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

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.


Introduction
Many results of di erential equations can be easily generalized to di erence equations, while other results seem to be completely di erent from their continuous counterparts. A major task of mathematics today is to harmonize continuous and discrete analysis. The theory of time scale, which was rst 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 di erence 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 di erent 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 in uence 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:

) = a(t) − b(t) exp{x(t)} + c(t) exp{y(t)}, y ∆ (t) = d(t) − e(t) exp{y(t)} − q(t)E(t) exp{y(t)} E(t) + m(t) exp{y(t)} ,
where x(t), y(t) are the density of species x, y at time t ∈ T (T is a time scale). x ∆ , y ∆ express the delta derivative of the functions x(t), y(t). E(t) denotes the harvesting e ort and q(t) is the catch ability coe cient. The coe cients 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 x (t), y(t) = ln y (t), if T = R + , then system (1.1) is reduced to a continuous version: if T = Z + , then system (1.1) can be simpli ed as the following discrete system: ( . ) The rest of this paper is arranged as follows. The next part we present some notations. After that, su cient 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.

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 + µ(t)p(t) ≠ 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 de ne the set R + = R + (T, R) = {p ∈ R: + µ(t)p(t) > for all t ∈ T}.
If p is a regressive function, then the generalized exponential function ep is de ned by For further reading we refer to the book by Bohner and Peterson [2].
De nition 2.1 (see [2]). Let T be a time scale. For t ∈ T we de ne the forward and backward jump operators σ, ρ : T → T and the graininess function µ : T → R + by and µ = inf t∈T + µ(t), µ = sup t∈T + µ(t).
De nition 2.2 (see [6]). A time scale T is called an almost periodic time scale if De nition 2.3 (see [6]). Let T be an almost periodic time scale. A function x ∈ C(T, R n ) is called an almost periodic function if the ε-translation set of x is a relatively dense set in T for all ε > , that is, for any given ε > , there exists a constant l(ε) > such that each interval of length l(ε) contains a τ(ε) ∈ E{ε, x} such that De nition 2.4 (see [6]). Let T be an almost periodic time scale and D denotes an open set in R n . A function f ∈ C(T × D, R n ) is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε-translation set of f is a relatively dense set in T for all ε > and for each compact subset S of D, that is, for any given ε > and each compact subset S of D, there exists a constant l(ε, S) > such that each interval of length l(ε, S) contains a τ(ε, S) ∈ E{ε, f , S} such that Lemma 2.5 (see [7]). Assume that a > , b > and −a ∈ R + . Then and its associate product system where f : T + × S H → R n , S H = {x ∈ R n : 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) de ned on T + × S H × S H satisfying the following conditions 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.
then any positive solution (x(t), y(t)) of system (1.1) satis es Proof. From the second equation of system (1.1) it follows By using Lemma 2.5 we get For a su ciently small ε > , from (2.5) and (2.8), there exists a t ∈ T + such that From (2.9) and the rst equation of system (1.1), we have By using Lemma 2.5 again, we have

Lemma 2.8 Under the hypothesis (2.5) and
then any positive solution (x(t), y(t)) of system (1.1) satis es ( . ) Proof. Lemma 2.7 means that for any ε > , there exists a t > t (the de nition of t in Lemma 2.7) such that It follows from the second equation of system (1.1) that We claim that for t ≥ t , Suppose that there exists at ≥ t such that Then and for any t ∈ [t ,t) T + , which implies y ∆ (t) < . It is a contradiction, so that (2.17) holds, i.e., For above ε and (2.24), there exists a t > t such that It follows from the rst equation of system (1.1) and above inequation that By analyzing (2.26) similar to (2.17)-(2.24), one has
( . ) Therefore, one has Also, the assumption (iii) of Lemma 2.6 is satis ed. 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: 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. Let Ω = {y(t) : y(t) is a solution of (3.27) and < N y ≤ y(t) ≤ M y }. It is obvious that Ω is an invariant set.

Numerical Simulations
We give the following examples to illustrate the feasibility of our main results. > ,

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

Discussion
In this paper, the su cient 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 uni ed well on time scales system.  Figure 4: Dynamic behaviors of the solutions y * (t) of system (4.9) with the initial conditions y * ( ) = . , . and , respectively.