Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

Yalong Xue; Xiangdong Xie; Qifa Lin

doi:10.1515/math-2019-0134

Article Open Access

Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

Yalong Xue , Xiangdong Xie and Qifa Lin

Published/Copyright: December 13, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

Keywords: almost periodic solutions; commensal symbiosis model; Lyapunov functional; Michaelis-Menten type harvesting; time scales

MSC 2010: 34C25; 92D25; 34D20; 34D40

1 Introduction

Many results of differential equations can be easily generalized to difference equations, while other results seem to be completely different from their continuous counterparts. A major task of mathematics today is to harmonize continuous and discrete analysis. The theory of time scale, which was first introduced by Stefan Hilger in his PhD thesis [1], can handle this problem well. For example, it can model insect populations that are continuous while in season (and many follows a difference scheme with variable), die out in (say) winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population [2]. More generally, time scales calculus can be applied to the system whose time domains are more complex. A good example can be found in economics: a consumer receives income at one point in time, asset holdings are adjusted at a different point in time, and consumption takes place at yet another point in time [3]. The time scales calculus has a tremendous potential for applications (see [4, 5, 6, 7, 8, 9, 10]).

Many scholars have recently studied the influence of the harvesting to predator-prey or competition system. Some of them (e.g., [11, 12, 13, 14]) argued that nonlinear harvesting is more feasible. Also consider that the almost periodic phenomenon and non-autonomous model are more accurate to describe the actual situation (e.g., [15, 16]). Therefore, we investigate the following commensalism system incorporating Michaelis-Menten type harvesting:

xΔ(t)=a(t)−b(t)exp⁡{x(t)}+c(t)exp⁡{y(t)},yΔ(t)=d(t)−e(t)exp⁡{y(t)}−q(t)E(t)exp⁡{y(t)}E(t)+m(t)exp⁡{y(t)}, (1.1)

where x(t), y(t) are the density of species x, y at time t ∈ T (T is a time scale). x^Δ, y^Δ express the delta derivative of the functions x(t), y(t). E(t) denotes the harvesting effort and q(t) is the catch ability coefficient. The coefficients are bounded positive almost periodic functions and we use the notations g^l = inf_t∈T⁺ g(t), g^u = sup_t∈T⁺ g(t).

Obviously, let x(t) = ln x₁(t), y(t) = ln y₁(t), if T = R⁺, then system (1.1) is reduced to a continuous version:

dx1dt=x1(t)(a(t)−b(t)x1(t)+c(t)y1(t)),dy1dt=y1(t)(d(t)−e(t)y1(t))−q(t)E(t)y12(t)E(t)+m(t)y1(t), (1.2)

if T = Z⁺, then system (1.1) can be simplified as the following discrete system:

x1(t+1)=x1(t)exp⁡(a(t)−b(t)x1(t)+c(t)y1(t)),y1(t+1)=y1(t)exp⁡(d(t)−e(t)y1(t)−q(t)E(t)y1(t)E(t)+m(t)y1(t)). (1.3)

The rest of this paper is arranged as follows. The next part we present some notations. After that, sufficient conditions for the uniformly asymptotic stability of unique almost periodic solution are established. We end this paper with two examples to verify the validity of our criteria.

2 Preliminaries

A time scale T is an arbitrary nonempty closed subset of the real numbers. A point t ∈ T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = t, and right-scattered if σ(t) > t. If T has a left-scattered maximum m, then T^k = T ∖ {m}; otherwise T^k = T. If T has a right-scattered minimum m, then T_k = T∖{m}; otherwise T_k = T.

A function p : T → R is called regressive provided 1 + μ(t)p(t) ≠ 0 for all t ∈ T^k. The set of all regressive and rd-continuous functions p : T → R will be denoted by 𝓡 = 𝓡(T) = 𝓡(T, R). We define the set 𝓡⁺ = 𝓡⁺(T, R) = {p ∈ 𝓡 : 1 + μ(t)p(t) > 0 for all t ∈ T}.

If p is a regressive function, then the generalized exponential function e_p is defined by

ep(t,s)=exp⁡{∫tsξμ(τ)(p(τ))Δτ}

for all s, t ∈ T, with the cylinder transformation ξh(z)=Log(1+hz)hifh≠0,z ifh=0. For further reading we refer to the book by Bohner and Peterson [2].

Definition 2.1

(see [2]). Let T be a time scale. For t ∈ T we define the forward and backward jump operators σ, ρ : T → T and the graininess function μ : T → R⁺ by

σ(t)=inf{s∈T:s>t},ρ(t)=sup{s∈T:s<t},μ(t)=σ(t)−t,

and μ₁ = inf_t∈T⁺μ(t), μ₂ = sup_t∈T⁺μ(t).

Definition 2.2

(see [6]). A time scale T is called an almost periodic time scale if

∏:={τ∈R:T+τ∈T,∀t∈T}≠{0}.

Definition 2.3

(see [6]). Let T be an almost periodic time scale. A function x ∈ C(T, Rⁿ) is called an almost periodic function if the ε-translation set of x

E{ε,x}={τ∈∏:|x(t+τ)−x(t)|<ε,∀t∈T}

is a relatively dense set in T for all ε > 0, that is, for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ(ε) ∈ E{ε, x} such that

|x(t+τ)−x(t)|<ε,∀t∈T.

Definition 2.4

(see [6]). Let T be an almost periodic time scale and D denotes an open set in Rⁿ. A function f ∈ C(T × D, Rⁿ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε-translation set of f

E{ε,f,S}={τ∈∏:|f(t+τ,x)−f(t,x)|<ε,∀(t,x)∈T×S}

is a relatively dense set in T for all ε > 0 and for each compact subset S of D, that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ(ε, S) ∈ E{ε, f, S} such that

|f(t+τ,x)−f(t,x)|<ε,∀(t,x)∈T×S.

Lemma 2.5

(see [7]). Assume that a > 0, b > 0 and –a ∈ 𝓡⁺. Then

yΔ(t)≥(≤)b−ay(t),y(t)>0,t∈[t0,∞)T (2.1)

implies

y(t)≥(≤)ba[1+(ay(t0)b−1)e(−a)(t,t0)],t∈[t0,∞)T. (2.2)

Consider the following system

xΔ(t)=f(t,x) (2.3)

and its associate product system

xΔ(t)=f(t,x),zΔ(t)=f(t,z) (2.4)

where f : T⁺ × S_H → Rⁿ, S_H = {x ∈ Rⁿ : ∥x∥ < H}, f(t, x) is almost periodic in t uniformly for x ∈ S_H and is continuous in x.

Lemma 2.6

(see [8]). Suppose that there exists a Lyapnov function V(t, x, z) defined on T⁺ × S_H × S_H satisfying the following conditions

a(∥x – z∥) ≤ V(t, x, z) ≤ b(∥x – z∥), where a, b ∈ K, K = {α ∈ C(R⁺, R⁺) : α(0) = 0 and α is increasing};
|V(t, x, z) – V(t, x₁, z₁)| ≤ L(∥x – x₁∥ + ∥z – z₁∥), where L > 0 is a constant;
D⁺ V(2.4)Δ (t, x, z) ≤ –cV(t, x, z), where c > 0, –c ∈ ℜ⁺.

Moreover, if there exists a solution x(t) ∈ S of system (2.3) for t ∈ T⁺, where S ⊂ S_H is a compact set, then there exists a unique almost periodic solution q(t) ∈ S of system (2.3), which is uniformly asymptotically stable.

Lemma 2.7

du>el,au+cuexp⁡{My}>bl, (2.5)

then any positive solution (x(t), y(t)) of system (1.1) satisfies

lim supt→+∞y(t)≤My=def(du−el)/el,lim supt→+∞x(t)≤Mx=def(au+cuexp⁡{My}−bl)/bl. (2.6)

Proof

From the second equation of system (1.1) it follows

yΔ(t)≤d(t)−e(t)exp⁡{y(t)}≤d(t)−e(t)(y(t)+1)≤(du−el)−ely(t). (2.7)

By using Lemma 2.5 we get

lim supt→+∞y(t)≤(du−el)/el=defMy. (2.8)

For a sufficiently small ε > 0, from (2.5) and (2.8), there exists a t₁ ∈ T⁺ such that

y(t)≤My+ε,∀t>t1.au+cuexp⁡{My+ε}−bl>0, (2.9)

From (2.9) and the first equation of system (1.1), we have

xΔ(t)≤a(t)+c(t)exp⁡{My+ε}−b(t)exp⁡{x(t)}≤(au+cuexp⁡{My+ε}−bl)−blx(t). (2.10)

By using Lemma 2.5 again, we have

lim supt→+∞x(t)≤(au+cuexp⁡{My+ε}−bl)/bl. (2.11)

Setting ε → 0, one has

lim supt→+∞x(t)≤(au+cuexp⁡{My}−bl)/bl=defMx. (2.12)

Lemma 2.8

Under the hypothesis (2.5) and

El(dl−eu)(eu−dl)ml+quEu>exp⁡{My}>exp⁡{Ny}>bu−alcl, (2.13)

then any positive solution (x(t), y(t)) of system (1.1) satisfies

lim inft→+∞y(t)≥Ny=defln⁡dl(El+mlexp⁡{My})−quEuexp⁡{My}eu(El+mlexp⁡{My}),lim inft→+∞x(t)≥Nx=defln⁡al+clexp⁡{Ny}bu. (2.14)

Proof

Lemma 2.7 means that for any ε > 0, there exists a t₂ > t₁ (the definition of t₁ in Lemma 2.7) such that

y(t)≤My+ε,x(t)≤Mx+ε,∀t>t2. (2.15)

It follows from the second equation of system (1.1) that

yΔ(t)≥dl−euexp⁡{y(t)}−quEuexp⁡{My+ε}El+mlexp⁡{My+ε},∀t>t2. (2.16)

We claim that for t ≥ t₂,

dl−euexp⁡{y(t)}−quEuexp⁡{My+ε}El+mlexp⁡{My+ε}≤0. (2.17)

Suppose that there exists a t̂ ≥ t₂ such that

dl−euexp⁡{y(t^)}−quEuexp⁡{My+ε}El+mlexp⁡{My+ε}>0 (2.18)

and for any t ∈ [t₂, t̂)_T⁺

dl−euexp⁡{y(t)}−quEuexp⁡{My+ε}El+mlexp⁡{My+ε}≤0. (2.19)

Then

y(t^)<ln⁡dl(El+mlexp⁡{My+ε})−quEuexp⁡{My+ε}eu(El+mlexp⁡{My+ε}) (2.20)

and for any t ∈ [t₂, t̂)_T⁺,

y(t)≥ln⁡dl(El+mlexp⁡{My+ε})−quEuexp⁡{My+ε}eu(El+mlexp⁡{My+ε}), (2.21)

which implies y^Δ(t) < 0. It is a contradiction, so that (2.17) holds, i.e.,

y(t)≥ln⁡dl(El+mlexp⁡{My+ε})−quEuexp⁡{My+ε}eu(El+mlexp⁡{My+ε}), (2.22)

thereby

lim inft→+∞y(t)≥ln⁡dl(El+mlexp⁡{My+ε})−quEuexp⁡{My+ε}eu(El+mlexp⁡{My+ε}). (2.23)

Setting ε → 0,

lim inft→+∞y(t)≥ln⁡dl(El+mlexp⁡{My})−quEuexp⁡{My}eu(El+mlexp⁡{My})=defNy. (2.24)

For above ε and (2.24), there exists a t₃ > t₂ such that

y(t)≥Ny−ε,∀t>t3. (2.25)

It follows from the first equation of system (1.1) and above inequation that

xΔ(t)≥al−buexp⁡{x(t)}+clexp⁡{Ny−ε}. (2.26)

By analyzing (2.26) similar to (2.17)-(2.24), one has

lim inft→+∞x(t)≥ln⁡al+clexp⁡{Ny}bu=defNx. (2.27)

3 Positive almost periodic solution

From Lemma 2.7 and Lemma 2.8, let Ω = {(x(t), y(t)) : (x(t), y(t)) is a solution of (1.1) and 0 < N_x ≤ x(t) ≤ M_x, 0 < N_y ≤ y(t) ≤ M_y}. Obviously, Ω is an invariant set.

Theorem 3.1

Under the hypothesis (2.5) and (2.13), the Ω ≠ ∅.

Proof

Since the coefficients are almost periodic sequences, there exists a sequence {τ_k} ⊆ T⁺ with τ_k → ∞ as k → ∞ such that

a(t+τk)→a(t),b(t+τk)→b(t),c(t+τk)→c(t),d(t+τk)→d(t),e(t+τk)→e(t),q(t+τk)→q(t),m(t+τk)→m(t),E(t+τk)→E(t). (3.1)

From Lemma 2.7 and Lemma 2.8, for sufficiently small ε > 0, there exists a t₄ ∈ T⁺ such that

Nx−ε≤x(t)≤Mx+ε,Ny−ε≤y(t)≤My+ε,∀t>t4. (3.2)

Write x_k(t) = x(t + τ_k) and y_k(t) = y(t + τ_k) for t > t₄ – τ_k and k = 1, 2, ⋅⋅⋅. For arbitrary q ∈ N⁺, it is easy to see that there exists sequences {x_k(t) : k ≥ q} and {y_k(t) : k ≥ q} such that the sequence {x_k(t)} and {y_k(t)} has a subsequence, denoted by {xk∗(t)}(xk∗(t)=x(t+τk∗)) and {yk∗(t)}(yk∗(t)=y(t+τk∗)), respectively, converging on any finite interval of T⁺ as k → ∞. Therefore there exist two almost periodic sequences {z(t)} and {w(t)} such that for t ∈ T⁺,

xk∗(t)→z(t),yk∗(t)→w(t),ask→∞. (3.3)

Apparently the above sequence { τk∗ } ⊆ T⁺ with τk∗ → ∞ as k → ∞ such that

a(t+τk∗)→a(t),b(t+τk∗)→b(t),c(t+τk∗)→c(t),d(t+τk∗)→d(t),e(t+τk∗)→e(t),q(t+τk∗)→q(t),m(t+τk∗)→m(t),E(t+τk∗)→E(t). (3.4)

which, together with (3.3) and

xk∗Δ(t)=a(t+τk∗)−b(t+τk∗)exp⁡{xk∗(t)}+c(t+τk∗)exp⁡{yk∗(t)},yk∗Δ(t)=d(t+τk∗)−e(t+τk∗)exp⁡{yk∗(t)}−q(t+τk∗)E(t+τk∗)exp⁡{yk∗(t)}E(t+τk∗)+m(t+τk∗)exp⁡{yk∗(t)}, (3.5)

yields

zΔ(t)=a(t)−b(t)exp⁡{z(t)}+c(t)exp⁡{w(t)},wΔ(t)=d(t)−e(t)exp⁡{w(t)}−q(t)E(t)exp⁡{w(t)}E(t)+m(t)exp⁡{w(t)}. (3.6)

Obviously, (z(t), w(t)) is a solution of system (1.1) and

Nx−ε≤z(t)≤Mx+ε,Ny−ε≤w(t)≤My+ε,∀t∈T+. (3.7)

Since ε is arbitrary, thus

Nx≤z(t)≤Mx,Ny≤w(t)≤My,∀t∈T+. (3.8)

Theorem 3.2

Assume that (2.5) and (2.13) are hold. If λ = min{A, B} > 0 and –λ ∈ 𝓡⁺, then system (1.1) admits a unique almost periodic solution (x(t), y(t)), which is uniformly asymptotically stable and (x(t), y(t)) ∈ Ω, where

A=2(μ1+1)blexp⁡{Nx}−(μ22+μ2)(bu)2exp⁡{2Mx}−1,B=2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}−μ2[eu+quEu(El+mlexp⁡{Ny})2]2exp⁡{2My}−(μ2+1)(cu)2exp⁡{2My}, (3.9)

Proof

By Theorem 3.1, there exists a solution (x(t), y(t)) such that

Nx≤x(t)≤Mx,Ny≤y(t)≤My,∀t∈T+. (3.10)

Define

∥(x(t),y(t))∥=|x(t)|+|y(t)|. (3.11)

Suppose that U₁(t) = (x(t), y(t)), U₂(t) = (z(t), w(t)) are arbitrary two positive solutions of system (1.1), then ∥U₁(t)∥ ≤ M_x + M_y, ∥U₂(t)∥ ≤ M_x + M_y. Consider the product system of (1.1)

xΔ(t)=a(t)−b(t)exp⁡{x(t)}+c(t)exp⁡{y(t)},yΔ(t)=d(t)−e(t)exp⁡{y(t)}−q(t)E(t)exp⁡{y(t)}E(t)+m(t)exp⁡{y(t)},zΔ(t)=a(t)−b(t)exp⁡{z(t)}+c(t)exp⁡{w(t)},wΔ(t)=d(t)−e(t)exp⁡{w(t)}−q(t)E(t)exp⁡{w(t)}E(t)+m(t)exp⁡{w(t)}. (3.12)

Construct the Lyapunov functional V(t, U₁(t), U₂(t)) on T⁺ × Ω × Ω

V(t,U1(t),U2(t))=(x(t)−z(t))2+(y(t)−w(t))2. (3.13)

The norm

∥U1(t)−U2(t)∥=|x(t)−z(t)|+|y(t)−w(t)| (3.14)

is equivalent to

∥U1(t)−U2(t)∥∗=[(x(t)−z(t))2+(y(t)−w(t))2]12, (3.15)

i.e., there exist constants B₁ > 0 and B₂ > 0 such that

B1∥U1(t)−U2(t)∥≤∥U1(t)−U2(t)∥∗≤B2∥U1(t)−U2(t)∥, (3.16)

thus we have

(B1∥U1(t)−U2(t))2∥≤V(t,U1(t),U2(t))≤(B2∥U1(t)−U2(t)∥)2. (3.17)

Let a, b ∈ C(R⁺, R⁺), a(x) = B12x2,b(x)=B22x2, then the assumption (i) of Lemma 2.6 is satisfied.

On the other side,

|V(t,U1(t),U2(t))−V(t,U1∗(t),U2∗(t))|=|(x(t)−z(t))2+(y(t)−w(t))2−(x∗(t)−z∗(t))2−(y∗(t)−w∗(t))2|≤|(x(t)−z(t))−(x∗(t)−z∗(t))|×|(x(t)−z(t))+(x∗(t)−z∗(t))|+|(y(t)−w(t))−(y∗(t)−w∗(t))|×|(y(t)−w(t))+(y∗(t)−w∗(t))|≤|(x(t)−z(t))−(x∗(t)−z∗(t))|×(|x(t)|+|z(t)|+|x∗(t)|+|z∗(t))|)+|(y(t)−w(t))−(y∗(t)−w∗(t))|×(|y(t)|+|w(t)|+|y∗(t)|+|w∗(t)|)≤γ(|x(t)−x∗(t)|+|y(t)−y∗(t)|+|z(t)−z∗(t)|+|w(t)−w∗(t)|)=γ(∥U1(t)−U1∗(t)∥+∥U2(t)−U2∗(t)∥), (3.18)

where U1∗ (t) = (x^*(t), y^*(t)), U2∗ (t) = (z^*(t), w^*(t)), γ = 4 max{M_x, M_y}. Hence, the assumption (ii) of Lemma 2.6 is also satisfied.

Calculating the D⁺V^Δ along the system (3.12),

D+V(3.12)Δ(t,U1(t),U2(t))=(x(t)−z(t))Δ[(x(t)−z(t))+(x(σ(t))−z(σ(t)))]+(y(t)−w(t))Δ[(y(t)−w(t))+(y(σ(t))−w(σ(t)))]=(x(t)−z(t))Δ[(x(t)−z(t))+(μ(t)xΔ(t)+x(t)−μ(t)zΔ(t)+z(t))]+(y(t)−w(t))Δ[(y(t)−w(t))+(μ(t)yΔ(t)+y(t)−μ(t)wΔ(t)+w(t))]=(x(t)−z(t))Δ[2(x(t)−z(t))+μ(t)(x(t)−z(t))Δ]+(y(t)−w(t))Δ[2(y(t)−w(t))+μ(t)(y(t)−w(t))Δ]=V1+V2, (3.19)

where

V1=(x(t)−z(t))Δ[2(x(t)−z(t))+μ(t)(x(t)−z(t))Δ],V2=(y(t)−w(t))Δ[2(y(t)−w(t))+μ(t)(y(t)−w(t))Δ]. (3.20)

From (3.12), one has

(x(t)−z(t))Δ=−b(t)(exp⁡{x(t)}−exp⁡{z(t)})+c(t)[exp⁡{y(t)}−exp⁡{w(t)}],(y(t)−w(t))Δ=−e(t)(exp⁡{y(t)}−exp⁡{w(t)})−q(t)E(t)[exp⁡{y(t)}E(t)+m(t)exp⁡{y(t)}−exp⁡{w(t)}E(t)+m(t)exp⁡{w(t)}]. (3.21)

By the mean value theorem,

exp⁡{x(t)}−exp⁡{z(t)}=exp⁡{ξ1(t)}(x(t)−z(t)),exp⁡{y(t)}−exp⁡{w(t)}=exp⁡{ξ2(t)}(y(t)−w(t)),exp⁡{y(t)}E(t)+m(t)exp⁡{y(t)}−exp⁡{w(t)}E(t)+m(t)exp⁡{w(t)}=exp⁡{ξ3(t)}(y(t)−w(t))(E(t)+m(t)exp⁡{ξ3(t)})2, (3.22)

where ξ₁(t) lies between x(t) and z(t) and ξ_i(t) (i = 2, 3) lie between y(t) and w(t). Thus (3.21) can be expressed as follows

(x(t)−z(t))Δ=−b(t)exp⁡{ξ1(t)}(x(t)−z(t))+c(t)exp⁡{ξ2(t)}(y(t)−w(t)),(y(t)−w(t))Δ=−e(t)exp⁡{ξ2(t)}(y(t)−w(t))−[q(t)E(t)exp⁡{ξ3(t)}(y(t)−w(t))(E(t)+m(t)exp⁡{ξ3(t)})2], (3.23)

which together with (3.20) yield

V1={−b(t)exp⁡{ξ1(t)}(x(t)−z(t))+c(t)exp⁡{ξ2(t)}(y(t)−w(t))}[2(x(t)−z(t))+μ(t)(−b(t)exp⁡{ξ1(t)}(x(t)−z(t))+c(t)exp⁡{ξ2(t)}(y(t)−w(t)))]=[−2b(t)exp⁡{ξ1(t)}+μ(t)b2(t)exp⁡{2ξ1(t)}](x(t)−z(t))2+2c(t)exp⁡{ξ2(t)}[1−μ(t)b(t)exp⁡{ξ1(t)}](x(t)−z(t))(y(t)−w(t))+μ(t)c2(t)exp⁡{2ξ2(t)}(y(t)−w(t))2≤[−2(μ(t)+1)b(t)exp⁡{ξ1(t)}+(μ2(t)+μ(t))b2(t)exp⁡{2ξ1(t)}+1](x(t)−z(t))2+(μ(t)+1)c2(t)exp⁡{2ξ2(t)}(y(t)−w(t))2≤[−2(μ1+1)blexp⁡{Nx}+(μ22+μ2)(bu)2exp⁡{2Mx}+1](x(t)−z(t))2+(μ2+1)(cu)2exp⁡{2My}(y(t)−w(t))2, (3.24)

analogously

V2≤{−2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}+μ2[eu+quEu(El+mlexp⁡{Ny})2]2exp⁡{2My}}(y(t)−w(t))2 (3.25)

Therefore, one has

D+V(3.12)Δ(t,U1(t),U2(t))≤{−2(μ1+1)blexp⁡{Nx}+(μ22+μ2)(bu)2exp⁡{2Mx}+1}(x(t)−z(t))2+{(μ2+1)(cu)2exp⁡{2My}−2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}+μ2[eu+quEu(El+mlexp⁡{Ny})2]2exp⁡{2My}}(y(t)−w(t))2=−A(x(t)−z(t))2−B(y(t)−w(t))2≤−λV(t,U1(t),U2(t)). (3.26)

Also, the assumption (iii) of Lemma 2.6 is satisfied.

By Lemma 2.6, there exists a unique uniformly asymptotically stable almost periodic solution (x(t), y(t)) ∈ Ω of system (1.1).

Now we consider the following single specie model with Michaelis-Menten type harvesting on time scales:

yΔ(t)=d(t)−e(t)exp⁡{y(t)}−q(t)E(t)exp⁡{y(t)}E(t)+m(t)exp⁡{y(t)}, (3.27)

For system (3.27), when we conduct the similar analysis of Lemma 2.7, Lemma 2.8, Theorem 3.1 and Theorem 3.2, one can easily obtain the following results and we omit the proof details here.

Lemma 3.3

If d^u > e^l, then positive solution y(t) of system (3.27) satisfies

lim supt→+∞y(t)≤My′=def(du−el)/el. (3.28)

Lemma 3.4

If Lemma 3.3 and the following inequality

El(dl−eu)>((eu−dl)ml+quEu)exp⁡{My′} (3.29)

hold, then any positive solution y(t) of system (3.27) satisfies

lim inft→+∞y(t)≥Ny′=defln⁡dl(El+mlexp⁡{My′})−quEuexp⁡{My′}eu(El+mlexp⁡{My′}). (3.30)

Let Ω^′ = {y(t) : y(t) is a solution of (3.27) and 0 < Ny′ ≤ y(t) ≤ My′ }. It is obvious that Ω^′ is an invariant set.

Theorem 3.5

Assume that Lemma 3.3 and Lemma 3.4 hold, then Ω^′ ≠ ∅. Moreover, if C > 0, then system (3.27) admits a unique uniformly asymptotically stable almost periodic solution y(t), and y(t) ∈ Ω^′, where

C=2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}−μ2[eu+quEu(El+mlexp⁡{Ny})2]2exp⁡{2My}. (3.31)

4 Numerical Simulations

We give the following examples to illustrate the feasibility of our main results.

Example 4.1

Consider the continuous version:

dx1dt=x1(t)(2+0.25sin⁡(2t)−(1.5−0.5cos⁡(5t))x1(t)+(0.6+0.15cos⁡(5t))y1(t)),dy1dt=y1(t)(3.15+0.05sin⁡(3t)−(2+sin⁡(3t))y1(t))−(0.08+0.06sin⁡(6t))y12(t)1+(1.2+0.2cos⁡(3t))y1(t), (4.1)

By calculating, one has

du−el=1.2>0,au+cuexp⁡{My}−bl=2.6166>0,El(dl−eu)(eu−dl)ml+quEu=2.5>exp⁡{My}=1.8221>exp⁡{Ny}=1.0032>bu−alcl=0.5556. (4.2)

Obviously, the assumption in (2.5) and (2.13) are satisfied. We obtain from Example 1.2 in [2] that μ(t) ≡ 0, moreover, from (3.9) we have μ₁ ≡ 0, μ₂ ≡ 0. Thus

A=2blexp⁡{Nx}−1=1.2014>0,B=2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}−(cu)2exp⁡{2My}=2.1484>0, (4.3)

λ = min{A, B} > 0 and –λ ∈ 𝓡⁺. From Figure 1, it is easy to see that for system (4.1) there exists a positive almost periodic solution denoted by (x1∗(t),y1∗(t)).

$Figure 1 Dynamic behaviors of the solutions (x1∗(t),y1∗(t)) $\begin{array}{} (x^{*}_{1}(t), y^{*}_{1}(t)) \end{array}$ of system (4.1) with the initial conditions (x1∗(0),y1∗(0))=(1,1.5), $\begin{array}{} (x^{*}_{1}(0), y^{*}_{1}(0))=(1, 1.5), \end{array}$ (2, 3) and (3, 5), respectively.$

Figure 1

Dynamic behaviors of the solutions (x1∗(t),y1∗(t)) of system (4.1) with the initial conditions (x1∗(0),y1∗(0))=(1,1.5), (2, 3) and (3, 5), respectively.

Example 4.2

Consider the discrete version:

x1(t+1)=x1(t)exp⁡(0.72+0.02sin⁡(2t)−(0.72+0.02cos⁡(3t))x1(t)+(0.07+0.01sin⁡(5t))y1(t)),y1(t+1)=y1(t)exp⁡(1+0.02cos⁡(2t)−(0.25−0.01sin⁡(3t))y1(t) −(0.02+0.001cos⁡(3n))(1.5+0.5sin⁡(2t))y1(t)1.5+0.5sin⁡(2t)+(1.3+0.1cos⁡(6t))y1(t)). (4.4)

By calculating, one has

du−el=0.13>0,au+cuexp⁡{My}−bl=0.1326>0,El(dl−eu)(eu−dl)ml+quEu=2.0833>exp⁡{My}=1.1573>exp⁡{Ny}=1.03>bu−alcl=0.6667. (4.5)

Obviously, the assumption in (2.5) and (2.13) are satisfied. We obtain from Example 1.2 in [2] that μ(t) ≡ 1, moreover, from (3.9) we have μ₁ ≡ 1, μ₂ ≡ 1. Thus

A=4blexp⁡{Nx}−2(bu)2exp⁡{2Mx}−1=0.2829>0,B=2[el+qlEl(Eu+muexp⁡{My})2]exp⁡{Ny}−[eu+quEu(El+mlexp⁡{Ny})2]2exp⁡{2My}−2(cu)2exp⁡{2My}=0.6277>0, (4.6)

λ = min{A, B} > 0 and –λ ∈ 𝓡⁺. From Figure 2, it is easy to see that for system (4.4) there exists a positive almost periodic solution denoted by (x1∗(t),y1∗(t)).

$Figure 2 Dynamic behaviors of the solutions (x1∗(t),y1∗(t)) $\begin{array}{} (x^{*}_{1}(t), y^{*}_{1}(t)) \end{array}$ of system (4.4) with the initial conditions (x1∗(0),y1∗(0))=(1,1.5), $\begin{array}{} (x^{*}_{1}(0), y^{*}_{1}(0))=(1, 1.5), \end{array}$ (2, 2.5) and (3, 3.5), respectively.$

Figure 2

Dynamic behaviors of the solutions (x1∗(t),y1∗(t)) of system (4.4) with the initial conditions (x1∗(0),y1∗(0))=(1,1.5), (2, 2.5) and (3, 3.5), respectively.

In addition, we perform numerical simulation on the system (3.27).

Example 4.3

Consider the continuous version:

dy1dt=y1(t)(3.2+0.1sin⁡(5t)−(2.8+0.2cos⁡(3t))y1(t))−(0.11+0.03sin⁡(6t))(0.9+0.1sin⁡(3))y12(t)0.9+0.1sin⁡(3)+(1.2+0.2cos⁡(2t))y1(t) (4.7)

We obtain from Example 1.2 in [2] that μ(t) ≡ 0, moreover, from (3.9) we have μ₁ ≡ 0, μ₂ ≡ 0. Thus, by calculating one has

du−el=0.7>0,C=5.2554>0,El(dl−eu)=0.08>((eu−dl)ml+quEu)exp⁡{My}=0.0262. (4.8)

From Figure 3, it is easy to see that for system (4.7) there exists a positive almost periodic solution denoted by y1∗ (t).

$Figure 3 Dynamic behaviors of the solutions y1∗ $\begin{array}{} y^{*}_{1} \end{array}$(t) of system (4.7) with the initial conditions y1∗ $\begin{array}{} y^{*}_{1} \end{array}$(0) = 1.5, 1 and 0.5, respectively.$

Figure 3

Dynamic behaviors of the solutions y1∗ (t) of system (4.7) with the initial conditions y1∗ (0) = 1.5, 1 and 0.5, respectively.

Example 4.4

Consider the following discrete system:

y1(t+1)=y1(t)exp⁡(1.1+0.1cos⁡(2t)−(0.89−0.01sin⁡(3t))y1(t) −(0.02+0.01cos⁡(5t))(1.4+0.1sin⁡(2t))y1(t)1.4+0.1sin⁡(2t)+(0.6+0.2cos⁡(6t))y1(t)). (4.9)

We obtain from Example 1.2 in [2] that μ(t) ≡ 0, moreover, from (3.9) we have μ₁ ≡ 0, μ₂ ≡ 0. Thus, by calculating one has

du−el=0.32>0,C=0.1593>0,El(dl−eu)=0.13>((eu−dl)ml+quEu)exp⁡{My}=0.0072. (4.10)

From Figure 4, it is easy to see that for system (4.9) there exists a positive almost periodic solution denoted by y1∗ (t).

$Figure 4 Dynamic behaviors of the solutions y1∗ $\begin{array}{} y^{*}_{1} \end{array}$(t) of system (4.9) with the initial conditions y1∗ $\begin{array}{} y^{*}_{1} \end{array}$(0) = 1.8, 1.5 and 1, respectively.$

Figure 4

Dynamic behaviors of the solutions y1∗ (t) of system (4.9) with the initial conditions y1∗ (0) = 1.8, 1.5 and 1, respectively.

5 Discussion

In this paper, the sufficient conditions of existence and stability of positive almost periodic solutions for system (1.1) on time scale are obtained. Our results shows that the continuous system and discrete system can be unified well on time scales system.

Acknowledgements

The research is supported by the Project for Young and Middle-aged Teachers in Ningde Normal University (2018Q104), Fujian Educational Research Projects (JT180602) and Fujian Science and Technology Project (2019J01841).

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Received: 2019-04-03

Accepted: 2019-11-02

Published Online: 2019-12-13

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

Articles in the same Issue

https://doi.org/10.1515/math-2019-0134

Keywords for this article

almost periodic solutions; commensal symbiosis model; Lyapunov functional; Michaelis-Menten type harvesting; time scales

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