Abstract
We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results.
1. Introduction
Let denote the set of all nonnegative integers. For any bounded sequence , set and
In the real world, there are many types of interactions between two species. Competitive relations are among the most common ecological interactions. As we all know, the competitive system has been established and was accepted by many scientists and now it became the most important means to explain the ecological phenomenon. During the last decade, the study of the dynamic behaviors of competitive system with toxic substance or feedback control have been discussed by many authors; see, for example, [1–16]. However, most of the studies are based on the traditional Lotka-Volterra competitive system [5, 7, 8, 17–20]; seldom did scholars consider the nonlinear case [1–4, 8, 9, 11, 12, 15, 16, 21–25].
In [1], Li and Chen studied the extinction property of the following two species competitive system: where , , , , , are assumed to be continuous and bounded above and below by positive constants and , are population density of species and at time , respectively.
In fact, when the size of the population is relatively small, the discrete time models governed by difference equations are more appropriate than the continuous ones. Therefore, Li and Chen [2] and Guo et al. [3] studied the following discrete Lotka-Volterra competition system: where ,,, and ,, are bounded nonnegative sequences defined on . In [2], Li and Chen showed that if the coefficients of system (2) satisfy species will be driven to extinction. In [3], Guo et al. introduced the average growth rate and showed that if the coefficients of system (2) satisfy the following inequality: then the same conclusion holds. Obviously, condition is weaker than that of .
Since conditions and are all sufficient conditions, one of the interesting problems is whether the results still hold under the weaker condition. Now let us consider the following example.
Example 1. Consider the following system: In this case By simple computation, one can see that and (6) and (7) show that neither nor holds; hence one could not draw any conclusion about the dynamic behaviors of the system. However, Figure 1 shows species will be driven to extinction in this case. This motivates us to revisit the extinction property of system (2).
On the other hand, Gilpin and Ayala [21] conducted experiment on fruit fly dynamics to test the validity of 10 models of competitions. One of the models accounting best for the experimental results is given by
Fan and Wang [22] studied the dynamic behaviors of the following nonautonomous -species Gilpin-Ayala competitive system: where , , and , , are continuous for are positive constants.
Chen et al. [23] studied a discrete -species Gilpin-Ayala competitive system where , , , are all positive sequences bounded above and below by positive constants. are positive constants.
Recently, stimulated by the works of [1, 21, 22], Chen et al. [24] proposed the following, a nonautonomous nonlinear competition system: Chen et al. showed that if the coefficients of system (10) satisfy the second species will be driven to extinction while the first one will stabilize at a certain solution of the system
Stimulated by the works of [1–3, 21–24], we propose the following a nonlinear discrete two species competition system: We introduce the following assumptions: are bounded sequence defined on ; , , and , , are bounded nonnegative sequences defined on ; , , are positive constants. There exists positive integer such that for each
From the point of view of biology, we assume that , ; then system (14) has a positive solution passing through
The aim of this paper is, by developing the analysis technique of Li and Chen [2], Chen et al. [4, 24], and Xu et al. [6], to study the extinction property of system (14).
The organization of this paper is as follows. In Section 2, sufficient conditions for the permanence of system (14) are obtained. In Section 3, we study the extinction of species . In Section 4, we study the global stability of species when species is eventual extinction. Examples are presented in Section 5 to show the feasibility of our main results.
2. Permanence
Lemma 2 (see [26]). Assume that satisfy and where and are nonnegative sequences bounded above and below by positive constants. Then
Lemma 3 (see [26]). Assume that satisfies , and , where and are nonnegative sequences bounded above and below by positive constants and . Then
Lemma 4. Assume that ; every positive solution of system (14) satisfies where .
Proof. By the first equation of system (14), we have Suppose ; then
From (21), we have That is, Applying Lemma 2 such that hence
Lemma 5. Assume that ; every positive solution of system (14) satisfieswhere .
Proof. The proof of Lemma 5 is similar to that of Lemma 4, so we omit the detail here.
Lemma 6. Assume that holds; every positive solution of system (14) satisfies where , , and .
Proof. In view of (26), for each , there exists a such that By the first equation of system (14), we have Suppose ; then
From (28), we have That is, where .
Applying Lemma 3 such that where and .
Hence Note that Thus And soHence
Lemma 7. Assume that holds; every positive solution of system (14) satisfies where , , and .
Proof. The proof of Lemma 7 is similar to that of Lemma 6, so we omit the detail here.
Lemma 8. Assume that hold; then system (14) is permanent. That is, for every solution of system (14), one has
3. Extinction
Theorem 9. Assume that hold; assume further that .Let be any positive solution of system (14); then as
Proof. By Lemma 2 we know that there exists such that By , there exist positive constants and such that By , we can choose positive constants , , and such that Thus, there exists a , such that for all Let ; then From (43) and (44), it follows that For any , we choose an integer such that . Integrating (46) from to , from (43), we have where and
(47) implies that for all . Hence, exponentially as
4. Global Stability
In Section 3, we prove that species will be driven to extinction if the conditions , , and hold. Now we investigate the stability property of species under the same conditions.
Before we state the main result of this section, we first introduce some lemmas.
Lemma 10. Assume that , , , and hold; let be any positive solution of system (14); then where
Proof. Under the assumption conditions , , and , it follows from Theorem 9 that From Lemma 4, we have By Lemma 10, it is enough to show that In view of (50) and (51), for each , there exists an integer such that We consider the following two cases.
Case 1. We assume that there exists an such that . Note that for In particular, with , we obtain which implies that From (54) and (56), it follows that Let Note that and thus ; also, for arbitrary , or We claim that By way of contradiction, assume that there exists a such that . Then . Let be the smallest integer such that . Then . The above argument produces that , a contradiction. This proves the claim.
Case 2. We assume that for ; then . We claim that By way of contradiction, assume that Taking limit in the first equation in system (14) gives which is a contradiction since This proves the claim; then we have Combining Case 1 and Case 2, we see that Setting , note that Now, we can easily see that (52) holds. This completes the proof of Lemma 10.
We consider a discrete equationwhere and are bounded nonnegative sequences; similarly to the proof of Lemma 10, we can obtain the following lemma.
Lemma 11. For any positive solution of (70), one has where and .
Now, we state the main result of this section.
Theorem 12. Assume that , , , and hold; assume further that then for any positive solution of system (14) and any positive solution of system (70), one has
Proof. Since , , and hold, it follows from Theorem 12 that To prove , let It follows from the first equation of system (14) and (74) that Using the Mean Value Theorem, we get Then the first equation of system (14) is equivalent to where .
To complete the proof, it suffices to show that We first assume that and then we can choose positive constant small enough such that For above , according to Lemmas 10 and 11 and (73), there exists an integer such that It follows from (81) that Note that implies that lies between and . From (77) and (80)–(82), we get This implies that Since and is arbitrary small, we obtain ; it means that (78) holds when .
Note that and thus, is equivalent to or Now, we can conclude that (78) is satisfied as holds, and so . This completes the proof of Theorem 12.
As a direct corollary of Theorems 9 and 12, for system (2), we have the following result.
Corollary 13. Assume that , , , and hold; assume further that then for any positive solution of system (2) and any positive solution of one has
5. Examples
The following examples show the feasibility of our main results.
Example 14. Now let us consider Example 1; in this case, one can easily check that hence Also Equations (91)–(93) show that all the conditions of Corollary 13 hold; then species will be driven to extinction while species will be globally attractive with any positive solution of the following discrete equation:
Example 15. Consider the following system: In this case, corresponding to system (2), , , , , , , , , , and . By simple computation, one can see that Equations (97) and (99) show that all the conditions of Theorem 12 hold; thus species is driven to extinction while species is asymptotic to any positive solution of Figure 2 shows the dynamic behaviors of system (95).
6. Conclusion
In this paper, we consider a nonlinear discrete two species competition system with the effect of toxic substances. In Theorem 9, by constructing a suitable Lyapunov-type function, we obtain a set of sufficient conditions which ensure species will be driven to extinction. Our results improve and generalize Theorem 2.1 of [2] and Theorem 1.1 of [3].
Competing Interests
The authors declare that there are no competing interests.
Acknowledgments
The research was supported by the Natural Science Foundation of Fujian Province (2015J010121, 2015J01019).