Research Article Dynamic Behaviors of a General Discrete

We study the dynamic behaviors of a general discrete nonautonomous system of plankton allelopathy with delays. We first show that under some suitable assumption, the system is permanent. Next, by constructing a suitable Lyapunov functional, we obtain a set of sufficient conditions which guarantee the global attractivity of the two species. After that, by constructing an extinction-type Lyapunov functional, we show that under some suitable assumptions, one species will be driven to extinction. Finally, two examples together with their numerical simulations show the feasibility of the main results.


Introduction
The aim of this paper is to investigate the dynamic behaviors of the following general discrete nonautonomous system of plankton allelopathy with delay: together with the initial condition N i −l ≥ 0, N i 0 > 0, i 1, 2; l 0, 1, . . ., m, 1.2 where m is a positive integer, N i k represent the densities of population i at the kth generation, r i k are the intrinsic growth rate of population i at the kth generation, a il k measure the intraspecific influence of the k − l th generation of population i on the density of own population, b il k stand for the interspecific influence of the k − l th generation of population i on the density of own population, and c il k stand for the effect of toxic inhibition of population i by population j at the k − l th generation, i, j 1, 2 and i / j.Also, {r i k }, {a il k }, {b il k } and {c il k } are all bounded nonnegative sequences defined for k ∈ N, denoted by the set of all nonnegative integers, and l ∈ {0, 1, . . ., m} such that here, for any bounded sequence {f k }, define As was pointed out by Chattopadhyay 1 the effects of toxic substances on ecological communities are an important problem from an environmental point of view.Chattopadhyay 1 and Maynard-Smith 2 proposed the following two species Lotka-Volterra competition system, which describes the changes of size and density of phytoplankton: where x 1 t and x 2 t denote the population density of two competing species at time t for a common pool of resources.The terms b 1 x 1 t x 2 t and b 2 x 1 t x 2 t denote the effect of toxic substances.Here, they made the assumption that each species produces a substance toxic to the other, only when the other is present.Noticing that the production of the toxic substance allelopathic to the competing species will not be instantaneous, but delayed by different discrete time lags required for the maturity of both species, thus, Mukhopadhyay et al. 3 also incorporated the discrete time delay into the above system.Tapaswi and Mukhopadhyay 4 also studied a two-dimensional system that arises in plankton allelopathy involving discrete time delays and environmental fluctuations.They assumed that the environmental parameters are assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density.They focus on the dynamic behavior of the stochastic system and the fluctuations in population.For more works on system 1.5 , one could refer to 1-3, 5-24 and the references cited therein.Since the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, and discrete time models can also provide efficient computational models of continuous models for numerical simulations, corresponding to system 1.5 , Huo and Li 25 argued that it is necessary to study the following discrete two species competition system: where x 1 k and x 2 k are the population sizes of the two competitors at generation k, b 1 k and b 2 k have respectively, shown that each species produces a toxic substance to the other but the other only is present.In 25 , sufficient conditions were obtained to guarantee the permanence of the above system, they also investigated the existence and stability property of the positive periodic solution of system 1.6 .Recently, Li and Chen 26 further investigated the dynamic behaviors of the system 1.6 .For general nonatonomous case, they obtain a set of sufficient conditions which guarantee the extinction of species x 2 and the global stability of species x 1 when species x 2 is eventually extinct.For periodic case, the other set of sufficient conditions, which concerned with the average condition of the coefficients of he system, were obtained to ensure the eventual extinction of species x 2 and the global stability of positive periodic solution of species x 1 when species x 2 is eventually extinct.For more works on discrete population dynamics, one could refer to 7, 10, 25-45 .
Liu and Chen 32 argued that for a more realistic model, both seasonality of the changing environment and some of the past states, that is, the effects of time delays, should be taken into account in a model of multiple species growth.They proposed and studied the system 1.1 , which is more general than system 1.6 .By applying the coincidence degree theory, they obtained a set of sufficient conditions for the existence of at least one positive periodic solution of system 1.1 -1.2 .Zhang and Fang 46 also investigated the periodic solution of the system 1.1 , they showed that under some suitable assumption, system 1.1 could admit at least two positive periodic solution.As we can see, the works 32, 46 are all concerned with the positive periodic solution of the system.However, since few things in the nature are really periodic, it is nature to study the general nonautonomous system 1.1 , in this case, it is impossible to study the periodic solution of the system, however, such topics as permanence, extinction, and stability become the most important things.In this paper, we will further investigate the dynamics behaviors of the system 1.1 .More precisely, by developing the analysis technique of Liu 31 and Muroya 35, 36 , we study the permanence, global attractivity and extinction of system 1.1 -1.2 .
The organization of this paper is as follows.We study the persistence property of the system in Section 2 and the stability property in Section 3. Then in Section 4, by constructing a suitable Lyapunov functional, sufficient conditions which ensure the extinction of species N 2 of system 1.1 -1.2 are studied.In Section 5, two examples together with their numeric simulations show the feasibility of main results.For more relevant works, one could refer to 2, 3, 5-9, 12, 13, 27-30, 33, 34, 37-45 and the references cited therein.

Permanence
In this section, we study the persistent property of system 1.1 -1.2 .

Lemma 2.1. For any positive solution
where Proof.Let { N 1 k , N 2 k } be any positive solution of system 1.1 -1.2 , in view of the system 1.1 for all k ∈ N, we have This completes the proof of Lemma 2.1.

Lemma 2.2. Assume that
hold, where B 1 and B 2 are defined in 2.2 .Then for any positive solution where

2.7
Proof.In view of 2.5 , we can choose a constant ε > 0 small enough such that In view of 2.1 , for above ε > 0, there exists an integer k 0 ∈ N such that We consider the following two cases.
Case (i).We assume that there exists an integer

2.11
So we can obtain It follows from 2.8 that 2.13 Then we have

2.18
We can claim that By way of contradiction, assume that there exists an integer p 0 ≥ l 0 such that N 1 p 0 < N 1ε .Then p 0 ≥ l 0 2. Let p 0 ≥ l 0 2 be the smallest integer such that N 1 p 0 < N 1ε .Then N 1 p 0 − 1 > N 1 p 0 .The above argument produces that N 1 p 0 ≥ N 1ε , a contradiction.Thus 2.19 proved.
Case (ii).We assume that We can claim that

2.20
By the way of contradiction, assume that

7
Taking limit in the first equation of 1.1 gives

2.23
The claim is thus proved.From 2.20 , we see that Combining Cases i and ii , we see that

2.26
So we can easily see that lim inf From the second equation of 1.1 , similar to above analysis, we have lim inf where A 2 is defined in 2.6 .This completes the proof of Lemma 2.2.
It immediately follows from Lemmas 2.1 and 2.2 that the following theorem holds.

Global attractivity
This section devotes to study the stability property of the positive solution of system 1.1 -1.2 .
Theorem 3.1.Assume that there exists a constant η > 0 such that where, for i, j 1, 2, i / j, B i and

then for any two positive solutions
Proof.First, let Then from the first equation of 1.1 , we have

3.6
Substituting 3.6 into 3.4 leads to

3.7
So it follows that

3.8
According to 2.1 , for any constant ε > 0, there exists an integer k 0 ∈ N such that So for all k ≥ k 0 m, l 0, 1, . . ., m, it follows that

3.10
Discrete Dynamics in Nature and Society So for all k ≥ k 0 m, it follows from 3.4 that

3.11
Next, let 3.12 and we can obtain

3.13
Now, we define V 1 by

3.14
So for all k ≥ k 0 m, it follows from 3.6 and 3.9 that

3.15
Similar to above arguments, we can define where

3.17
Then for all k ≥ k 0 m, we can obtain where θ 2 k lies between N 2 k and N * 2 k .Now, we define V by

3.19
It is easy to see that V k ≥ 0 for all k ∈ Z and V k 0 m < ∞.For the arbitrariness of ε > 0 and by H 0 , we can choose ε > 0 small enough such that for i 1, 2, min a L i0 ,

3.20
So for all k ≥ k 0 m, it follows from 3.15 and 3.18 that

3.21
So we have Yaoping Chen et al. 13 This completes the proof of Theorem 3.1.

Extinction of species N 2
This section devotes to study the extinction of the species N 2 .

Lemma 4.1. For any positive solution
Proof.By 2.1 , there exists a constant B > 0 such that In view of 1.1 for all k > k 0 m, i 1, 2, it follows that So we have For all k > k 0 m, i, j 1, 2, i / j, it follows from 1.1 and 4.4 that 4.6 so we have

4.8
Note that for all k > k 0 , x k N 1 k N 2 k < 2B, so similar to the proof of Lemma 2.2 of Chen 27 , we have lim inf Then, there is a positive constant σ > 0 such that lim inf This completes the proof of Lemma 4.1.

Theorem 4.2. Assume that
Corollary 4.3.Assume that for all l 0, 1, . . ., m, the following inequalities For above ε > 0 from 2.1 , there is an integer K ∈ N such that for i 1, 2, Lemma 4.1 also implies that there exists K 1 > K such that

4.15
So for all k > K 2 > K 1 m, it follows from 1.1 , 4.13 , and 4.14 that

4.16
That is, for all k > K 2 ,

4.17
So from the definition of u k it follows that

4.18
The above analysis shows that lim This completes the proof of Theorem 4.2.

Examples
The following two examples show the feasibility of our results.
Example 5.1.Consider the following system One could easily see that

5.2
Clearly, conditions 2.5 are satisfied.From Theorem 2.3, it follows that system 5.1 is permanent.Also, by simple computation, we have

5.3
The above inequality shows that H 0 is fulfilled.From Theorem 3.1, it follows that   The above inequality shows that H * 1 is fulfilled.From Theorem 4.2, it follows that lim k → ∞ N 2 k 0. Numeric simulation of the dynamic behaviors of system 5.5 with the initial conditions N 1 k , N 2 k 0.42, 0.6 , k −1, 0 is presented in Figure 3.
Remark 5.3.In the above two examples, we can take sin k , cos k as the perturbation terms.
Our numeric simulations show that if the perturbation terms are large enough, then those terms will greatly influence the dynamic behaviors of the system, and in some cases, may lead to the extinction of the species.
where B 1 is defined in 2.2 .Let { N 1 k , N 2 k } be any positive solution of system 1.1 -1.2 , then N 2 k → 0 as k → ∞.Proof ofCorollary 4.3.Obviously, if condition H * 1 holds, one could easily see that condition H 1 holds, thus, the conclusion of Corollary 4.3 follows from Theorem 4.2.The proof is complete.Proof of Theorem 4.2.It follows from H 1 that we can choose a constant ε > 0 small enough such that