Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays

Mengxin He; Zhong Li; Fengde Chen

doi:10.1515/math-2019-0062

Article Open Access

Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays

Mengxin He , Zhong Li and Fengde Chen

Published/Copyright: July 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we consider a nonautonomous two-species impulsive competitive system with infinite delays. By the impulsive comparison theorem and some mathematical analysis, we investigate the permanence, extinction and global attractivity of the system, as well as the influence of impulse perturbation on the dynamic behaviors of this system. For the logistic type impulsive equation with infinite delay, our results improve those of Xuxin Yang, Weibing Wang and Jianhua Shen [Permanence of a logistic type impulsive equation with infinite delay, Applied Mathematics Letters, 24(2011), 420-427]. For the corresponding nonautonomous two-species impulsive competitive system without delays, we discuss its permanence, extinction and global attractivity, which weaken and complement the results of Zhijun Liu and Qinglong Wang [An almost periodic competitive system subject to impulsive perturbations, Applied Mathematics and Computation, 231(2014), 377-385].

Keywords: Competitive system; Impulses; Permanence; Extinction; Infinite delays

MSC 2010: 34D23; 34A37

1 Introduction

The logistic system is considered to be one of the most important systems in mathematical ecology, and a great deal of research works have been done based on this system. Because of the seasonal fluctuations in the environment and hereditary factors, many scholars have investigated the logistic system with time delays (see [1, 2, 3, 4, 5, 6, 7, 8]). Noticing that the disturbance of environmental factors at certain time moments can give rise to instantaneous and changes of population density, many scholars have investigated the dynamic behaviors of impulsive differential equations (see [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]). Especially, Yang [21] investigated the following logistic system with infinite delay

x˙(t)=x(t)(a(t)−b(t)∫0+∞K(s)x(t−s)ds),t≥0,t≠tk,x(tk+)=hkx(tk),k=1,2,⋯, (1.1)

with the initial condition x(t) = ϕ(t), t ≤ 0, which is continuous and bounded on (−∞, 0] to [0, +∞) with ϕ(0) > 0. Here a(t) and b(t) are continuous functions, bounded above and below by positive constants; K : [0, +∞) → (0, +∞) is a continuous kernel such that ∫0+∞ K(s)ds = 1; t_k (k = 1, 2, …) are impulse points with lim_k→+∞ t_k = +∞; the impulse perturbations {h_k : k = 1, 2, …} is positive sequences bounded above and below by positive constants. The authors discussed the permanence and global asymptotical stability of system (1.1) under the following condition

0<infk≥1hk≤hk≤1(k=1,2,⋯)andinfk≥1(hk−hk−1)>0,

which implies h_k ≤ 1 is an increase sequence.

On the other hand, competition for limited resources among ecologically similar species has been intensively investigated by many scholars due to its extensive prevalence and its importance on determining the structure of animal and plant communities, the diversity and the evolution of species. The famous Lotka-Volterra competition system has been studied extensively (see [22, 23, 24]). Naturally, impulse perturbations have been introduced into competitive systems and many excellent results have been obtained (see [13, 18, 25, 26, 27, 28, 29, 30, 31, 32]). Recently, Liu and Wang [32] considered an almost periodic impulsive competitive system of the form

x1˙(t)=x1(t)(r1(t)−a1(t)x1(t)−b1(t)x2(t)1+x2(t)),x2˙(t)=x2(t)(r2(t)−a2(t)x2(t)−b2(t)x1(t)1+x1(t)),t≥0,t≠tk,x1(tk+)=h1kx1(tk),x2(tk+)=h2kx2(tk),k=1,2,⋯. (1.2)

For any given continuous function f(t), let f_L and f_M denote inf_0≤t<+∞ f(t) and sup_0≤t<+∞ f(t), respectively. The authors discussed the permanence of system (1.2) under the following conditions:

Π_{0<t_k<t} h_ik, i = 1, 2, are bounded above and below by positive constants for all t > 0;
r_iL − b_iM > 0, i = 1, 2.

But the authors did not consider its competition exclusion, global attractivity and extinction. For the permanence of system (1.2), we also want to know whether conditions (H1) and (H2) can be weakened? To answer this question, we first introduce the following example.

Example 1.1

For system (1.2), let r1(t)=0.2t+0.4t+1, r2(t)=t+22t+1, a1(t)=5+4sin⁡2t, a2(t) = 11.5 + 8.5 sin 2t, b₁(t) = 2 + sin t, b₂(t) = 1 + 0.5 sin t, h_1k = 5.5 − 0.5 cos k, h_2k = 3.5 + 0.5 sin 2k and t_k = k + 1k . Obviously, condition (H2) does not hold, but Figure 1 shows that system (1.2) is permanent.

$Figure 1 System (1.2) with the initial conditions (0.01, 0.01)T and (0.03, 0.03)Trespectively$

Figure 1

System (1.2) with the initial conditions (0.01, 0.01)^T and (0.03, 0.03)^Trespectively

This example gives a certain answer to the above question. So it requires us to give its strict mathematical verification and to discuss the competition exclusion, global attractivity and extinction of (1.2). Our results improve and complement the corresponding results of Liu and Wang [32].

Motivated by the above papers, in this paper we consider the following system

x1˙(t)=x1(t)(r1(t)−a1(t)∫0+∞K1(s)x1(t−s)ds−b1(t)x2(t)1+x2(t)),x2˙(t)=x2(t)(r2(t)−a2(t)∫0+∞K2(s)x2(t−s)ds−b2(t)x1(t)1+x1(t)),t≥0,t≠tk,x1(tk+)=h1kx1(tk),x2(tk+)=h2kx2(tk),k=1,2,⋯, (1.3)

under an initial condition

xi(u)=ϕi(u)≥0,foru∈(−∞,0]withϕi∈C((−∞,0],[0,+∞))andϕi(0)>0. (1.4)

Here x₁(t) and x₂(t) are population densities of species x₁ and x₂ at time t respectively; r₁(t) > 0 and r₂(t) > 0 are the growth rates; a₁(t) > 0 and a₂(t) > 0 are the effects of intra-specific competition; r_i(t) and a_i(t) are continuous functions, bounded above and below by positive constants for all t > 0; the continuous functions b₁(t) ≥ 0 and b₂(t) ≥ 0 are the rates of inter-specific competition, which are bounded for all t > 0; K_i : [0, +∞) → (0, +∞) (i = 1, 2) are continuous kernels such that ∫0+∞ K_i(s)ds = 1; 0 < t₁ < t₂ < … < t_k < t_k+1 < … are impulse points with lim_k→+∞ t_k = +∞; the impulse perturbations {h_ik : k = 1, 2, …} (i = 1, 2) are positive sequences bounded above and below by positive constants.

2 Preliminaries

In this section, we present the following definitions and lemmas which are useful in proving our main results.

Let PC([0, +∞), R²) = {ϕ : [0, +∞) → R⁺ × R⁺, ϕ is continuous for t ≠ t_k. Also ϕ(tk−) and ϕ(tk+) exist, and ϕ(tk−) = ϕ(t_k), k = 1, 2, …}. By the basic theories of impulsive differential equations in [9][10], system (1.3) has a unique solution X(t) = X(t, X₀) ∈ PC([0, +∞), R⁺ × R⁺). For any sequence {h_k}, let h_L and h_M denote inf_k∈Z h_k and sup_k∈Z h_k, respectively. For the sequence {t_k}, denote sup_k∈Z tk1 = sup_k∈Z(t_k+1 − t_k) = η and inf_k∈Z tk1 = θ. Obviously η ≥ θ > 0.

Define G_k = (t_k−1, t_k) × R⁺ × R⁺, k = 1, 2, …; G = ⋃k=1+∞ G_k; V₀ = {V ∈ C[G, R⁺], there exists the limits V(tk−,X0), V(tk+,X0),V(tk−,X0)=V(tk,X0) , and V is locally Lipschitz continuous}.

Definition 2.1

Let V ∈ V₀. For any (t, X(t)) ∈ [t_k−1, t_k) × R⁺ × R⁺, the right-hand derivative D⁺ V(t, X(t)) along the solution X(t, X₀) of system (1.3) is defined by

D+V(t,X(t))=lim infh→0+1h[V(t+h,X(t+h))−V(t,X(t))].

Lemma 2.1

(see [10]) Assume that m ∈ PC[R⁺, R] with points of discontinuity at t = t_k is left continuous at t = t_k, k = 1, 2, …, and that

D+m(t)≤g(t,m(t)),t≠tk,k=1,2,⋯,m(tk+)≤ϕk(m(tk)),t=tk,k=1,2,⋯, (2.1)

where g ∈ C[R⁺ × R⁺, R], ϕ_k ∈ C[R, R] and ϕ_k(u) is nondecreasing in u for each k = 1, 2, …. Let r(t) be the maximal solution of the scalar impulsive differential equation

u˙=g(t,u),t≠tk,k=1,2,⋯,u(tk+)=ϕk(u(tk))≥0,t=tk,tk>t0,k=1,2,⋯,u(t0+)=u0, (2.2)

existing on [t₀, +∞), then m(t0+) ≤ u₀ implies m(t) ≤ r(t), t ≥ t₀.

Remark 2.1

(see [10]) In Lemma 2.1, assume inequalities (2.1) reverse. Let p(t) be the minimal solution of (2.1) existing on [t₀, +∞), then p(t0+) ≥ u₀ implies p(t) ≥ r(t), t ≥ t₀.

Consider the following impulsive system

y˙(t)=y(t)(a−by(t)),t≠tk,y(tk+)=hky(tk),k=1,2,⋯, (2.3)

where a and b are positive constants.

Lemma 2.2

(see [17]), Let y(t) be any positive solution of system (2.3). It follows that:

If h_L ≥ 1, then

aη+ln⁡hLbηhL≤lim inft→+∞y(t)≤lim supt→+∞y(t)≤(aθ+ln⁡hM)hMbθ.
If h_L < 1, h_M < 1 and aθ + ln h_L > 0, then

(aθ+ln⁡hL)hLbθ≤lim inft→+∞y(t)≤lim supt→+∞y(t)≤aη+ln⁡hMbηhM.
If h_L < 1, h_M ≥ 1 and aθ + ln h_L > 0, then

(aθ+ln⁡hL)hLbθ≤lim inft→+∞y(t)≤lim supt→+∞y(t)≤(aθ+ln⁡hM)hMbθ.

Lemma 2.3

Let y(t) be any positive solution of system (2.3). Assume that aη + ln h_M ≤ 0. Then limt→+∞ y(t) = 0.

Proof

Let z(t) = 1/y(t), then system (2.3) is transformed into

z˙(t)=−az(t)+b,t≠tk,z(tk+)=1hkz(tk),k=1,2,⋯,

According to [9], for any T > 0, we can obtain

z(t)=∏T≤tk<t1hke−a(t−T)z(T+)+b∫Tt∏s≤tk<t1hke−a(t−s)ds.

First we consider aη + ln h_M = 0, that is eahM1/η = 1 and h_M < 1. According to [17], we obtain

z(t)≥(1hM)t−Tη−1e−a(t−T)z(T+)+b∫Tt(1hM)t−sη−1e−a(t−s)ds=hM(z(T+)+bt−bT)→+∞,t→+∞.

Next consider aη + ln h_M < 0, that is eahM1/η < 1 and h_M < 1, then

z(t)≥(1hM)t−Tη−1e−a(t−T)z(T+)+bηhMaη+ln⁡hM[1−(1hM)t−Tηe−a(t−T)]≥(hMz(T+)−bηhMaη+ln⁡hM)(1eahM1/η)t−T+bηhMaη+ln⁡hM→+∞,t→+∞,

because of bηhMaη+ln⁡hM < 0. Therefore, it follows from the positivity of y(t) and the relationship between z(t) and y(t) that limt→+∞ y(t) = 0. This completes the proof of Lemma 2.3.

Lemma 2.4

Let (x₁(t), x₂(t))^T be any solution of system (1.3) with (1.4), then x_i(t) > 0, i = 1, 2, for all t ≥ 0.

Proof

From the ith equation of (1.3) with (1.4) (i = 1, 2), we can obtain

xi(t)=ϕi(0)(∏0<tk<thik)exp∫0t(ri(u)−ai(u)∫0+∞Ki(s)xi(u−s)ds−bi(u)xj(u)1+xj(u))du>0,

where 1 ≤ j ≤ 2, i ≠ j, which completes the proof of Lemma 2.4.

Lemma 2.5

For any y ∈ PC([0, +∞), R⁺), let k : [0, +∞) → (0, +∞) be a continuous kernel such that ∫0+∞ k(s)ds = 1. Then

lim inft→+∞y(t)≤lim inft→+∞∫0+∞k(s)y(t−s)ds≤lim supt→+∞∫0+∞k(s)y(t−s)ds≤lim supt→+∞y(t).

The proof is similar to that of Lemma 3 in [24], so we omit it.

3 Main results

In this section, we present the main results of this paper. First we study the coexistence of system (1.3).

Theorem 3.1

Let (x₁(t), x₂(t))^T be any solution of system (1.3) with (1.4), i = 1, 2. Assume that

(riL−biMMj1+Mj)θ+ln⁡hiL>0,1≤i,j≤2,i≠j,riL−biMMj1+Mj>0, (3.1)

then mi≤lim inft→+∞xi(t)≤lim supt→+∞xi(t)≤Mi , i = 1, 2, where

Mi=max{(riMθ+ln⁡hiM)hiM2aiLθR1i,riMη+ln⁡hiMaiLhiM2ηR2i},mi=min{(riL−biMMj1+Mj)η+ln⁡hiLaiMH1iηhiL2,[(riL−biMMj1+Mj)θ+ln⁡hiL]hiL2aiMH2iθ}

with

R1i=∫0+∞(hiM1θeriM)−sKi(s)ds<+∞,R2i=∫0+∞(hiM1ηeriM)−sKi(s)ds<+∞,H1i=∫0+∞(1hiL)sηexp⁡{−(riL−aiMMi−biMMj1+Mj)s}Ki(s)ds<+∞,H2i=∫0+∞(1hiL)sθexp⁡{−(riL−aiMMi−biMMj1+Mj)s}Ki(s)ds<+∞.

Proof

From (1.3), we can obtain for i = 1, 2 that

xi˙(t)≤riMxi(t),xi(tk+)=hikxi(tk).

Then according to Lemma 2.1, we have

xi(t−s)≥(∏t−s≤tk<t1hik)e−riMsxi(t).

For i = 1, 2, substituting this into the ith equation of (1.3), we obtain

xi˙(t)≤xi(t)[riM−aiL∫0+∞(∏t−s≤tk<t1hik)e−riMsKi(s)dsxi(t)]. (3.2)

If h_iM ≥ 1, it follows that

xi˙(t)≤xi(t)[riM−aiL∫0+∞(1hiM)sθ+1e−riMsKi(s)dsxi(t)]≤xi(t)[riM−aiLR1ihiMxi(t)],

where R1i=∫0+∞(hiM1θeriM)−sKi(s)ds . According to Lemma 2.2, we obtain that

lim supt→+∞xi(t)≤(riMθ+ln⁡hiM)hiM2aiLR1iθ,i=1,2.
If h_iM < 1, we have

xi˙(t)≤xi(t)[riM−aiL∫0+∞(1hiM)sη−1exp⁡{−riMs}Ki(s)dsxi(t)]≤xi(t)(riM−aiLhiMR2ixi(t)),

where R2i=∫0+∞(hiM1ηeriM)−sKi(s)ds . Again from Lemma 2.2 we have

lim supt→+∞xi(t)≤riMη+ln⁡hiMaiLhiM2R2iη.

All the above analysis show that

lim supt→+∞xi(t)≤max{(riMθ+ln⁡hiM)hiM2aiLR1iθ,riMη+ln⁡hiMaiLhiM2R2iη}≜Mi,i=1,2. (3.3)

Therefore for any given ε > 0 satisfying

(riL−biMMj+ε1+Mj+ε)θ+ln⁡hiL>0,1≤i,j≤2,i≠j,riL−biMMj+ε1+Mj+ε>0, (3.4)

there exists a T > 0 such that for t > T, x_i(t) ≤ M_i + ε, i = 1, 2.

Substituting this into system (1.3), it follows from Lemma 2.5 that, for 1 ≤ i, j ≤ 2 and i ≠ j

xi˙(t)≥xi(t)[riL−aiM(Mi+ε)−biM(Mj+ε)1+Mj+ε],xi(tk+)=hikxi(tk).

We can easily obtain that

xi(t−s)≤(∏t−s≤tk<t1hik)exp⁡{−(riL−aiM(Mi+ε)−biM(Mj+ε)1+Mj+ε)s}xi(t).

Substituting this into the ith equation of system (1.3) gives rise to

xi˙(t)≥xi(t)[riL−biM(Mj+ε)1+Mj+ε−aiM∫0+∞(∏t−s≤tk<t1hik)exp⁡{−(riL−aiM(Mi+ε)−biM(Mj+ε)1+Mj+ε)s}Ki(s)dsxi(t)].

Next we prove lim inft→+∞ x_i(t) ≥ m_i.
If h_iL ≥ 1, we deduce that

xi˙(t)≥xi(t)[riL−biM(Mj+ε)1+Mj+ε−aiM∫0+∞(1hiL)sη−1exp⁡{−(riL−aiM(Mi+ε)−biM(Mj+ε)1+Mj+ε)s}Ki(s)dsxi(t)].

By setting ε → 0, it follows from Lemma 2.2 that

lim inft→+∞xi(t)≥(riL−biMMj1+Mj)η+ln⁡hiLaiMH1ihiL2η,

where

H1i=∫0+∞(1hiL)sηexp⁡{−(riL−aiMMi−biMMj1+Mj)s}Ki(s)ds.
If h_iL < 1, we obtain

xi˙(t)≥xi(t)[riL−biM(Mj+ε)1+Mj+ε−aiM∫0+∞(1hiL)sθ+1exp⁡{−(riL−aiM(Mi+ε)−biM(Mj+ε)1+Mj+ε)s}Ki(s)dsxi(t)].

By setting ε → 0, it follows from Lemma 2.2 that

lim inft→+∞xi(t)≥[(riL−biMMj1+Mj)θ+ln⁡hiL]hiL2aiMH2iθ,

where

H2i=∫0+∞(1hiL)sθexp⁡{−(riL−aiMMi−biMMj1+Mj)s}Ki(s)ds.

Thus,

lim inft→+∞xi(t)≥min{[(riL−biMMj1+Mj)θ+ln⁡hiL]hiL2aiMθH2i,(riL−biMMj1+Mj)η+ln⁡hiLaiMηhiL2H1i}≜mi,i=1,2.

This proves the permanence of (1.3).

Theorem 3.2

Suppose that the conditions of Theorem 3.1 holds, and there exist σ_i > 0 and ρ_i > 0 such that

∫0+∞sKi(s)ds=σi,i=1,2 (3.5)

and

2a1LM1ρ1−b1Mρ1m12−b2Mρ2−2a1M2ρ1σ1−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2>0,2a2Lρ2M2−b1Mρ1−b2Mρ2m22−2a2M2ρ2σ2−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2>0 (3.6)

where M_i and m_i (i = 1, 2) are defined in Theorem 3.1. Then for any two solutions (x₁(t), x₂(t))^T and (y₁(t), y₂(t))^T of system (1.3) with (1.4), there are

limt→+∞|xi(t)−yi(t)|=0,fori=1,2.

Proof

Let (x₁(t), x₂(t))^T and (y₁(t), y₂(t))^T be any two solutions of system (1.3) with (1.4). From Theorem 3.1, for any ε₁ > 0 satisfying 0 < ε₁ < min{m₁, m₂}, there exist δ > 0 such that

2a1Lρ1M1+ε1−b1Mρ1(m1−ε1)2−b2Mρ2−2a1M2ρ1σ1−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2≥δ,2a2Lρ2M2+ε1−b1Mρ1−b2Mρ2(m2−ε1)2−2a2M2ρ2σ2−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2≥δ, (3.7)

and T₁ > 0 such that for t > T₁,

mi−ε1≤xi,yi≤Mi+ε1,i=1,2. (3.8)

Define a Lyapunov function as follows

V1i(t)=(ln⁡xi(t)−ln⁡yi(t)−∫0+∞∫t−stKi(s)ai(v+s)(xi(v)−yi(v))dvds)2,i=1,2.

For t > T₁ and t ≠ t_k, k = 1, 2, …, calculating the upper right derivatives of V_1i(t) with 1 ≤ i, j ≤ 2 and i ≠ j, we have

D+V1i(t)=2(ln⁡xi(t)−ln⁡yi(t)−∫0+∞∫t−stKi(s)ai(v+s)(xi(v)−yi(v))dvds)×[−(xi(t)−yi(t))∫0+∞ai(t+s)Ki(s)ds−bi(t)(xj(t)1+xj(t)−yj(t)1+yj(t))]≤−2aiL(xi(t)−yi(t))(ln⁡xi(t)−ln⁡yi(t))+2biM(1+ξj(t))2|ln⁡xi(t)−ln⁡yi(t)||xj(t)−yj(t)|+2[aiM2|xi(t)−yi(t)|+biMaiM(1+ξj(t))2|xj(t)−yj(t)|]∫0+∞∫t−stKi(s)|xi(v)−yi(v)|dvds≤−2aiL(xi(t)−yi(t))(ln⁡xi(t)−ln⁡yi(t))+2biM|ln⁡xi(t)−ln⁡yi(t)||xj(t)−yj(t)|+aiM2σi|xi(t)−yi(t)|2+biMaiMσi|xj(t)−yj(t)|2+(aiM2+biMaiM)∫0+∞∫t−stKi(s)|xi(v)−yi(v)|2dvds,

where ξ_j(t) lies between x_j(t) and y_j(t), j = 1, 2.

For i = 1, 2, define

V2i(t)=(aiM2+biMaiM)∫0+∞∫t−st∫vtKi(s)|xi(u)−yi(u)|2dudvds.

For t > T₁ and t ≠ t_k, k = 1, 2, …, calculating the upper right derivatives of V_2i(t), it follows that

D+V2i(t)=(aiM2+biMaiM)(σi|xi(t)−yi(t)|2−∫0+∞∫t−stKi(s)|xi(v)−yi(v)|2dvds).

Denote V_i(t) = V_1i(t) + V_2i(t) for i = 1, 2. Therefore, for t > T₁ and t ≠ t_k, k = 1, 2, …,

D+V(t)=D+(ρ1V1(t)+ρ2V2(t))≤∑i=12ρi[−2aiL(xi(t)−yi(t))(ln⁡xi(t)−ln⁡yi(t))+2biM|ln⁡xi(t)−ln⁡yi(t)||xj(t)−yj(t)|+(2aiM2+biMaiM)σi|xi(t)−yi(t)|2+biMaiMσi|xj(t)−yj(t)|2]=−2a1Lρ1(x1(t)−y1(t))(ln⁡x1(t)−ln⁡y1(t))+2b1Mρ1|ln⁡x1(t)−ln⁡y1(t)||x2(t)−y2(t)|−2a2Lρ2(x2(t)−y2(t))(ln⁡x2(t)−ln⁡y2(t))+2b2Mρ2|ln⁡x2(t)−ln⁡y2(t)||x1(t)−y1(t)|+(2a1M2ρ1σ1+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x1(t)−y1(t)|2+(2a2M2ρ2σ2+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x2(t)−y2(t)|2≤−2a1Lρ1ξ11(t)|x1(t)−y1(t)|2+b1Mρ1ξ122(t)|x1(t)−y1(t)|2+b1Mρ1|x2(t)−y2(t)|2−2a2Lρ2ξ21(t)|x2(t)−y2(t)|2+b2Mρ2ξ222(t)|x2(t)−y2(t)|2+b2Mρ2|x1(t)−y1(t)|2+(2a1M2ρ1σ1+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x1(t)−y1(t)|2+(2a2M2ρ2σ2+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x2(t)−y2(t)|2≤(−2a1Lρ1M1+ε1+b1Mρ1(m1−ε1)2+b2Mρ2+2a1M2ρ1σ1+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x1(t)−y1(t)|2(−2a2Lρ2M2+ε1+b2Mρ2(m2−ε1)2+b1Mρ1+2a2M2ρ2σ2+b1Ma1Mρ1σ1+b2Ma2Mρ2σ2)|x2(t)−y2(t)|2,≤−δ(|x1(t)−y1(t)|2+|x2(t)−y2(t)|2),

where ξ_ij(t) (1 ≤ i, j ≤ 2; i ≠ j) lies between x_i(t) and y_i(t), i = 1, 2.

For t = t_k, we can easily verify that V(tk+) = V(t_k). Integrating both sides of the above inequality from T₁ to t, we obtain

V(t)+δ∫T1t(|x1(s)−y1(s)|2+|x2(s)−y2(s)|2)ds≤V(T1+)<+∞.

Therefore, V(t) is bounded on [T₁, +∞) and there is

∫T1+∞(|x1(s)−y1(s)|2+|x2(s)−y2(s)|2)ds<+∞.

Similarly to the analysis of [17], it is obvious that

limt→+∞|x1(t)−y1(t)|=limt→+∞|x2(t)−y2(t)|=0.

This completes the proof of Theorem 3.2.

Next, we consider the competition exclusion of system (1.3).

Theorem 3.3

Let (x₁(t), x₂(t))^T be any solution of system (1.3) with (1.4). Assume that

r1Lθ+ln⁡h1L>0, (3.9)

r2Mη+ln⁡h2M≤0, (3.10)

then the species x₁ is permanent but the species x₂ is extinct, that is

m¯1≤lim inft→+∞x1(t)≤lim supt→+∞x1(t)≤M1andlimt→+∞x2(t)=0,

where M₁ is defined in Theorem 3.1 and

m¯1=min{r1Lη+ln⁡h1La1MH¯11ηh1L2,(r1Lθ+ln⁡h1L)h1L2a1MH¯21θ},

with

H¯11=∫0+∞(1h1L)sηexp⁡{−(r1L−a1MM1)s}K1(s)ds<+∞,H¯21=∫0+∞(1h1L)sθexp⁡{−(r1L−a1MM1)s}K1(s)ds<+∞.

Proof

Since (3.9) implies that r_1M θ + ln h_1L > 0, according to the proof of Theorem 3.1 there is lim supt→+∞ x₁(t) ≤ M₁. Condition (3.10) implies h_2M ≤ 1. Again from the proof of Theorem 3.1, we obtain

x2˙(t)≤x2(t)[r2M−a2Lh2M∫0+∞(h2M1ηer2M)−sK2(s)dsx2(t)].

According to Lemmas 2.1 and 2.3, we have

limt→+∞x2(t)=0.

Then for any ε₂ > 0 satisfying (r1L−b1Mε21+ε2)θ+ln⁡h1L > 0, there exists a T₂ > 0 such that for t > T₂,

x1(t)<M1+ε2,x2(t)<ε2. (3.11)

Substituting this into system (1.3), it follows from Lemma 2.5 that

x1˙(t)≥x1(t)[r1L−a1M(M1+ε2)−b1Mε21+ε2],xi(tk+)=hikxi(tk).

Similarly we have

x1˙(t)≥x1(t)[r1L−b1Mε21+ε2−a1M∫0+∞(∏t−s≤tk<t1h1k)exp⁡{−(r1L−a1M(M1+ε2)−b1Mε21+ε)s}K1(s)dsx1(t)].

Then similarly to the analysis of Lemma 2.2, by setting ε₂ → 0 we can easily obtain

lim inft→+∞x1(t)≥min{r1Lη+ln⁡h1La1MH¯11ηh1L2,(r1Lθ+ln⁡h1L)h1L2a1MH¯21θ}≜m¯1

with

H¯11=∫0+∞(1h1L)sηexp⁡{−(r1L−a1MM1)s}K1(s)ds<+∞,H¯21=∫0+∞(1h1L)sθexp⁡{−(r1L−a1MM1)s}K1(s)ds<+∞.

This completes the proof of the theorem.

Consider the following impulsive system

x˙(t)=x(t)(r1(t)−a1(t)∫0+∞K1(s)x(t−s)ds),x(tk+)=h1kx(tk),k=1,2,⋯, (3.12)

Theorem 3.4

Under the assumptions of Theorem 3.3, we further suppose that there exists a σ₁ > 0 such that

∫0+∞sK1(s)ds=σ1anda1LM1−a1M2σ1>0.

Then for any positive solution (x₁(t), x₂(t))^T of system (1.3), and any positive solution x(t) of system (3.12), there is limt→+∞ |x₁(t) − x(t)| = 0.

Proof

Let (x₁(t), x₂(t))^T be any positive solution of system (1.3), and x(t) be any positive solution of system (3.12). From the condition of Theorem 3.6, there exists a δ₁ > 0 such that

a1LM1−a1M2σ1≥δ1.

According to Theorem 3.5, for any 0 < ε₃ < m̄₁ small enough, there exists a T₃ > 0 such that for t > T₃,

m¯1−ε3≤x1≤M1+ε3.

Define a Lyapunov function as follows

V¯1(t)=(ln⁡x1(t)−ln⁡x(t)−∫0+∞∫t−stK1(s)a1(v+s)(x1(v)−x(v))dvds)2.

Similarly to the analysis of Theorem 3.2, for t > T₃ and t ≠ t_k, k = 1, 2, …, calculating the upper right derivatives of V̄₁(t), we can obtain

Define

V¯2(t)=a1M2∫0+∞∫t−st∫vtK1(s)|x1(u)−x(u)|2dudvds+2b1Ma1Mε31+ε3∫0+∞∫t−st∫vtK1(s)|x1(u)−x(u)|dudvds.

For t > T₃ and t ≠ t_k, k = 1, 2, …, calculating the upper right derivatives of V̄₂(t) and denoting V̄(t) = V̄₁(t) + V̄₂(t), it follows that

By the boundedness of x₁(t) and x(t) and setting ε₃ → 0, we educe that

D+V¯(t)≤(−2a1LM1+2a1M2σ1)|x1(t)−x(t)|2<−δ1|x1(t)−x(t)|2.

For t = t_k, we can easily verify that V¯(tk+) = V̄(t_k). Integrating both sides of the above inequality from T₃ to t, we obtain

V¯(t)+δ1∫T3t|x1(s)−x(s)|2ds≤V¯(T3+)<+∞.

Therefore, V̄(t) is bounded on [T₃, +∞) and there is

∫T3+∞|x1(s)−x(s)|2ds<+∞.

Similarly to the analysis of [17], it is obvious that

limt→+∞|x1(t)−x(t)|=0.

This completes the proof of Theorem 3.4.

Now we discuss the extinction of system (1.3).

Theorem 3.5

Let (x₁(t), x₂(t))^T be any positive solution of system (1.3). Assume that

riMη+ln⁡hiM≤0,1≤i≤2,

then system (1.3) is extinct, that is limt→+∞x1(t)=limt→+∞x2(t)=0 .

Proof

The proof of the theorem is similar to the corresponding part of Theorem 3.3, so we omit the detail.

In the following part of this section, based on the above theorems, we gives some corresponding results for systems (1.1) and (1.2) respectively. First for system (1.1), similarly to the analysis of Theorems 3.1 and 3.2, we can easy obtain the following theorem.

Theorem 3.6

Let x(t) and y(t) be any two positive solutions of system (1.1). Assume that

aLθ+ln⁡hL>0,∫0+∞sK(s)ds=σandbL>bM2Mσ.

Then system (1.1) is permanent and globally attractive, that is

m≤lim inft→+∞x(t)≤lim supt→+∞x(t)≤Mandlimt→+∞|x(t)−y(t)|=0,

where

M=max{(aMθ+ln⁡hM)hM2bLR1θ,aMη+ln⁡hMbLhM2R2η}andm=min{aLη+ln⁡hLbMH1ηhL2,(aLθ+ln⁡hL)hL2bMH2θ}

with

R1=∫0+∞(hM1θeaM)−sK(s)ds,R2=∫0+∞(hM1ηeaM)−sK(s)ds,H1=∫0+∞(1hL)sηexp⁡{−(aL−bMM)s}K(s)ds<+∞,H2=∫0+∞(1hL)sθexp⁡{−(aL−bMM)s}K(s)ds<+∞.

Remark 3.1

In Corollary 3.1, we prove the global attractivity of (1.1), but under some weaker conditions than those in Yang [21]; especially, our result does not require the following unreasonable condition:

0<infk≥1hk≤hk≤1(k=1,2,⋯)andinfk≥1(hk−hk−1)>0.

Next for system (1.2), similarly to the proof of Theorem 3.1, we can easily prove the following theorem.

Theorem 3.7

Let (x₁(t), x₂(t))^T be any solution of system (1.2) with x_i(0) > 0, i = 1, 2. Assume that

(riL−biMMj′1+Mj′)θ+ln⁡hiL>0,1≤i,j≤2,i≠j,riL−biMMj′1+Mj′>0, (3.13)

then mi′≤lim inft→+∞xi(t)≤lim supt→+∞xi(t)≤Mi′ , i = 1, 2, where

Mi′=max{(riMθ+ln⁡hiM)hiMaiLθ,riMη+ln⁡hiMaiLhiMη},mi′=min{(riL−biMMj′1+Mj′)η+ln⁡hiLaiMhiLη,[(riL−biMMj′1+Mj′)θ+ln⁡hiL]hiLaiMθ}.

Theorem 3.8

Under the conditions of Theorem 3.7, we further assume that there exist ρ₁ > 0 and ρ₂ > 0 such that

a1Lρ1−b2Mρ2>0anda2Lρ2−b1Mρ1>0,

Then for any two positive solutions (x₁(t), x₂(t))^T and (y₁(t), y₂(t))^T of system (1.2), there are limt→+∞ |x_i(t) − y_i(t)| = 0, for i = 1, 2.

Proof

Let (x₁(t), x₂(t))^T and (y₁(t), y₂(t))^T be any two positive solutions of system (1.2). From Theorem 3.7, for any ε₄ > 0 small enough, there exist δ₂ > 0 satisfying mi′ − ε₄ > 0 and T₄ > 0 such that for t > T₄

a1Lρ1−b2Mρ2≥δ2anda2Lρ2−b1Mρ1≥δ2,mi′−ε4≤xi,yi≤Mi′+ε4,i=1,2.

Define a Lyapunov function as follows

V~(t)=∑i=12ρi|ln⁡xi(t)−ln⁡yi(t)|.

For t > T₄ and t ≠ t_k, k = 1, 2, …, calculating the upper right derivatives of Ṽ(t), for j = 1, 2 and j ≠ i, we have

where ζ_j(t) lies between x_j(t) and y_j(t).

For t = t_k, we can easily verify that V~(tk+) = Ṽ(t_k). Integrating both sides of the above inequality from T₄ to t, we obtain

V~(t)+δ2∫T4t(|x1(s)−y1(s)|+|x2(s)−y2(s)|)ds≤V~(T4+)<+∞.

Therefore, Ṽ(t) is bounded on [T₄, +∞) and there is

∫T4+∞(|x1(s)−y1(s)|+|x2(s)−y2(s)|)ds<+∞.

Similarly to the analysis of [17], it is obvious that

limt→+∞|x1(t)−y1(t)|=limt→+∞|x2(t)−y2(t)|=0.

This completes the proof of Theorem 3.8.

Consider the following impulsive system

x˙(t)=x(t)(r1(t)−a1(t)x(t)),x(tk+)=h1kx(tk),k=1,2,⋯, (3.14)

Similarly to the analysis of Theorems 3.7 and 3.8, we can easily prove the following theorem.

Theorem 3.9

Let (x₁(t), x₂(t))^T be any positive solution of system (1.2), x(t) be any positive solution of system (3.14). Assume that

r1Lθ+ln⁡h1L>0andr2Mη+ln⁡h2M≤0

Then the species x₁ is permanent and globally attractive but the species x₂ is extinct, that is

m1″≤lim inft→+∞x1(t)≤lim supt→+∞x1(t)≤M1′andlimt→+∞|x1(t)−x(t)|=0;limt→+∞x2(t)=0,

where M1′ is defined in Theorem 3.7 and

m1″=min{r1Lη+ln⁡h1La1Mh1Lη,(r1Lθ+ln⁡h1L)h1La1Mθ}.

Theorem 3.10

Let (x₁(t), x₂(t))^T be any solution of system (1.2) with x_i(0) > 0, i = 1, 2. Assume that

riMη+ln⁡hiM≤0,1≤i,≤2,

then system (1.2) is extinct, that is limt→+∞x1(t)=limt→+∞x2(t)=0 .

Proof

By impulsive comparison theorem and Lemma 2.3, these results can be easily obtained, so we omit the detail.

Remark 3.2

Obviously, condition (H₁) implies (3.13), but not vice versa. Thus Theorem 3.7 weakens Lemma 2.4 in [32]. Also Theorems 3.8-3.10 complement the results of [32].

4 Numerical simulation

In this section, we present some numerical simulations to show the influence of impulse perturbations on the dynamic behaviors of systems.

In Table 1, by calculation we have r_1L = 0.8, r_1M = 0.88, a_1L = 0.27, a_1M = 0.29, b_1L = 0.01, b_1M = 0.012, h_1L = 0.9, h_1M = 1.3, r_2L = 0.3, r_2M = 0.36, a_2L = 0.1, a_2M = 0.12, b_2L = 0.02, b_2M = 0.022, h_2L = 1.01, h_2M = 1.09, θ = 0.5, η = 1. Choose ρ₁ = 9 and ρ₂ = 5. Therefore,

Table 1

Parameter values of system (1.3)

Parameter	Interpretation	Value
r₁(t)	Growth rate of species x₁	0.8t+0.88t+1
r₂(t)	Growth rate of species x₂	0.3t+0.36t+1
a₁(t)	Intra-specific competition of species x₁	0.28+0.01sin 2 t
a₂(t)	Intra-specific competition of species x₂	0.11 + 0.01 sin 2t
b₁(t)	Interspecific competition o of species x₂ on x₁	0.011 + 0.001 sin t
b₂(t)	Interspecific competition o of species x₁ on x₂	0.021 + 0.001 sin t
K₁(t)	Kernel function of species x₁	10e^−10t
K₂(t)	Kernel function of species x₂	8e^−8t
h_1k	Impulse perturbations on species x₁	1.1 − 0.2 cos k
h_2k	Impulse perturbations on species x₂	1.05 + 0.04 sin 2k
t_k	Impulse points	k + 1k

M1=(r1Mθ+ln⁡h1M)h1M2a1Lθ∫0+∞(h1M1θer1M)−sK1(s)ds≈10.0277,M2=(r2Mθ+ln⁡h2M)h2M2a2Lθ∫0+∞(h2M1θer2M)−sK2(s)ds≈6.7458,

H21=∫0+∞(1h1L)sθexp⁡{−(r1L−a1MM1−b1MM21+M2)s}K1(s)ds≈1.3036,H12=∫0+∞(1h2L)sηexp⁡{−(r2L−a2MM2−b2MM11+M1)s}K2(s)ds≈1.0695,σ1=∫0+∞sK1(s)ds=110andσ2=∫0+∞sK2(s)ds=18.

We can easily verify that

r1L−b1MM21+M2≈0.7895>0,(r1L−b1MM21+M2)θ+ln⁡h1L≈0.2894>0,r2L−b2MM11+M1≈0.2800>0,(r2L−b2MM11+M1)θ+ln⁡h2L≈0.1499>0,m1=[(r1L−b1MM21+M2)θ+ln⁡h1L]h1L2a1MH21θ≈1.2402,m2=(r2L−b2MM11+M1)η+ln⁡h2La2MH12ηh2L2≈2.2148,2a1LM1ρ1−b1Mρ1m12−b2Mρ2−2a1M2ρ1σ1−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2≈0.0603>0,2a2Lρ2M2−b1Mρ1−b2Mρ2m22−2a2M2ρ2σ2−b1Ma1Mρ1σ1−b2Ma2Mρ2σ2≈0.0430>0.

Thus all the conditions of Theorem 3.2 are satisfied. Therefore both species x₁ and x₂ are permanent and globally attractive, which is shown in Figure 2.

$Figure 2 System (1.3) with (ϕ1(t), ϕ2(t)) = (2.6, 2.8)T and (3.8, 3.6)T for t ≤ 0 respectively.$

Figure 2

System (1.3) with (ϕ₁(t), ϕ₂(t)) = (2.6, 2.8)^T and (3.8, 3.6)^T for t ≤ 0 respectively.

Furthermore, we keep the growth rates, the intra-specific competition and the kernel functions of all species unchanged in Table 1, but adjust the values of the impulse perturbations given in Table 2, then simulations (see Figures 3-5) show that the permanence and extinction of the species are significantly changed, which are in accordance with the results of Theorems 3.4 and 3.5, here we can verify the corresponding conditions similarly to those in Table 1.

$Figure 3 System (1.3) with (ϕ1(t), ϕ2(t)) = (3.8, 0.6)T and (2.6, 1.8)T for t ≤ 0 respectively.$

Figure 3

System (1.3) with (ϕ₁(t), ϕ₂(t)) = (3.8, 0.6)^T and (2.6, 1.8)^T for t ≤ 0 respectively.

$Figure 4 System (1.3) with (ϕ1(t), ϕ2(t)) = (2.6, 3.8)T and (1.8, 8.6)T for t ≤ 0 respectively.$

Figure 4

System (1.3) with (ϕ₁(t), ϕ₂(t)) = (2.6, 3.8)^T and (1.8, 8.6)^T for t ≤ 0 respectively.

$Figure 5 System (1.3) with (ϕ1(t), ϕ2(t)) = (0.1, 0.8)T and (0.8, 0.2)T for t ≤ 0 respectively.$

Figure 5

System (1.3) with (ϕ₁(t), ϕ₂(t)) = (0.1, 0.8)^T and (0.8, 0.2)^T for t ≤ 0 respectively.

Table 2

Simulations of system (1.3)

Case	h_1k	h_2k	Species x₁	Species x₂	Figure
1	1.1 − 0.2 cos k	0.2 + 0.1 sin 2k	Permanence	Extinction	Figure 3
2	0.2 − 0.2 cos k	1.5 + 0.1 sin 2k	Extinction	Permanence	Figure 4
3	0.3 − 0.1 cos k	0.5 + 0.1 sin 2k	Extinction	Extinction	Figure 5

5 Conclusion

In this paper, we are devoted to obtaining the major factors that affect the coexistence, competition exclusion and extinction of system (1.3). Table 1 shows that we can choose some suitable values of parameters of system (1.3) to guarantee the coexistence of both species. However, when we change the values of the impulse perturbations shown in Table 2, there is a significant variation of the survival of each species. When choo sing the impulse perturbations h_ik < 1 small enough and keeping the value of the growth rate unchanged, it is hard to maintain the permanence of the species x_i. Moreover, this can result in the extinction of both species, which is different from the continuous system. The impulse perturbation plays an important role in the survival of the species and can deduce more situations of real ecosystems. Furthermore, for the logistic type impulsive equation with infinite delay, our results improve those of [21] and remove its unreasonable condition. For the corresponding nonautonomous two-species impulsive competitive system without delays, our results weaken and complement the results of [32].

Acknowledgements

The research was supported by the Scientic Research Foundation of Fuzhou University under Grant GXRC-18062.

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Received: 2018-09-26

Accepted: 2019-05-02

Published Online: 2019-07-31

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

Articles in the same Issue

https://doi.org/10.1515/math-2019-0062

Keywords for this article

Competitive system; Impulses; Permanence; Extinction; Infinite delays

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BY 4.0