Abstract
A discrete allelopathic phytoplankton model with infinite delays and feedback controls is studied in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the extinction of the system are obtained. Our results extend and supplement some known results and show that the feedback controls and toxic substances play a crucial role on the permanence and extinction of the system.
1. Introduction
Given a bounded sequence of real numbers , let and denote and , respectively.
Many real-world phenomena are studied through discrete mathematical models governed by difference equations which are more appropriate than the continuous ones when the populations have nonoverlapping generations; the study of the dynamic behaviors of discrete time models becomes the subject of intense research in mathematics biology, such topics as permanence and extinction, and existence of positive periodic solution (almost periodic solution), have been extensively studied by many scholars (see [1–23] and the references cited therein).
Recently, some scholars believed that a more appropriate competition model should be considered with nonlinear interinhibition terms. Qin et al. [1] and Wang et al. [2] considered the following discrete two species competitive system with nonlinear interinhibition terms:where are assumed to be bounded positive sequences, and represent the density of species and at the kth generation, respectively, is the intrinsic growth rate of two species. is the rate of intraspecific competition of the first and second species, respectively. is the rate of interspecific competition of the first and second species, respectively. Qin et al. [1] obtained sufficient conditions which ensure the permanence, existence, and global stability of positive periodic solutions of system (1). As for the almost periodic case, Wang and Liu [2] further investigated the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1).
In recent years, many scholars have made great achievements in the competition system with the effect of toxic substance [3, 4, 24–26]. Yue [3] investigated system (1) with the second species which could be toxic, while the other one is nontoxic. He studied system (2) and gave the sufficient conditions of the extinction of one species and the global attractive of the other one.
Note that ecosystems are easily disturbed by human activities, such as planting and harvesting, which can give rise to changes of population density. In order to give a better description of such a system, scholars introduced feedback control variables into ecosystems. Many researchers have done research on the systems with feedback control variables [6–8, 11, 14, 16, 17]. Recently, Wang and Liu [6] studied the competitive system with feedback controls as follows:where , denotes the feedback control variables. Wang and Liu [6] studied the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (3). Yu [7] further investigated the influence of feedback control variables on the extinction of the system. He gave a set of sufficient conditions for the extinction of one species.
As we all know, due to seasonal fluctuations in the environment and hereditary factors, time delays have been introduced into the biological system (see [11, 14, 21, 27–36]). Zhao et al. [8] further considered a discrete Lotka–Volterra competition system with infinite delays and single feedback control variable as follows:
By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained.
Through the above discussion, we find that, based on system (2), we can further consider the infinite delay and feedback control. So, we propose and study the following discrete competition system with infinite delays and feedback control variables:
In system (4), is the density of species at the kth generation, and , is the feedback control variable.
Throughout this paper, we assume that
() are bounded sequences of real numbers defined on Z such that
, and () are nonnegative bounded sequences such that
According to the biological background of system (5), we only consider the solution of system (5) with the following initial conditions:where . It is easy to prove that the solution of system (5) which satisfies the initial conditions (8) is positive.
Here, we mention that this is the first time such kind of model be proposed and studied, and as far as system (5) is concerned, whether the feedback control variables and toxic substances have influence on the permanence and extinction of the system or not is an interesting problem. The aim of this paper is to investigate the permanence and extinction of system (5); specially, we will find out the answer to the above problem.
The paper is structured in the following way. Some useful Lemmas are presented in Section 2. In Sections 3 and 4, by using the methods of Zhao et al. [8], we investigate the permanence and extinction of system (5). Three examples are presented to show the feasibility of the main results in Section 5. We end this paper by a brief discussion.
2. Lemmas
Now, let us consider the following difference equation:where are positive constants.
Lemma 1 (see [37]). Assume that , for any initial value , there exists a unique solution of equation (9), which can be expressed as follows:where . Thus, for any solution y(k) of system (9), we have
Lemma 2 (see [37]). Let , . For any fixed k, is a nondecreasing function with respect to r, and for , the following inequalities hold:If , then for all .
Lemma 3 (see [38]). Assume that , x(n) satisfies andfor , where a is a positive constant. Then,
Lemma 4 (see [38]). Assume that , satisfies andfor , and , where a and are positive constants such that . Then,
Lemma 5 (see [38]). Let x: be a nonnegative bounded sequence, and let H: be a nonnegative sequence such that , then
3. Permanence
Concerned with the permanence of system (5), we have the following result.
Theorem 1. Assume thathold; then, for any positive solution of system (5), we havewhere
Proof. From the first and second equations of system (5), we haveHence, from Lemma 3, we can obtainAccording to Lemma 5 and the above inequality, we haveFor any , there exists a positive integer such thatBy the third and fourth equations of system (5), we haveHence, by applying Lemmas 1 and 2 to (25), we obtainSetting , it follows thatCondition (18) implies that, for enough small positive constant , the following inequalities hold:For above , it follows from (22) and (27) that there exists a positive integer such thatThus, for all , from (28), (29), and the first two equations of system (5), we havewhereNotice thatthenHence, according to Lemma 4, we haveSetting , it follows thatwhereAccording to Lemma 5, from (35) and (36), we have, for any small enough (without loss of generality, assume that , i = 1, 2), there exists , such thatFor , from (38) and the third and fourth equations of system (5), we haveHence, by applying Lemmas 1 and 2 to (39), we obtainSetting , it follows thatThis ends the proof of Theorem 3.1.
4. Extinction
Concerned with the extinction property of system (5), we could establish the following results.
Theorem 2. Assume thatholds. Let be any positive solution of system (5), then
Theorem 3. Assume thathold. Let be any positive solution of system (5), then
Proof of Theorem 2. Condition is equivalent toFrom (46), one could choose positive constants , and enough small positive ε such thatThat is,Let be a positive solution of system (5). For above ε, from Theorem 1, there exists a enough large , such thatConsider the following discrete Lyapunov functionalBy calculating, we obtainFrom inequalities (48) and (51), we can obtainTherefore,From (22) and (27), we know that there exists a such thatHence,On the contrary, we also haveCombining inequalities (53), (55), and (56), we havewhereHence, we obtain thatSimilar to the corresponding proof of Theorem 1 by Chen et al. [7], we can easily obtain that . This ends the proof of Theorem 2.
Proof of Theorem 3. Condition is equivalent toFrom and (60), there exist positive constants and enough small positive ε such thatThat is,Let be a positive solution of system (5). For above ε, from Theorem 1, there exists an enough large , such thatDefine the following Lyapunov functional as follows:Similar to the analysis of (51)–(58), we haveThis ends the proof of Theorem 3.
From Theorems 2 and 3, we know that under some suitable assumption, one of the species in the system may be driven to extinction; in this case, one interesting problem is to investigate the stability property of the rest species.
Consider the following discrete equations:
Theorem 4. Assume that () holds; also,holds; then, for any positive solution of system (5) and any positive solution of system (66), we havewhere .
Theorem 5. Assume that () and () hold; also,holds; then, for any positive solution of system (5) and any positive solution of system (67), we havewhere
Proof of Theorem 4. By conditions (68), we can choose positive constants such thatThus, there exists enough small positive constant ω and ε such thatwhere .
Now, we define a Lyapunov functional as follows:One could easily see that for all Also, for any fixed from (54), one could see thatIt follows from the second equation of system (5) and the mean value theorem thatwhere .
Summating both sides of the above inequality from to k, we haveHence,Then, from (75) and (57), we haveTherefore,which means thatConsequently,This completes the proof of Theorem 4.
Proof of Theorem 5. The proof of Theorem 5 is similar to that of Theorem 4, and we omit the detail here.
5. Examples
In this section, we shall give three examples to illustrate the feasibility of main results.
Example 1. Consider the following equations:One could easily see that conditions () and () are satisfied. Also, by calculating, one hasClearly, condition (18) is satisfied, and so from Theorem 1, we havewhere is any positive solution of system (83).
Figure 1 shows the dynamic behaviors of system (83), which strongly supports the above assertions.
Example 2. Consider the following equations:One could easily see that conditions () and () are satisfied. Also, by calculating, one hasClearly, condition () and (68) are satisfied, and so from Theorem 2 and 4 we know that species will be driven to extinction, while species is global attractive.
Figure 2 shows the dynamic behaviors of system (86), which strongly support our results.
Example 3. Consider the following equations:One could easily see that conditions () and () are satisfied. Also, by calculating, one hasClearly, conditions (), (), and (70) are satisfied, and so from Theorem 3 and 5, we know that species will be driven to extinction, while species is global attractive.
Figure 3 shows the dynamics behaviors of system (88), which strongly supports our results.
6. Discussion
(1) From the conditions of Theorems 2–5, we can easily find that the feedback control variables and toxic substances play a crucial role on the extinction of system (5). We find that, by choosing suitable feedback control variable, one of the species will be extinct or permanent, that is, feedback control variable, which represents the biological control or some harvesting procedures, is an unstable factor of the system. From the conditions of Theorem 2, we also find that, despite the second species could produce toxicity, if the toxic rate is very low such that inequality holds; then, the second species is still driven to extinction; in other words, that lower rate of toxic production has no influence on the extinction property of system (5).(2) From the conditions of Theorem 1, we find that the feedback control variables and toxic substances play an important role on the permanence of the system; only the rate of toxic production and feedback control variables are small enough such that inequality (18) holds, and the toxic substances and feedback controls have no effect on the permanence of the system.(3) Yu [7] obtained a set of sufficient conditions that guarantees the extinction of system (3), as direct corollaries of Theorems 2–5; one could also obtain other conditions for the extinction of system (3), which supplements and complements the results of Yu [7].(4) Now, concerned with the extinction property of system (2), with some minor revisions to the proof of Theorems 2 and 3, we could obtain the following results.
Corollary 1. Assume thatholds, and let be any positive solution of system (2), then
Corollary 2. Assume thathold; let be any positive solution of system (2), then
It is interesting to investigate the stability property of the rest species under the assumption that one of the species in system (2) is driven to extinction. As direct corollaries of Theorems 4 and 5, by simple computation, we have
Corollary 3. Assume that (90) holds; also,holds; then, for any positive solution of system (2), we havewhere is any positive solution of the system .
Corollary 4. Assume that (92) and (93) hold; also,holds; then, for any positive solution of system (2), we havewhere is any positive solution of the system .
Obviously, Theorems 2.1, 2.5, 3.1, and 3.2 obtained by Yue [3] are corollaries 1–4 of Theorems 2–5, so corollaries 1–4 extend the results of Yue [3].
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
The research was supported by the Natural Science Foundations of China (no. 11771082) and the Scientific Research Development Fund of Young Researchers of Guangxi University of Finance and Economics (no. 2019QNB09).