Analysis of a Periodic Single Species Population Model Involving Constant Impulsive Perturbation

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.

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, 2 ( )/( + 2 ( )), 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 ( ) = (t)/( + ( )) 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 2 Journal of Applied Mathematics (3) and established the following periodic system with linear impulsive perturbations: where (0) > 0, Δ ( ) = ( + ) − ( ), is an impulsive point for every , and 0 < 1 < 2 < ⋅ ⋅ ⋅ < < ⋅⋅⋅ and N 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 > 0 such that + = , + = + . We have investigated the permanence and global attractivity of system (4a) and (4b) and two results can be redescribed as follows.
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.

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].
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.

Permanence and Global Attractivity
We first give the main result on the permanence of system (7a) and (7b). Proof. It follows from (7a) and (7b) that we obtain Obviously, if (0 + ) = (0) > 0, then ( ) > 0 for > 0. 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 By (1) in Lemma 2 we know that system (15) is permanent, which implies that there exist constants > 0 and > 0 such that Let ( ) be the solution of (15) with (0) = (0). 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 stableperiodic solution, denoted byV( ). The asymptotic property ofV( ) implies that there exist * ≥ and > 0 such that 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 .
The proof of Theorem 6 is complete.

Remark 7.
The positive impulse has the effect on the global attractivity of system (7a) and (7b).

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, ℎ + = .
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.