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

Fengde Chen; Xinyu Guan; Xiaoyan Huang; Hang Deng

doi:10.1515/math-2019-0082

Artikel Open Access

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

Fengde Chen , Xinyu Guan , Xiaoyan Huang und Hang Deng

Veröffentlicht/Copyright: 8. November 2019

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Abstract

A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.

Keywords: predator; prey; Allee effect; global stability; density dependent birth rate

MSC 2010: 34C25; 92D25; 34D20

1 Introduction

As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already, the influence of the Allee effect [2, 3, 4, 5, 6], the influence of the mutual interferences [7, 8], the influence of the stage structure [9, 10, 11, 12, 13], the stability of the positive equilibrium [12, 13, 14, 15, 16, 17], the existence and stability of the almost periodic solution [18], the existence of the positive periodic solution [19, 20], the persistent of the system [21] have been extensively studied, and many excellent results were obtained.

Allee effect, which reflects the fact that the population growth rate is reduced at low population size, due to its importance, the ecosystem subject to Allee effect has recently been extensively studied by many scholars, see [2, 3, 4, 5, 6, 22, 23, 24, 25, 26] and the references cited therein.

Hüseyin Merdan [2] investigated the influence of the Allee effect on the Lotka-Volterra type predator-prey system. To do so, the author proposed the following predator-prey with Allee effect system

dxdt=rxβ+xx(1−x)−axy,dydt=ay(x−y). (1.1)

Hüseyin Merdan showed that if r − a β > 0 hold, the model (1.1) has three steady-state solutions: A(0, 0), B(1, 0) and C(x^*, y^*). the first two are locally unstable, while the third one is locally asymptotically stable. By carrying out a series of numeric simulations, the author found the following two phenomenon. (1) The system subject to an Allee effect takes a longer time to reach its steady-state solution; (2) The Allee effect reduces the population densities of both predator and prey at the steady-state.

In [17], Guan, Liu and Xie argued that "It seems interesting to consider the influence of the Allee effect on the predator species, since generally speaking, the higher the hierarchy in the food chain, the more likely it is to become extinct" and they proposed the following model with the Allee effect on the predator species:

dxdt=rx(1−x)−axy,dydt=ayβ+yy(x−y), (1.2)

where r, a are positive constants. They showed that if r > a holds, then system (1.2) admits a unique positive equilibrium, and the Allee effect has no influence on the final density of the species.

It bring to our attention that in system (1.1) and (1.2), without consider the influence of the predator species and the Allee effect, the prey species satisfies the traditional Logistic equation

dxdt=rx(1−x), (1.3)

where r is the intrinsic growth rate, which is equal to the birth rate minus death rate. Hence system (1.3) could be revised as

dxdt=x(a1−d1−e1x). (1.4)

where a₁ is the birth rate of the species and d₁ is the death rate of the species. Already, Brauer and Castillo-Chavez [27], Tang and Chen [28] and Berezansky, Braverman, et al. [29] had showed that in some case, the density dependent birth rate of the species is more suitable. If we take the famous Beverton-Holt function [29] as the birth rate, then system (1.4) should be revised to

dxdt=x(a1b1+c1x−d1−e1x). (1.5)

System (1.5) combines with the idea of Merdan [2] and Guan et al. [17], will lead to the following Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

dxdt=x(a1b1+c1x−d1−e1x)−axy,dydt=ayβ+yy(x−y). (1.6)

It is well known that in a more realistic model the delay effect should be an average over past populations. This results in an equation with a distributed delay or an infinite delay [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Here, if we incorporate the infinite delay to system (1.6), then we will have the following system

dxdt=x(a1b1+c1x−d1−e1x)−ax∫−∞tK1(t−s)y(s)ds,dydt=ayβ+yy(∫−∞tK2(t−s)x(s)ds−y). (1.7)

The delay kernels K_i : [0, +∞) → (0, +∞), i = 1, 2 are continuous functions such that

∫0+∞Ki(s)ds=1. (1.8)

We shall consider (1.7) together with the initial conditions

x(s)=ϕ(s),s∈(−∞,0],y(s)=ψ(s),s∈(−∞,0], (1.9)

where ϕ, ψ ∈ BC⁺. It is well known that by the fundamental theory of functional differential equations [37], system (1.7) has a unique solution (x(t), y(t)) satisfying the initial condition (1.9). We easily prove x(t) > 0, y(t) > 0 in maximal interval of existence of the solution. In this paper, the solution of system (1.7) satisfying the initial conditions (1.9) is said to be positive.

We mention here that to this day, though there are many scholars investigated the dynamic behaviors of the ecosystem with Allee effect [1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26], none of them considered the density dependent birth rate of the species. Also, to the best of the authors knowledge, to this day, still no scholars propose a ecosystem with infinite delay and Allee effect at the same time. It seems that this is the first time such kind of model are proposed and studied.

The paper is arranged as follows. In section 2 we investigate the persistent and extinct property of the system, based on this, we are able to investigate the locally stability property of the equilibrium solutions of system (1.6). In section 3, by applying the Dulac criterion, we are able to show that under some assumption, the positive equilibrium is globally asymptotically stable. Section 4 presents some numerical simulations concerning the stability of our model. We end this paper by a briefly discussion.

2 Persistence and local stability of the equilibria

We need several Lemmas to prove the persistent property of the system.

Lemma 2.1

[40] Consider the following equation

dydt=y(ab+cy−d−ey). (2.1)

Assume that a > bd, then the unique positive equilibrium y^* of system (2.1) is globally asymptotically stable, where

y∗=−(eb+dc)+(eb+dc)2−4ec(db−a)2ec.

Lemma 2.2

[22] Consider the following equation

dydt=ayβ+yy(b−y). (2.2)

The unique positive equilibrium y^* = b is global stability.

Theorem 2.1

Assume that

a1b1>d1+au∗ (2.3)

holds, where u^* is defined by (2.6), then system (1.6) is permanent.

Proof

It follows from (2.3) that there exists a ε > 0 enough small such that

a1b1>d1+a(u∗+ε). (2.4)

Let (x(t), y(t)) be any positive solution of system (1.6). From system (1.6) it follows that

dxdt≤x(a1b1+c1x−d1−e1x).

Consider the equation

du1dt=u1(a1b1+c1u1−d1−e1u1). (2.5)

It follows from Lemma 2.1 that (2.5) admits a unique globally stable positive equilibrium u^*, where

u∗=−(e1b1+d1c1)+(e1b1+d1c1)2−4e1c1(d1b1−a1)2e1c1. (2.6)

By using the differential inequality theory, any solution of (2.5) satisfies

lim supt→+∞x(t)≤limt→+∞u(t)=u∗. (2.7)

Hence, there exists a T₁ > 0 such that

x(t)<u∗+ε2. (2.8)

For t > T₁, it follows from the second equation of system (1.6) that

dydt≤ayβ+yy(u∗+ε2−y). (2.9)

Consider the equation

du2dt=au2β+u2u2(u∗+ε2−u2). (2.10)

It follows from Lemma 2.2 that (2.10) admits a unique globally stable positive equilibrium

u2∗=u∗+ε2. (2.11)

By differential inequality theory, any solution of (2.9) satisfies

lim supt→+∞y(t)≤u∗+ε2. (2.12)

Hence, there exists a T₂ > T₁ such that

y(t)<u∗+ε. (2.13)

For t > T₂, it follows from the first equation of system (1.6) that

dxdt≥x(a1b1+c1x−d1−e1x−a(u∗+ε)). (2.14)

Now let’s consider the equation

dv1dt=v1(a1b1+c1v1−d1−e1v1−a(u∗+ε)). (2.15)

Since

a1>b1(d1+a(u∗+ε)),

it follows from Lemma 2.1 that system (2.15) admits a unique positive equilibrium v1∗ , which is globally asymptotically stable. Applying the differential inequality theory to (2.14) leads to

lim inft→+∞x(t)≥limt→+∞v(t)=v1∗.

It follows from above inequality that there exists an enough large T₃ > T₂ such that

x(t)>v1∗−ε4forallt≥T3,

and so, from the second equation of system (1.6), we have

dydt≥ayβ+yy(v1∗−ε4−y). (2.16)

Consider the equation

dv2dt=av2β+v2v2(v1∗−ε4−v2). (2.17)

It follows from Lemma 2.2 that (2.17) admits a unique globally stable positive equilibrium

v2∗=v1∗−ε4. (2.18)

By using the differential inequality theory, any solution of (2.16) satisfies

lim inft→+∞y(t)≥limt→+∞v2(t)=v1∗−ε4. (2.19)

(2.7), (2.12), (2.15) and (2.19) show that system (1.6) is permanent. This ends the proof of Theorem 2.1.

Remark 2.1

By using the software Maple, for the fixed coefficients, one could always compute u^* easily, however, condition (2.3) could be replaced by some more restricted but easily verified condition, indeed, we could have the following results.

Corollary 2.1

Assume that

a1b1>d1+aa1b1−d1e1 (2.20)

holds, then system (1.6) is permanent.

One interesting problem is to investigate the extinction property of system (1.6), for this, we have the following result.

Theorem 2.2

Assume that

a1b1<d1

holds, then

limt→+∞x(t)=0,limt→+∞y(t)=0.

Proof

From the first equation of system (1.6) we have

dxdt=x(a1b1+c1x−d1−e1x)−axy≤x(a1b1+c1x−d1−e1x)≤x(a1b1−d1).

Hence

x(t)≤x(0)exp⁡{(a1b1−d1)t}→0ast→+∞.

For any positive constant ε > 0 enough small, there exists a T > 0 such that

x(t)<εforallt≥T.

Hence, from the second equation of system (1.6), we have

dydt≤ayβ+yy(ε−y).

Consider the equation

dudt=auβ+uu(ε−u).

It follows from Lemma 2.2 that above equation admits a unique globally stable positive equilibrium u^* = ε. By using the differential inequality theory, we have

lim supt→+∞y(t)≤ε.

Hence

0≤lim inft→+∞y(t)≤lim supt→+∞y(t)≤ε.

Since ε is any small positive constant, setting ε → 0 in above inequality leads to

limt→+∞y(t)=0.

This ends the proof of Theorem 2.2.

Now we are in the position of investigate the stability property of steady-state solutions of the model (1.6). Defining

f(x,y):=x(a1b1+c1x−d1−e1x)−axy,g(x,y):=ayβ+yy(x−y).

The steady-state solutions of (1.6) are obtained by solving the equations f(x, y) = 0 and g(x, y) = 0. The model has three steady-state solutions: A(0, 0), B(u^*, 0) and C(x^*, y^*).

Theorem 2.3

If a₁ > b₁d₁ holds, then C(x^*, y^*) is non-negative equilibrium and it is locally asymptotically stable. If inequality (2.3) holds, then A(0, 0) and B(u^*, 0) is unstable.

Proof

The variation matrix of the continuous-time system (1.6) at an equilibrium solution (x, y) is

J(x,y)=fx(x,y)fy(x,y)gx(x,y)gy(x,y)=K1−axay2β+yK2,

where

K1=a1c1x+b1−d1−e1x+x−a1c1c1x+b12−e1−ay,K2=2ayx−yβ+y−ay2β+y−ay2x−yβ+y2.

Noting that (x^*, y^*) satisfies the equation

(a1b1+c1x∗−d1−e1x∗)−ay∗=0,ay∗β+y∗y∗(x∗−y∗)=0.

Hence, at C(x^*, y^*)

J(x∗,y∗)=−x∗(a1c1(c1x∗+b1)2+e1)−ax∗a(y∗)2β+y∗−a(y∗)2β+y∗.

Noting that

tr(J(x∗,y∗))=−x∗(a1c1(c1x∗+b1)2+e1)−a(y∗)2β+y∗<0,

and

det(J(x∗,y∗))=(a1c1(c1x∗+b1)2+e1)ax∗(y∗)2β+y∗+a2x∗(y∗)2β+y∗>0.

So that both eigenvalues of J(x^*, y^*) have negative real parts, and hence this steady-state solution is locally asymptotically stable.

From Theorem 2.1 we know that under the assumption (2.3) holds, system (1.6) is permanent, hence no solution could approach to A(0, 0) and B(u^*, 0), which means that A(0, 0) and B(u^*, 0) are locally unstable.

This ends the proof of Theorem 2.3.

3 Global stability

We had showed that the positive equilibrium is locally stable, in this section, we further give sufficient conditions to ensure the global stability of the positive equilibrium.

Theorem 3.1

Assume that (2.3) holds, then the unique positive equilibrium is globally asymptotically stable.

Proof

Set

P1=x(a1b1+c1x−d1−e1x)−axy,Q1=ayβ+yy(x−y). (3.1)

From Theorem 2.2 system (1.6) admits an unique local stable positive equilibrium C(x^*, y^*). Also, from Theorem 2.3, A(0, 0) and B(u^*, 0) is unstable. To ensure C(x^*, y^*) is globally asymptotically stable, we consider the Dulac function u₁(x, y) = x⁻¹y⁻², then

∂(u1P1)∂x+∂(u1Q1)∂y=1xy2a1c1x+b1−d1−e1x+x−a1c1c1x+b12−e1)−ay)−aβ+yx−ax−yβ+y2x−1x2y2xa1c1x+b1−d1−e1x−axy=−1x(c1x+b1)2y2(β+y)2K(x,y),

where

K(x,y)=aβc12x2y2+ac12x3y2+β2c12e1x3+2βc12e1x3y+c12e1x3y2+2ab1βc1xy2+2ab1c1x2y2+2b1β2c1e1x2+4b1βc1e1x2y+2b1c1e1x2y2+ab12βy2+ab12xy2+b12β2e1x+2b12βe1xy+b12e1xy2+a1β2c1x+2a1βc1xy+a1c1xy2.

Hence

∂(u1P1)∂x+∂(u1Q1)∂y<0forallx>0,y>0.

By Dulac Theorem [41], there is no closed orbit in area R2+ . So C(x^*, y^*) is globally asymptotically stable. This completes the proof of Theorem 3.1.

4 Global attractivity of system (1.7)

As far as system (1.7) is concerned, one of the most important topics is to obtain a set of sufficient conditions to ensure the global attractivity of the positive equilibrium, since which means the stale coexistence of the two species. Before we state and prove the main result of this section, we need to introduce two lemmas.

Lemma 4.1

[35] Let x : R → R be a bounded nonnegative continuous function, and let k : [0, +∞) → (0, +∞) be a continuous kernel such that ∫0∞k(s)ds=1. Then

lim inft→+∞x(t)≤lim inft→+∞∫−∞tk(t−s)x(s)ds≤lim supt→+∞∫−∞tk(t−s)x(s)ds≤lim supt→+∞x(t).

Lemma 4.2

[35] If a > 0, b > 0 and ẋ ≥ x(b − ax), when t ≥ 0 and x(0) > 0, we have

lim inft→+∞x(t)≥ba.

If a > 0, b > 0 and ẋ ≤ x(b − ax), when t ≥ 0 and x(0) > 0, we have

lim supt→+∞x(t)≤ba.

Lemma 4.3

Assume that a1b1>d1, then equation F(x)=a1b1+c1x−d1−e1x=0 admits unique positive solution x^*, also, x^* is the decreasing function of d₁.

Proof

One could easily see that the equation F(x) = 0 admits a unique positive solution

x∗=12−b1e1−c1d1+Δc1e1,

where

Δ=b12e12−2b1c1d1e1+c12d12+4a1c1e1.

It immediately follows from the fact

dx∗dd1=−12b1e1−c1d1+ΔΔe1<0

that x^* is the decreasing function of d₁. This ends the proof of Lemma 4.3.

Concerned with the global attractivity of the positive equilibrium of system (1.7), we have the following result.

Theorem 4.1

Assume that

a1b1>d1+au∗

holds, where u^* is defined by (2.6), then system (1.7) admits a unique positive equilibrium which is globally attractive.

Proof

The positive solution of system (1.7) satisfies the equation

a1b1+c1x−d1−e1x=ay=0,x=y. (4.1)

Obviously, under the assumption of Theorem 4.1, system (4.1) admits a unique positive solution C(x^*, y^*).

To end the proof of Theorem 4.1, it is enough to show that C(x^*, y^*) is globally attractive.

It follows from (4.1) that there exists a ε > 0 enough small such that

a1b1>d1+a(u∗+ε). (4.2)

Let (x(t), y(t)) be any positive solution of system (1.7). From system (1.7) it follows that

dxdt≤x(a1b1+c1x−d1−e1x).

Consider the equation

du1dt=u1(a1b1+c1u1−d1−e1u1). (4.3)

It follows from Lemma 2.1 that (4.3) admits a unique globally stable positive equilibrium u^*, where u^* is defined by (2.6). By using the differential inequality theory, any positive solution of (1.7) satisfies

lim supt→+∞x(t)≤limt→+∞u(t)=u∗, (4.4)

and so, from Lemma 4.1 we have

lim supt→+∞∫−∞tK2(t−s)x(s)ds≤u∗. (4.5)

Hence, there exists a T₁₁ > 0 such that

x(t)<u∗+ε2=defM1(1), (4.6)

and

∫−∞tK2(t−s)x(s)ds<u∗+ε2=defM1(1). (4.7)

For t > T₁₁, it follows from the second equation of system (1.7) and (4.7) that

dydt≤ayβ+yy(M1(1)−y). (4.8)

Consider the equation

du2dt=au2β+u2u2(M1(1)−u2). (4.9)

It follows from Lemma 2.2 that (4.9) admits a unique globally stable positive equilibrium

u2∗=u∗+ε2. (4.10)

By differential inequality theory, any positive solution of (1.7) satisfies

lim supt→+∞y(t)≤M1(1), (4.11)

and so, from Lemma 4.1 we have

lim supt→+∞∫−∞tK1(t−s)y(s)ds≤M1(1). (4.12)

Hence, there exists a T₁₂ > T₁₁ such that

y(t)<M1(1)+ε2=defM2(1), (4.13)

and

∫−∞tK1(t−s)y(s)ds<M1(1)+ε2=defM2(1). (4.14)

For t > T₁₂, it follows from the first equation of system (1.7) and (4.14) that

dxdt≥x(a1b1+c1x−d1−e1x−aM2(1)). (4.15)

Now let’s consider the equation

dv1dt=v1(a1b1+c1v1−d1−e1v1−aM2(1)). (4.16)

Since

a1>b1(d1+a(u∗+ε))=b1(d1+aM2(1)),

it follows from Lemma 2.1 that system (4.16) admits a unique positive equilibrium v1∗ , which is globally asymptotically stable. Applying the differential inequality theory to (4.15) leads to

lim inft→+∞x(t)≥limt→+∞v(t)=v1∗,

and so, from Lemma 4.1 we have

lim inft→+∞∫−∞tK2(t−s)x(s)ds≥v1∗.

It follows from above inequality that there exists an enough large T₁₃ > T₁₂ such that for all t ≥ T₁₃, the following inequalities hold.

x(t)>v1∗−ε4=defm1(1), (4.17)

∫−∞tK2(t−s)x(s)ds>v1∗−ε4=defm1(1). (4.18)

From the second equation of system (1.7), for t ≥ T₁₃, we have

dydt≥ayβ+yy(m1(1)−y). (4.19)

Consider the equation

dv2dt=av2β+v2v2(m1(1)−v2). (4.20)

It follows from Lemma 2.2 that (4.20) admits a unique globally stable positive equilibrium

v2∗=m1(1). (4.21)

By using the differential inequality theory, any solution of (4.19) satisfies

lim inft→+∞y(t)≥limt→+∞v2(t)=m1(1),

and so, from Lemma 4.1 we have

lim inft→+∞∫−∞tK1(t−s)y(s)ds≥m1(1).

It follows from above inequality that there exists an enough large T₁₄ > T₁₃ such that for all t ≥ T₁₄, the following inequalities hold

y(t)>m1(1)−ε2=defm2(1), (4.22)

∫−∞tK1(t−s)y(s)ds>m1(1)−ε2=defm2(1). (4.23)

For t > T₁₄, it follows from (4.23) and the first equation of system (1.7) that

dxdt≤x(a1b1+c1x−d1−e1x)−axm2(1).

Consider the equation

du1dt=u1(a1b1+c1u1−d1−am2(1)−e1u1). (4.24)

It follows from Lemma 2.1 that (4.24) admits a unique globally stable positive equilibrium um2(1)∗ , from Lemma 4.3, one could see that um2(1)∗ < u^*. By using the differential inequality theory, any positive solution of (1.7) satisfies

lim supt→+∞x(t)≤limt→+∞u(t)=um2(1)∗, (4.25)

and so, from Lemma 4.1 we have

lim supt→+∞∫−∞tK2(t−s)x(s)ds≤um2(1)∗. (4.26)

Hence, there exists a T₂₁ > 0 such that

x(t)<um2(1)∗+ε4=defM1(2), (4.27)

and

∫−∞tK2(t−s)x(s)ds<um2(1)∗+ε4=defM1(2). (4.28)

For t > T₂₁, it follows from the second equation of system (1.7) and (4.28) that

dydt≤ayβ+yy(M1(2)−y). (4.29)

Consider the equation

du2dt=au2β+u2u2(M1(2)−u2). (4.30)

It follows from Lemma 2.2 that (4.30) admits a unique globally stable positive equilibrium M1(2) . By using the differential inequality theory, any positive solution of (1.7) satisfies

lim supt→+∞y(t)<M1(2), (4.31)

and so, from Lemma 4.1 we have

lim supt→+∞∫−∞tK1(t−s)y(s)ds≤M1(2). (4.32)

Hence, there exists a T₂₂ > T₂₁ such that

y(t)<M1(2)+ε=defM2(2), (4.33)

and

∫−∞tK1(t−s)y(s)ds<M1(2)+ε=defM2(2). (4.34)

For t > T₂₂, it follows from the first equation of system (1.7) and (4.34) that

dxdt≥x(a1b1+c1x−d1−e1x−aM2(2)). (4.35)

Now let’s consider the equation

dv1dt=v1(a1b1+c1v1−d1−e1v1−aM2(2)). (4.36)

Since

M2(2)<M2(1),

it follows from (4.1) that

a1>b1(d1+aM2(2)).

Hence, applying Lemma 2.1 to system (4.36), one could see that (4.36) admits a unique positive equilibrium vM2(2)∗ , which is globally asymptotically stable. Also, from Lemma 4.3, we have

vM2(2)∗>v1∗.

Applying the differential inequality theory to (4.35) leads to

lim inft→+∞x(t)≥limt→+∞v(t)=vM2(2)∗,

and so, from Lemma 4.1 we have

lim inft→+∞∫−∞tK2(t−s)x(s)ds≥vM2(2)∗.

It follows from above inequality that there exists an enough large T₁₃ > T₁₂ such that for all t ≥ T₁₃, the following inequalities hold.

x(t)>vM2(2)∗−ε4=defm1(2), (4.37)

∫−∞tK2(t−s)x(s)ds>vM2(2)∗−ε4=defm1(2). (4.38)

From the second equation of system (1.7), we have

dydt≥ayβ+yy(m1(2)−y). (4.39)

Consider the equation

dv2dt=av2β+v2v2(m1(2)−v2). (4.40)

It follows from Lemma 2.2 that (4.40) admits a unique globally stable positive equilibrium m1(2) . By using the differential inequality theory, any solution of (4.39) satisfies

lim inft→+∞y(t)≥limt→+∞v2(t)=m1(2), (4.41)

and so, from Lemma 4.1 we have

lim inft→+∞∫−∞tK1(t−s)y(s)ds≥m1(2).

It follows from above inequality that there exists an enough large T₂₄ > T₂₃ such that for all t ≥ T₂₄, the following inequalities hold.

y(t)>m1(2)−ε2=defm2(2), (4.42)

∫−∞tK1(t−s)y(s)ds>m1(2)−ε2=defm2(2). (4.43)

One could easily see that

M1(2)=um2(1)∗+ε4<u∗+ε2=M1(1);M2(2)=M1(2)+ε<u∗+ε=M2(1);m1(2)=vM2(2)∗−ε4>v1∗−ε4=m1(1);m2(2)=m1(2)−ε2>m1(1)−ε2=m2(1). (4.44)

Repeating the above procedure, we get four sequences Mi(n),mi(n), i = 1, 2, n = 1, 2, ⋯, such that for n ≥ 2

a1b1+c1(M1(n)−ε2n)−d1−e1(M1(n)−ε2n)−am2(n−1)=0;M2(n)=M1(n)+ε;a1b1+c1(m1(n)+ε2n)−d1−e1(m1(n)+ε2n)−aM2(n)=0;m2(n)=m1(n)−ε2. (4.45)

Obviously

mi(n)<Ni(t)<Mi(n),fort≥T2n,i=1,2.

We claim that sequences Mi(n) , i = 1, 2 are non-increasing, and sequences mi(n) , i = 1, 2 are non-decreasing. To prove this claim, we will carry out by induction. Firstly, from (4.44) we have

Mi(2)<Mi(1),mi(2)>mi(1),i=1,2.

Let us assume now that our claim is true for n, that is,

Mi(n)<Mi(n−1),mi(n)>mi(n−1),i=1,2.

Then, by Lemma 4.3, we immediately obtain

M1(n+1)<M1(n);M2(n+1)<M2(n);m1(n+1)>m1(n);m2(n+1)>m2(n).

Therefore

limt→+∞M1(n)=x¯,limt→+∞M2(n)=y¯,limt→+∞m1(n)=x_,limt→+∞m2(n)=y_.

Letting n → +∞ in (4.45), we obtain

a1b1+c1x¯−d1−e1x¯−ay_=0;y¯=x¯;a1b1+c1x_−d1−e1x_−ay¯=0;y_=x_. (4.46)

(4.46) shows that (x, y) and (x, y) are solutions of (4.1), which (4.1) has a unique positive solution C(x^*, y^*). Hence, we conclude that

x¯=x_=x∗,y¯=y_=y∗,

that is

limt→+∞x(t)=x∗limt→+∞y(t)=y∗.

Thus, the unique interior equilibrium C(x^*, y^*) is globally attractive. This completes the proof of Theorem 4.1.

5 Numeric simulations

Now let’s consider the following four examples.

Example 5.1

dxdt=x(12+x−1−x)−xy,dydt=yy1+y(x−y). (5.1)

In this system, corresponding to system (1.6), we take a₁ = c₁ = d₁ = e₁ = a = β = 1, b₁ = 2, since a₁ < b₁d₁, it follows from Theorem 2.2 that the boundary equilibrium A(0, 0) is globally asymptotically stable. Figure 1 supports this assertion.

$Figure 1 Dynamic behavior of system (5.1), here the initial condition (x(0), y(0)) = (1, 1), (1, 0.3), (1, 0.1) and (1, 0.6), respectively.$

Figure 1

Dynamic behavior of system (5.1), here the initial condition (x(0), y(0)) = (1, 1), (1, 0.3), (1, 0.1) and (1, 0.6), respectively.

Example 5.2

dxdt=x(52+x−1−6x)−xy,dydt=yy1+y(x−y). (5.2)

In this system, corresponding to system (1.6), we take a₁ = c₁ = d₁ = e₁ = a = β = 1, b₁ = 2, e₁ = 6, since a1b1=52>1+512=d1+aa1b1−d1e1, it follows from corollary 2.1 that system (4.2) is permanent. Figure 2 supports this assertion.

$Figure 2 Dynamic behavior of system (5.2), here the initial condition (x(0), y(0)) = (0.05, 0.5), (0.05, 0.3), (0.05, 0.5), (0.3, 0.1), (0.3, 0.3), (0.3, 0.5) and (0.3, 0.2), respectively.$

Figure 2

Dynamic behavior of system (5.2), here the initial condition (x(0), y(0)) = (0.05, 0.5), (0.05, 0.3), (0.05, 0.5), (0.3, 0.1), (0.3, 0.3), (0.3, 0.5) and (0.3, 0.2), respectively.

Example 5.3

dxdt=x(21+x−1−x)−4xy,dydt=4yy1+y(x−y). (5.3)

In this system, corresponding to system (1.6), we take b₁ = c₁ = d₁ = e₁ = β = 1, a₁ = 2, a = 4, by computation, u^* = 2 − 1, and so, a1b1 = 2 < 1 + 4( 2 − 1) = d₁ + au^*, Hence, the conditions of Theorem 2.1 could not satisfied, however, numeric simulation (Figure 3) shows that the system also admits a unique positive equilibrium which is globally asymptotically stable.

$Figure 3 Dynamic behavior of system (5.3), here the initial condition (x(0), y(0)) = (0.3, 0.5), (0.3, 0.1), (0.3, 0.3), (0.3, 0.4), (0.3, 0.2), (0.1, 0.3), (0.1, 0.1) and (0.1, 0.5), respectively.$

Figure 3

Dynamic behavior of system (5.3), here the initial condition (x(0), y(0)) = (0.3, 0.5), (0.3, 0.1), (0.3, 0.3), (0.3, 0.4), (0.3, 0.2), (0.1, 0.3), (0.1, 0.1) and (0.1, 0.5), respectively.

6 Discussion

During the last decades, many scholars [2, 3, 4, 5, 6, 22, 23, 24, 25] investigated the influence of Allee effect on the dynamic behaviors of ecosystem. Also, there are several scholars [32, 33, 34, 35, 36, 37, 38] investigated the almost periodic solution of the ecosystem. However, all of those studies are based on the traditional Logistic model.

In this paper, we argued that the nonlinear birth rate of the prey species is more suitable, and take Beverton-Holt function [28] as the birth rate, this leads to system (1.6).

We showed that depending on the range of the birth rate parameter, the system maybe collapse or the two species could be coexist in a stable state. That is, the birth rate plays essential role on the dynamic behaviors of system (1.6).

For the system with infinite delay, by using the iterative method, we could able to show that inequality (2.3) is enough to ensure the globally attractive of the positive equilibrium. We mentioned here that with the nonlinear birth rate, the method used in the paper [34] and [36] could not be applied to our system directly, to overcome this difficulty, we developing some new analysis technique.

At the end of the paper, we would like to point out that the results obtained in this paper are the sufficient ones, as was shown in Example 4.3, there are still have room to improve our results, we leave this for future study. Also, it seems interesting to investigate the dynamic behaviors of the non-autonomous case of system (1.6), specially focus on the permanence, extinction and almost periodic solution, we also leave this for future investigation.

Acknowledgements

The authors would like to thank Dr. Yu Liu for useful discussion about the mathematical modeling. The research was supported by the National Natural Science Foundation of China under Grant (11601085) and the Natural Science Foundation of Fujian Province (2017J01400).

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Received: 2019-05-10

Accepted: 2019-06-08

Published Online: 2019-11-08

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https://doi.org/10.1515/math-2019-0082

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predator; prey; Allee effect; global stability; density dependent birth rate

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