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

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.

Permanence for an impulsively perturbed periodic single-species model considering Monod-Haldane Type functional response as predation term

“…In a single species model, the periodicity and stability under the influence of impulsive perturbations at fixed moments of time by considering the predation term as Holling Type-II functional response using comparison theorems and Lyapnuov function is discussed by Tan et al [16]. In continuation to this, the effect of linear and constant impulsive perturbations on single species population density considering Holling Type-II functional response is compared by Tan et al [15]. Liu et al [13] analyzed the constant and linear impulses at fixed moments of time by taking single species model in which Holling Type III functional response is taken as predation term and permanence has been established using comparison theorems of impulsive differential equations.…”

Permanence and almost periodic solutions for a single-species system with impulsive effects on time scales

“…If the density of species x is small, then the predation term h(x) drops rapidly. To investigate the effects of other specific forms of h(x), Murray [10] took h(x) = αx 2 (t) β+x 2 (t) and the authors of [13,14] took h(x) = cx(t) d+x(t) . Since, in reality, many natural and man-made factors (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding etc.)…”

“…If the density of species x is small, then the predation term h(x) drops rapidly. To investigate the effects of other specific forms of h(x), Murray [10] took h(x) = αx 2 (t) β+x 2 (t) and the authors of [13,14] took h(x) = cx(t) d+x(t) .…”

Multiwave solutions of fractional 4th and 5th order Burgers equations

“…In the literature, several methods are used to study FDEs. Examples include differential transform, Adomian's decomposition, variational iteration, homotopy perturbation, finite difference, finite element, fractional subequation, (G ′ /G)-expansion, and the first integral [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].…”