^{1}

^{*}

^{1}

In research [1], the authors investigate the dynamic behaviors of a discrete ecological system. The period-double bifurcations and chaos are found in the system. But no strategy is proposed to control the chaos. It is well known that chaos control is the first step of utilizing chaos. In this paper, a controller is designed to stabilize the chaotic orbits and enable them to be an ideal target one. After that, numerical simulations are presented to show the correctness of theoretical analysis.

Population dynamics in ecology are generally governed by discrete and continuous systems. In recent years, the study of discrete ecological systems has attracted extensive attentions [1-6]. This is because that some natural populations have non-overlapping generations, thus discrete models are more realistic than continuous ones to study these species. Another reason is that people always study population changes by one year (mouth, week or day). Such investigations are often required discrete models. Especially, using discrete models is more efficient for numerical simulations. Recently, Zhang and Li [

where x_{n}, y_{n} denote the two ecological species’ densities respectively in generation n; δ is the integral step size. The more meaning of system (1) can refer to the reference [1,2]. It is shown that the system (1) generates period-double bifurcations and chaos. But the authors did not investigate the chaos control of the system.

It is well known that chaos control is the first step of utilizing chaos. The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. Recently, some authors have investigated that such a method can be applied to population dynamics and even play a nontrivial evolutionary role in ecology [7-9]. In this paper, we design a proper controller to control the chaos of system (1).

In this section, chaotic orbits to an unstable fixed point are stabilized by utilizing some control techniques. Firstly, we introduce the following lemma which is useful to establish our results Lemma 1^{ }[

Consider the following map which is the feedback is applied to system (1)

where X_{n} = (x_{n}, y_{n})^{T}, μ_{n} is control variable and satisfies,. Evidently, map (2) degenerates to original system (1) only if μ_{n} = 0. We select the feedback variable μ_{n} in the range (–ε, ε), so that the orbit holds in the neighborhood of fixed point E as long as the control arises. The ergodic nature of the chaotic dynamics guarantees that the mode trajectory in the neighborhood of the wishful orbit. In the neighborhood of E, map (2) can be approximated by the following form:

where A is the Jacobian matrix at E and B is a column vector, and they are given by:

,

.

Let X^{*} = (x^{*}, y^{*})^{T} and suppose that μ_{n} is a linear function of X_{n}, which is expressed as μ_{n} = P^{T}(Xn – X^{*}),. Substitute the result into (3), we get

.

According to the study [^{T}) is asymptotically stable, that is to say, all its eigenvalues are less than 1 in modulus. Now, we make use of “pole placement technique” [^{T}). If system (1) is chaotic, we obtain

.

Then we choose ,

as the desired eigenvalues of the matrix (A – BP^{T}). The controllability matrix

has two rank. Thus the solution to the pole placement problem is obtained as

where Q = CW, p_{1} and p_{2} are the coefficients of characteristic polynomial of the matrix A, and ,;

q_{1} and q_{2} are the coefficients of characteristic polynomial of the matrix (A – BP^{T}),

and , q_{2} = 0.

After calculations, we get

Furthermore, the controller has the following form:

where,.

However, the above considerations only are fit for a local small neighbor of E. In view of the global situationwe can specify μ_{n} by making μ_{n} = 0 if

is too large. This is because the range of μ_{n} is restrained by and. Thus, we limit the number value

.

Therefore, in practice we take μ_{n} as

According to the above analysis, we get the following result.

Theorem 1. If then the control variable can stabilize chaotic trajectory of system (1) to the fixed point E, where P^{T} is given by Equation (4).

In the section, we use density-time diagrams and phase portraits to confirm the above theoretical analysis.

Let a = 2.21, b = 1.02, δ = 0.9666. At the condition, has the value 7.70732. According to Lemma 1, system (1) has and only has a positive fixed point E(x^{*}, y^{*}) = (2.16667, 0.53846). We adopt

.

When ε is given the value 0.03 and 0.09, Theorem 1 is satisfied. Density-time diagram of ecological specie x_{n} is given by

stabilizes to the fixed point (2.16667, 0.53846), which is simulated by Figures 2(c) and 2(d).

In this paper, we design a proper controller to control the chaos of system (1) which was firstly studied by Zhang and Li [^{*}, y^{*}) under the condition of

where P^{T} is given by Equation (4). Then simulations are presented to show the correctness of theoretical analysis.

This work is supported by the National Natural Science Foundation of China (No. 30970305), the Sichuan Provincial Natural Science Foundation (No. 10ZB136), the Sichuan Provincial Old Revolutionary Base Areas Foundation (No. SLQ2010C-17).