Abstract
A logistic model with impulsive Holling type-II harvesting is proposed and investigated in this paper. Here, the species is harvested at fixed moments. By using the techniques derived from the theory of impulsive differential equations, sufficient conditions for both permanence and extinction of the system are established, respectively. Sufficient conditions which ensure the existence, uniqueness, and global attractivity of a positive periodic solution of the system are obtained. Our study shows that impulsive controls play an important role in maintaining the sustainable development of the ecological system. Compared with the linear impulsive capture or continuous nonlinear-type capture, our study shows that the nonlinear impulsive capture could lead to more complicated dynamic behaviors. Numeric simulations are carried out to show the feasibility of the main results. The results obtained here maybe useful to the practical biological economics management.
Highlights
It brings to our attention that all the models are based on a single species model, while a logistic model is one of the basic single species models, it is the cornerstone of the mathematics biology
There are many scholars investigating the dynamic behaviors of the population models incorporating the harvesting, see [3, 4, 7, 28,29,30,31] and the references cited therein
We propose the following autonomous logistic model with regular harvest pulse: x(t) = x(t) r – ax(t), t = tk, x tk+ =
Summary
Many scholars [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39] proposed various single or multiple species modeling. If the inequality rθ – ln ξ = rθ + ln(1 – γ ) > 0 holds, it follows from Theorem 2.1 that (rθ + ln(1 – γ ))(1 – γ ) ≤ lim inf x(t) ≤ lim sup x(t) ≤ r/a, aθ which generalizes the simple dynamics of the linear impulsive logistic equation. It follows from Theorem 2.4 that system (1.4) has a unique globally attractive positive 1-periodic solution. Remark 4.2 For fixed values of r, a and α, β, θ (i.e., the population nature coefficients and harvesting cycle are fixed), if the capture intensity α γ > α – erθ , from Theorem 3.1, the species x will be driven to extinction in system (1.4), that is, this harvesting cycle cannot be sustained. For the fixed θ , as γ is increasing gradually, the time for the species to be extinct becomes shorter
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
