Lemma 2.4 in [33], it follows from (3.5) that

limt→+∞ u1(t)=L2ε,limt→+∞ u2(t)=L3ε. (3.6)

By comparison, we obtain that

lim supt→+∞ y1(t)≤L2ε,lim supt→+∞ y2(t)≤L3ε. (3.7)

Let ε → 0, it follows that

lim supt→+∞ y1(t)≤L2,lim supt→+∞ y2(t)≤L3. (3.8)

The proof is complete.

Lemma 3.2

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5), if (H1) and

r>μ(1−m)L31+cL3,   μη>e(β+d)[1+b(1−m)l1]β(1−m)l1 (3.9)

hold, then

lim inft→+∞ x(t)≥l1,  lim inft→+∞ y1(t)≥l2,  lim inft→+∞ y2(t)≥l3, (3.10)

where

l1=r(1+cL3)−μ(1−m)L3a(1+cL3),l2=βημ(1−m)l1−e(β+d)[1+b(1−m)l1]cβ(β+d),l3=βel2.

Proof

It follows from (H1) and condition (3.9) that l1=r(1+cL3)−μ(1−m)L3a(1+cL3)>0, l2=βημ(1−m)l1−e(β+d)[1+b(1−m)l1]cβ(β+d)> 0. Hence, there exists an enough small positive constant ε such that

l1ε=r(1+c(L3+ε))−μ(1−m)(L3+ε)a(1+c(L3+ε))>0,l2ε=βημ(1−m)(l1−ε)−e(β+d)[1+b(1−m)(l1−ε)]cβ(β+d)>0,l3ε=βel2ε>0.

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). For above ε > 0 sufficiently small, it follows from Lemma 3.1 that there exists a T2 > 0 such that if t > T2, y1(t) ≤ L2 + ε, y2(t) ≤ L3 + ε. Hence, we derive from the first equation of system (1.4) that for t > T2,

x˙(t)≥x(t)(r−ax(t)−μ(1−m)(L3+ε)1+c(L3+ε)).

According to condition (3.9) and similar to the proof of Lemma 3.1, we have

lim inft→+∞ x(t)≥r(1+cL3)−μ(1−m)L3a(1+cL3)=defl1>0. (3.11)

For above ε > 0 sufficiently small, there exists a T3 ≥ T2 such that if t > T3, x(t) ≥ l1 − ε. Therefore, it follows from the second and the third equations of system (1.4) that for t > T3 + τ,

y˙1(t)≥ημ(1−m)(l1−ε)y2(t−τ)1+b(1−m)(l1−ε)+cy2(t−τ)−βy1(t)−dy1(t),y˙2(t)=βy1(t)−ey2(t). (3.12)

Consider the following auxiliary equations:

u˙1(t)=ημ(1−m)(l1−ε)u2(t−τ)1+b(1−m)(l1−ε)+cu2(t−τ)−βu1(t)−du1(t),u˙2(t)=βu1(t)−eu2(t). (3.13)

Using a similar argument as that in the proof of Lemma 2.4 in [33], it follows from (3.13) that

limt→+∞ u1(t)≥l2ε,limt→+∞ u1(t)≥l3ε. (3.14)

By comparison, we obtain that

lim inft→+∞ y1(t)≥l2ε,lim inft→+∞ y2(t)≥l3ε. (3.15)

Let ε → 0, we conclude that

lim inft→+∞ y1(t)≥l2,lim inft→+∞ y2(t)≥l3. (3.16)

The proof is complete.

As a direct corollary of Lemmas 3.1 and 3.2 we have the following theorem.

Theorem 3.1

Assume that (H1) and (3.9) hold, then system (1.4) is permanent.

4 Global stability

In this section, we study the global stability of the predator-extinction equilibrium E1 and the global attractivity of the coexistence equilibrium E* of system (1.4). The strategy of proofs is to use Lyapunov functionals and the LaSalle invariance principle.

Theorem 4.1

If (H2) holds, then the predator-extinction equilibrium E1(r/a, 0, 0) of system (1.4) is globally asymptotically stable.

Proof

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Denote x0 = r/a.

Define

V11(t)=x−x0−x0lnxx0+ξ1y1+ξ2y2, (4.1)

where ξ1 = 1+b(1−m)x0η,  ξ2=μ(1−m)x0e .

Calculating the derivative of V11 along positive solutions of system (1.4), it follows that

ddtV11(t)=(1−x0x)[x(t)(r−ax(t))−μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t)]+ξ1[ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−βy1(t)−dy1(t)]+ξ2[βy1(t)−ey2(t)]. (4.2)

Substituting r = ax0 into (4.2), we obtain that

ddtV11(t)=−rx0(x(t)−x0)2−[1+b(1−m)x0]μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t)+μ(1−m)x0y2(t)−μ(1−m)cx0y22(t)1+b(1−m)x(t)+cy2(t)+ξ1ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−ξ1βy1(t)−ξ1dy1(t)+ξ2βy1(t)−ξ2ey2(t). (4.3)

Define

V1(t)=V11(t)+ξ1ημ(1−m)∫t−τtx(s)y2(s)1+b(1−m)x(s)+cy2(s)ds. (4.4)

We derive from (4.3) and (4.4) that

ddtV1(t)=−rx0(x(t)−x0)2−μ(1−m)cx0y22(t)1+b(1−m)x(t)+cy2(t)−[(β+d)[a+br(1−m)]ηa−βμr(1−m)ae]y1(t). (4.5)

If (H2) holds, it then follows from (4.5) that V̇1(t) ≤0. By Theorem 5.3.1 in [34], solutions limit to 𝓜, the largest invariant subset of {V̇1(t) = 0}. Obviously, we see from (4.5) that V̇1(t) = 0 if and only if x = x0, y2 = 0. Noting that 𝓜 is invariant, for each element in 𝓜, we have x(t) = x0, y2(t) = 0. It therefore follows from the third equation of system (1.4) that 0 = ẏ2(t) = βy1(t), which yields y1(t) = 0. Therefore, V̇1(t) = 0 if and only if (x, y1, y2) = (x0, 0, 0). Accordingly, the global asymptotic stability of E1 follows from LaSalle invariance principle. This completes the proof.

Theorem 4.2

Assume that (H1) and (3.9) are satisfied. If the following holds:

a(l1+x∗)−r≥ρ¯1,  l1x∗+b(1−m)l1x∗2≥ρ¯2, (4.6)

where ρ¯1=2μ(1−m)cL3y2∗(1+cy2∗)+μc(1−m)x∗y2∗[b(1−m)L1+cL3]2[1+b(1−m)l1+cl3][1+b(1−m)x∗+cy2∗], ρ¯2=x∗y2∗[b(1−m)L1+cL3]2 , and Li, li (i = 1, 2, 3) are defined as those in Theorem 3.1, then the positive equilibrium E∗(x∗,y1∗,y2∗) of system (1.4) is globally attractive.

Proof

Let (x(t), y1(t), y2(t)) be any positive solution of system (1.4) with initial conditions (1.5). Define

V21(t)=x−x∗−x∗lnxx∗+k1(y1−y1∗−y1∗lny1y1∗)+k2(y2−y2∗−y2∗lny2y2∗), (4.7)

where k1=1+b(1−m)x∗+cy2∗η, k2=k1(β+d)β .

Calculating the derivative of V11 along positive solutions of system (1.4), it follows that

ddtV21(t)=(1−x∗x)[x(t)(r−ax(t))−μ(1−m)x(t)y2(t)1+b(1−m)x(t)+cy2(t)]+k1(1−y1∗y1)[ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−βy1(t)−dy1(t)]+k2(1−y2∗y2)[βy1(t)−ey2(t)]. (4.8)

Substituting r=ax∗+μ(1−m)y2∗1+b(1−m)x∗+cy2∗ into (4.8), we derive that

ddtV21(t)=(1−x∗x)[x(r−ax)−x∗(r−ax∗)+μ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗]−μ(1−m)[[1+b(1−m)x∗+cy2∗]−x∗[1+b(1−m)x+cy2]x+c(x∗y2−xy2∗)x]×x(t)y2(t)1+b(1−m)x(t)+cy2(t)+k1(1−y1∗y1)×[ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−dy1(t)−βy1(t)]+k2(1−y2∗y2)[βy1(t)−ey2(t)]
=(1−x∗x)[x(r−ax)−x∗(r−ax∗)+μ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗]−μ(1−m)[1+b(1−m)x∗+cy2∗]×x(t)y2(t)1+b(1−m)x(t)+cy2(t)−μ(1−m)cx(t)y2(t)(x∗y2−xy2∗)x[1+b(1−m)x(t)+cy2(t)]+k1ημ(1−m)x(t−τ)y2(t−τ)1+b(1−m)x(t−τ)+cy2(t−τ)−k1ημ(1−m)y1∗x(t−τ)y2(t−τ)y1[1+b(1−m)x(t−τ)+cy2(t−τ)]+k1(β+d)y1∗−k2βy1∗y2∗y1∗y1(t)y2(t)+k2ey2∗. (4.9)

Define

V2(t)=V21(t)+k1ημ(1−m)∫t−τt[x(s)y2(s)1+b(1−m)x(s)+cy2(s)−x∗y2∗1+b(1−m)x∗+cy2∗−x∗y2∗1+b(1−m)x∗+cy2∗×ln [1+b(1−m)x∗+cy2∗]x(s)y2(s)x∗y2∗[1+b(1−m)x(s)+cy2(s)]]ds. (4.10)

It follows from (4.9) and (4.10) that

ddtV2(t)=(1−x∗x)[x(r−ax)−x∗(r−ax∗)+μ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗]−μ(1−m)cx(t)y2(t)(x∗y2−xy2∗)x[1+b(1−m)x(t)+cy2(t)]−k1ημ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗×y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]+k1(β+d)y1∗−k2βy1∗y2∗y1∗y1(t)y2(t)+k2ey2∗+k1ημ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗×ln [1+b(1−m)x(t)+cy2(t)]x(t−τ)y2(t−τ)x(t)y2(t)[1+b(1−m)x(t−τ)+cy2(t−τ)]. (4.11)

Noting that

k2ey2∗=k2βy1∗=k1(β+d)y1∗=k1ημ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗=μ(1−m)x∗y2∗,

and

(1−x∗x)μ(1−m)x∗y2∗1+b(1−m)x∗+cy2∗=μ(1−m)x∗y2∗(1−x∗[1+b(1−m)x(t)+cy2(t)]x(t)[1+b(1−m)x∗+cy2∗])+μ(1−m)cx∗y2∗(x∗y2−xy2∗)x(t)[1+b(1−m)x∗+cy2∗],

we derive from (4.11) that

ddtV2(t)=(x−x∗)2x[r−a(x+x∗)]−μ(1−m)cx(t)y2(t)(x∗y2−xy2∗)x(t)[1+b(1−m)x(t)+cy2(t)]+μ(1−m)cx∗y2∗(x∗y2−xy2∗)x(t)[1+b(1−m)x∗+cy2∗]−μ(1−m)x∗y2∗×[x∗[1+b(1−m)x+cy2]x∗[1+b(1−m)x∗+cy2∗]−1−ln x∗[1+b(1−m)x+cy2]x∗[1+b(1−m)x∗+cy2∗]]−μ(1−m)x∗y2∗×[y2∗y1(t)y1∗y2(t)−1−ln y2∗y1(t)y1∗y2(t)]−μ(1−m)x∗y2∗×[y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]−1−ln y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]]
≤−(x−x∗)2x[a(x+x∗)−r−ρ1]−μ(1−m)c(y−y2∗)2[xx∗+b(1−m)xx∗2−ρ2]x(t)[1+b(1−m)x(t)+cy2(t)][1+b(1−m)x∗+cy2∗]−μ(1−m)x∗y2∗×[x∗[1+b(1−m)x+cy2]x∗[1+b(1−m)x∗+cy2∗]−1−ln x∗[1+b(1−m)x+cy2]x∗[1+b(1−m)x∗+cy2∗]]−μ(1−m)x∗y2∗×[y2∗y1(t)y1∗y2(t)−1−ln y2∗y1(t)y1∗y2(t)]−μ(1−m)x∗y2∗×[y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]−1−ln y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]], (4.12)

where ρ1=2μ(1−m)cy2y2∗(1+cy2∗)+μc(1−m)x∗y2∗[b(1−m)x+cy2]2[1+b(1−m)x(t)+cy2(t)][1+b(1−m)x∗+cy2∗], ρ2=x∗y2∗[b(1−m)x+cy2]2 .

Hence, if (4.6) holds, it then follows from (4.12) that V̇1(t) ≤0, with equality if and only if x = x*, y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]=y2∗y1(t)y1∗y2(t) = 1. We now look for the invariant subset 𝓜 within the set M = {(x, y1, y2) : x = x*, y1∗[1+b(1−m)x∗+cy2∗]x(t−τ)y2(t−τ)x∗y2∗y1[1+b(1−m)x(t−τ)+cy2(t−τ)]=y2∗y1(t)y1∗y2(t) = 1}. Since x = x* on 𝓜 and due to 0 = ẋ(t) = x*(r − ax*) − μ(1−m)x∗y2(t)1+b(1−m)x∗+cy2(t) , we obtain y2(t) = y2∗ . From the third equation of model (1.4) that 0 = ẏ2(t) = βy1(t) − e y2∗ , which yields y1 = y1∗ . Therefore, the only invariant set in M is 𝓜 = {(x∗,y1∗,y2∗)} . Using the LaSalle invariance principle, then E* of system (1.4) is globally attractive.

5 The influence of prey refuge

In this section, we investigate the influence of prey refuge. Under the condition (H1), let us compute the derivative along the positive equilibrium E* with respect to m, that is

dx∗dm=[μη−be(1+dβ)]x∗aηcΔ>0,dy2∗dm=[μη−be(1+dβ)]x∗[(1−m)[μη−be(1+dβ)]−aηcΔ]aηc2e(1+dβ)Δ,dy1∗dm=[μη−be(1+dβ)]x∗[(1−m)[μη−be(1+dβ)]−aηcΔ]βaηc2(1+dβ)Δ.

Denote

R1=ηcr2+4ae(1+dβ)2r[μη−be(1+dβ)]>ae(1+dβ)r[μη−be(1+dβ)]=R2.

Due to the existence of E*, which implies that x* is a strictly increasing function of m, that is, increasing the constant amount of prey refuge m leads to the increase of prey densities. When 0 < m < 1 − ℜ1, dyi∗dm > 0 (i = 1, 2), it then yields that both y1∗  and  y2∗ are strictly increasing functions on m ∈ (0, 1 − ℜ1); when 1 − ℜ1 < m < 1 − ℜ2, dyi∗dm < 0 (i = 1, 2), which implies that both y1∗  and  y2∗ are strictly decreasing functions on m ∈ (1 − ℜ1, 1 − ℜ2); when m = 1 − ℜ1, the predator species reaches its maximum, and when m = 1 − ℜ2, the predator species goes to extinction.

6 Numerical simulations

Example 6.1

In (1.4), let m = τ = 0, then system (1.4) is degenerated into model (2.1) of [21]. Let r = 6, a = 3, μ = 20, b = d = e = 1, η = β = 2 and c = 4, then system (1.4) has a positive equilibrium E* = (0.5168, 1.5330, 3.0660). By calculation, μη = 40 > 1.5 = be(1 + dβ ), 0 < 0.9805 = 1 − ae(1+dβ)r[μη−be(1+dβ)] , r < 7.3835 = r2 and r < 17.7279 = r4. Thus E* is locally asymptotically stable.

Note that ηβ(1 + bx*) = 6.0671 > 0.4778 = b2x∗y2∗(β+d+e)1+cy2∗ . Therefore, condition (S3): ηβ(1 + bx*) < min e(β+d)(1+bx∗+cy2∗)2μx∗, b2x∗y2∗(β+d+e)1+cy2∗ in [21] which ensures the local stability of the positive equilibrium E* is not satisfied. This implies that our conditions are weaker that those in [21] (see Figs. 1 and 2).

Figure 1 m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) is locally asymptotically stable.
Figure 1

m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) is locally asymptotically stable.

Figure 2 m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) with different initial values is locally asymptotically stable.
Figure 2

m = τ = 0, E* = (0.5168, 1.5330, 3.0660) of system (1.4) with different initial values is locally asymptotically stable.

Example 6.2

In (1.4), let r = 6, a = 3, m = 0.2, b = d = e = 1, η = β = 2 and c = 4, then be(1 + dβ ) = 1.5.

  1. If μ = 1, τ = 10, it is easy to show that μη = 2 > be(1 + dβ ), 1 − m = 0.8 < 1.5 = ae(1+dβ)r[μη−be(1+dβ)] and the predator-extinction equilibrium E1 = (2, 0, 0). By Theorem 4.1, E1 is globally asymptotically stable (see Figs. 3 and 4).

    Figure 3 μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) is globally asymptotically stable.
    Figure 3

    μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) is globally asymptotically stable.

    Figure 4 μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) with different initial values is globally asymptotically stable.
    Figure 4

    μ = 1, τ = 10, E1 = (2, 0, 0) of system (1.4) with different initial values is globally asymptotically stable.

  2. If μ = 20, τ = 10, it is easy to show that μη = 40 > be(1 + dβ ), m < 0.9805 = 1 − ae(1+dβ)r[μη−be(1+dβ)] , r < 7.9443 = r1 and the positive equilibrium E* = (0.7953, 1.9162, 3.8323). By Theorem 2.4, E* is locally asymptotically stable (see Figs. 5 and 6).

    Figure 5 μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) is locally asymptotically stable.
    Figure 5

    μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) is locally asymptotically stable.

    Figure 6 μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) with different initial values is locally asymptotically stable.
    Figure 6

    μ = 20, τ = 10, E* = (0.7953, 1.9162, 3.8323) of system (1.4) with different initial values is locally asymptotically stable.

  3. If μ = 36, it is easy to show that μη = 72 > be(1 + dβ ), m < 0.9894 = 1 − ae(1+dβ)r[μη−be(1+dβ)] , r > 5.2798 = r1, r < 6.2736 = r2, r < 8.8358 = r4, and the positive equilibrium E* = (0.1302, 0.4868, 0.9735). Further, by calculation, we have τ0 = 0.9429. By Theorem 2.4, when τ < τ0, E* is locally asymptotically stable (see Fig. 7); when τ > τ0, the positive equilibrium E* of model (1.4) is unstable, which yields a periodic solution (see Fig. 8).

    Figure 7 τ = 0.9 < 0.9429 = τ0, μ = 36, E* = (0.1302, 0.4868, 0.9735) of system (1.4) is locally asymptotically stable, system (1.4) undergoes a Hopf bifurcation at E* when τ0.
    Figure 7

    τ = 0.9 < 0.9429 = τ0, μ = 36, E* = (0.1302, 0.4868, 0.9735) of system (1.4) is locally asymptotically stable, system (1.4) undergoes a Hopf bifurcation at E* when τ0.

    Figure 8 τ = 0.96 > 0.9429 = τ0, μ = 36, E* = (0.1302, 0.4868, 0.9735) of system (1.4) is locally asymptotically stable, system (1.4) undergoes a Hopf bifurcation at E* when τ0.
    Figure 8

    τ = 0.96 > 0.9429 = τ0, μ = 36, E* = (0.1302, 0.4868, 0.9735) of system (1.4) is locally asymptotically stable, system (1.4) undergoes a Hopf bifurcation at E* when τ0.

  4. If μ = 2, τ = 2, it is easy to show that μη = 4 > be(1 + dβ ), m < 0.7 = 1 − ae(1+dβ)r[μη−be(1+dβ)] , a(l1 + x*) − r = 5.5962 > 0.6417 = ρ1, l1 x* + b(1 − m)l1x*2 = 9.5580 > 1.2781 = ρ2, r > 0.25 = μ(1−m)L31+cL3 , μη > 2.4783 = e(β+d)[1+b(1−m)l1]β(1−m)l1 and the positive equilibrium E* = (1.9487, 0.1998, 0.3996). By Theorem 4.2, E* is globally attractive (see Fig. 9).

    Figure 9 μ = 2, τ = 2, E* = (1.9487, 0.1998, 0.3996) of system (1.4) with different initial values is globally attractive.
    Figure 9

    μ = 2, τ = 2, E* = (1.9487, 0.1998, 0.3996) of system (1.4) with different initial values is globally attractive.

Example 6.3

In (1.4), let r = 10, μ = 6, c = e = 0.5, a = b = d = 1, η = β = 2 and τ = 0. By simple computations, we find μη = 10 > 0.75 = be(1 + dβ ), 1 − ℜ1 = 0.4432, 1 − ℜ2 = 0.9919. Note that, if m ∈ [0, 0.9919), by calculation, we can obtain m < 1 − ae(1+dβ)r[μη−be(1+dβ)] , r < min{r2, r4}, then E* is locally asymptotically stable when τ = 0. Our simulations show that the constant prey refuge m plays an important role on the coexistence of prey-predator population (see Figs. 10 and 11). When m is small and increasing, the predator density increases, due to the fact that predators have sufficient preys available for predation, even though the refuge increases (see Fig. 12). But, as m crosses its threshold value, the predator density decreases with increasing m, the predators was unable to catch preys to sustain themselves and ultimately goes to extinction due to starvation (see Fig. 13).

Figure 10 Bifurcation diagram of the prey species of model (1.4), m as the bifurcation parameter.
Figure 10

Bifurcation diagram of the prey species of model (1.4), m as the bifurcation parameter.

Figure 11 Bifurcation diagram of the predator species of model (1.4), m as the bifurcation parameter.
Figure 11

Bifurcation diagram of the predator species of model (1.4), m as the bifurcation parameter.

Figure 12 τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from zero to 1 − ℜ1 = 0.4432, the predator species will increase.
Figure 12

τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from zero to 1 − ℜ1 = 0.4432, the predator species will increase.

Figure 13 τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from 1 − ℜ1 = 0.4432 to 0.9919 = 1 − ℜ2, the predator species will decrease.
Figure 13

τ = 0, E* is locally asymptotically stable. Change the values of the refuge m from 1 − ℜ1 = 0.4432 to 0.9919 = 1 − ℜ2, the predator species will decrease.

7 Conclusion

In this paper we investigate the influence of prey refuge on the dynamics of a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay. Sufficient conditions are derived to ensure the predator-extinction and the locally asymptotically stability of positive equilibrium. Compared with [21] we get more precise conditions. Also, we find that time delay can cause a stable equilibrium to become unstable one, even Hopf bifurcation to occur, when time delay passes through some critical values. Furthermore, the persistence is investigated. After that, we study the global stability of the predator-extinction and positive equilibrium by constructing some suitable Lyapunov functionals.

Also, we discuss the influence of prey refuge on the densities of predator species and prey species. When prey refuge m in the interval (0, 1 − ℜ1), the density of predators will increase with prey refuge m, due to predator species having sufficient food for their predation with sufficiently small prey refuge m. Predator population attains its maximum when the prey refuge m = 1 − ℜ1. In case of larger values of m (m > 1 − ℜ1), this implies that predators species are less likely to catch prey, and the predator species deceases with the increasing of prey refuge m. Eventually, the predator population will be extinct when prey refuge m = 1 − ℜ2.

  1. Competing interests The authors declare that there is no conflict of interests.

  2. Authors’ Contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (11601085), the Natural Science Foundation of Fujian Province (2017J01400) and the foundation of Education Department of Fujian Province (JAT160063).

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Received: 2018-07-03
Accepted: 2018-12-12
Published Online: 2019-03-26

© 2019 Xiao et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
  10. A new smoothing method for solving nonlinear complementarity problems
  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
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  23. Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
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  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
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  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
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  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
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  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
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  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
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  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
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  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
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  76. On Diophantine equations involving Lucas sequences
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  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
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  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
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  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
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  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
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  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
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https://doi.org/10.1515/math-2019-0014
Keywords for this article
Hopf bifurcation; Predator-prey; Beddington-DeAngelis; Stage structure; Global stability
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
  10. A new smoothing method for solving nonlinear complementarity problems
  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
  22. On split Lie color triple systems
  23. Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
  45. On Diophantine approximation by unlike powers of primes
  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
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