Abstract
This is a continuation of the work of Tan et al. (2012). In this paper a periodic single species model controlled by constant impulsive perturbation is investigated. The constant impulse is realized at fixed moments of time. With the help of the comparison theorem of impulsive differential equations and Lyapunov functions, sufficient conditions for the permanence and global attractivity are established, respectively. Also, by comparing the above results with corresponding known results of Tan et al. (2012) (i.e., the above model with linear impulsive perturbations), we find that the two different types of impulsive perturbations have influence on the above dynamics. Numerical simulations are presented to substantiate our analytical results.
1. Introduction
It is well known that many mathematical models can be expressed by impulsive differential equations, and in recent years many impulsive differential equations have been introduced in population dynamics and neural networks, such as harvesting or stocking [1–4], chemostat [5], vaccination [6], integrated pest management [7], chemotherapeutic treatment of disease [8], and neural networks [9]. Also, there has been a significant development in theory of impulsive differential equation. For example, [10–17] studied the existence and stability of systems. In this paper, we investigate the permanence and global attractivity of a periodic impulsive system (see (7a) and (7b)).
In 1978, Ludwig et al. [18] investigated the asymptotic behavior of the following single species autonomous system: where is the density of species at time , is the intrinsic growth rate, is the self-inhibition rate, and -term represents predation. Predation is an increasing function and usually saturates for large enough . The predation term drops rapidly if the density of species is small. To be specific Murray [19] took the form for suggested by Ludwig et al. [18], namely, , and the dynamics of is then governed by where the positive coefficients and are measures of saturation value. Murray [19] discussed the existence and stability of equilibria of system (2).
To investigate the effects of other specific forms of on system (1), similar to Murry [19], in [1] we took and established the following single species system: Obviously, is an increasing function with respect to and has a saturation value, and the positive coefficients and are measures of saturation value. In [1], we further consider the possible effects of periodically varying environment and impulsive perturbations on system (3) and established the following periodic system with linear impulsive perturbations:where , , is an impulsive point for every , and and is the set of positive integers. , , and are positive continuous -periodic functions. The intrinsic growth rate is not necessarily positive, since the environment fluctuates randomly; in a bad environment may be negative when the birth rate is smaller than the death rate. is a real sequence and there exists an integer such that , . We have investigated the permanence and global attractivity of system (4a) and (4b) and two results can be redescribed as follows.
Result A (see Lemma 3.2 in [1]). If assumption holds, then system (4a) and (4b) is permanent; that is, there exist positive constants , , and such that any positive solution of system (4a) and (4b) satisfies , for .
Result B (see Theorem 3.1 in [1]). In addition to (5), assume further that Then, system (4a) and (4b) is global attractive, where is described as Result A.
In this contribution, we continue to consider system (4a) with constant impulsive perturbation (i.e., ) and obtain the following system:The positive constant is the increased amount of species at which implies that species is subjected to impulsive increase at a constant rate .
In this paper, our main purpose is to establish sufficient conditions for the permanence and global attractivity of system (7a) and (7b) and compare them with the effects of the above two different types of impulsive perturbations on the above dynamics. The organization of this paper is as follows. In Section 2, we present some preliminary lemmas. In Section 3, two main results on the permanence and global attractivity of system (7a) and (7b) are presented. In Section 4, some examples together with their corresponding numerical simulations are presented to verify the validity of the proposed criteria.
2. Preliminaries
The following Lemmas 1 and 2 are useful for establishing the permanence of system (7a) and (7b). Lemma 1 shows that the Malthus growth model with constant impulse has a good asymptotic behavior and a detailed proof can be seen in Lemma 2.3 in [4].
Lemma 1 (see [2, 4]). Consider the following impulsive system: where , is a continuous -periodic function with , the constant , and there is an integer such that , . System (8) has a unique positive -periodic solution which is globally asymptotically stable.
Lemma 2 (see [4]). Consider the following impulsive single species system: where , and are continuous -periodic functions, and , the constant , and there exists an integer such that . Then,(1)species is permanent;(2)any two positives and satisfy .
The following Lemma 3 is useful for proving the global attractivity of system (7a) and (7b). We first give the following notations and spaces of functions.
Let and . is continuous for , , continuous from the left for and discontinuous of the first kind occurring at the point , and is continuously differential for , and exist, .
Lemma 3 (see [20]). If the function satisfies the inequalities where , , and are constants for , then, for , Analogously, we have for if all the inequalities of (10) are inverse.
3. Permanence and Global Attractivity
We first give the main result on the permanence of system (7a) and (7b).
Theorem 4. System (7a) and (7b) is permanent; that is, there exist positive constants , and such that any positive solution of system (7a) and (7b) satisfies for .
Proof. It follows from (7a) and (7b) that we obtain
Obviously, if , then for .
We first show that any positive solution of system (7a) and (7b) is uniformly ultimately upper bounded. To do this, we have from (7a) and (7b) that
Consider the comparison system of (14):
By (1) in Lemma 2 we know that system (15) is permanent, which implies that there exist constants and such that
Let be the solution of (15) with . By the comparison theorem of impulsive differential equations and (16), we obtain
Next, we prove that is uniformly ultimately lower bounded. It follows from (7a), (7b), and (17) that we have, for ,
where the positive constant satisfies
It follows from Lemma 1 that the comparison system of (18)
has a unique positive, globally asymptotically stable -periodic solution, denoted by . The asymptotic property of implies that there exist and such that
where the constant . Let be the solution of (20) with . By the comparison theorem of impulsive differential equations and (18)–(21), we have
From (17) and (22), one has
The proof of Theorem 4 is complete.
Remark 5. Theorem 4 shows that the permanence of system (7a) and (7b) is irrespective of the size of the positive impulse .
In the following, we continue to investigate the global attractivity of system (7a) and (7b).
Theorem 6. If system (7a) and (7b) satisfies then system (7a) and (7b) is global attractive, where and are described as in Theorem 4.
Proof. Let and be any two positive solutions of system (7a) and (7b). By Theorem 4, we know that there exists a large enough such that, for all ,
Consider a Lyapunov functional defined by
Calculating and estimating the upper right derivative of along the solutions of (7a) and (7b), we get, for and , that
From assumption (24), we obtain that there exists a constant such that
which, together with (28), yields
By the mean value theorem and (26), for any closed interval contained in , , and , one has
Therefore, for , , and , from (30) and (31), we have
where .
On the other hand, for , , we have
where is between and and .
It follows from (26) and (33) that, for , ,
Again, by the mean value theorem, we have
where is between and and .
Furthermore, we have from (26), (34), and (35) that, for ,
where
Recall assumption (25); we obtain that ; thus, we have from (34)–(36) that
Combining (32) and (38), we have
Note that is an impulsive point; by Lemma 3, we obtain that, for ,
Obviously, it follows from (26) and (27) that
which implies that is bounded.
Thus,
Therefore, from (40) and (42), we obtain that as ; that is,
By the mean value theorem, one has
The proof of Theorem 6 is complete.
Remark 7. The positive impulse has the effect on the global attractivity of system (7a) and (7b).
4. Discussion
In this paper, we have proposed a periodic single species model with constant impulsive perturbation which is a continuation of the work of [1]. From systems (4a), (4b), (7a), and (7b), we can see that the equation (4a) or (7a) describes the density variation of species in a periodically varying environment and the impulsive condition (4b) or (7b) reflects the possible external effects under which the population density changes very rapidly. Result A shows that the suitable large linear impulsive perturbations are favorable for the permanence of system (4a) and (4b) while Theorem 4 indicates that the permanence of system (7a) and (7b) is irrespective of the size of the positive constant impulse . As far as the global attractivity is concerned, Result B shows that the suitable large linear impulsive perturbations and self-inhibition rate are favorable for the global attractivity of system (4a) and (4b). Theorem 6 shows that the suitable large self-inhibition rate is also favorable for the global attractivity of system (7a) and (7b); meanwhile, the positive constant impulse satisfies a harsh condition; that is, .
To substantiate our analytical results, we will construct some specific examples. We first consider system (4a) and (4b) with , , , and . Let , ; it is obvious that and . The linear impulse perturbations satisfy (when ) and (when , , ). A simple calculation shows that assumption (5) of Result A holds, and from Figure 1 we may observe that system (4a) and (4b) with is permanent (here ). Furthermore, we can verify the validity of Result B, and hence system (4a) and (4b) is global attractive. From Figure 2, we observe that the positive solution with tends to the positive solution with . In the following, we verify the feasibilities of Theorems 4 and 6. Let us consider system (7a) and (7b) with , , , , and . Obviously, . Let , ; then, . From Figure 3 we see that system (7a) and (7b) with is permanent (here ). A calculation shows that assumptions of Theorem 6 are satisfied, and hence system (7a) and (7b) is global attractive (see Figure 4; the positive solution with tends to the positive solution with ). The above examples imply that the linear and constant impulsive perturbations play important roles and have effect on the above dynamics.
We would like to mention here that some interesting but challenging problems associated with the investigation of system (7a) and (7b) should be the dynamic behaviors if is replaced with or , or the impulse is random impulses. We leave them for future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for their valuable comments and suggestions that greatly improved the presentation of this work. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 212111), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1329), and the Key Subject of Hubei Province (Forestry).