A non-autonomous discrete Lotka-volterra commensal symbiosis model with Michaelis-Menten type harvesting is proposed and studied in this paper. Under some very simple and easily verified condition, we show that the system admits at least one positive periodic solution.

This article revisit the stability property of symbiotic model of commensalism and parasitism with harvesting in the commensal population. The model was proposed by Nurmaini Puspitasari, Wuryansari Muharini Kusumawinahyu, Trisilowati (Dynamic analysis of the symbiotic model of commensalism and parasitism with harvesting in commensal populations, Jurnal Teori dan Aplikasi Matematika, 2021, 5(1): 193-204). By establishing three powerful Lemmas, sufficient conditions which ensure the global stability of the equilibria are obtained.

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Of interest is the dynamics of the discrete-time amensalism model with a cover on the first species. We first obtain the existence and stability of fixed points and the conditions for the permanent coexistence of two species. Then we demonstrate the occurrence of flip bifurcation by using the central manifold theorem and bifurcation theory. A hybrid control strategy is used to control the flip bifurcation and stabilize unstable periodic orbits embedded in the complex attractor. Numerical simulation verifies the feasibility of theoretical analysis and reveals some novel and exciting dynamic phenomena.

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