Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting

Abstract

In this paper, we formulate a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting based on some management and capture methods of renewable resources. It is assumed that the population growth obeys the logistic equation in closed seasons and is captured following Michaelis-Menten type response function in open seasons. We define a length threshold of the closed season T¯⁎, which depends on the harvesting parameter. Under the extinct condition of the corresponding continuous harvesting model, by setting the closed season, theoretical results show that the origin is globally asymptotically stable if and only if T¯≤T¯⁎, and there exists a unique globally asymptotically stable T-periodic solution if and only if T¯>T¯⁎. In particular, under the critical conditions on special harvest parameters, it is found that the T-periodic solution still exists as long as an arbitrary positive close season is formulated. Numerical examples are carried out to confirm the obtained theoretical results. Brief conclusions and discussions on our findings are also provided.