Abstract
A cooperative system of May type incorporating partial closure for the populations and non-selective harvesting is proposed and studied in this paper. The locally stability property of the equilibria are determined by analyzing the Jacobian matrix of the system about the equilibria. By using the comparison theorem of the differential equation, sufficient conditions which ensure the global attractivity of the boundary equilibria are obtained. By using the iterative method, we are able to show that the conditions which ensure the existence of the unique positive equilibrium is enough to ensure its global attractivity. Our study shows that the intrinsic growth rate and the fraction of the stocks for the harvesting plays crucial role on the dynamic behaviors of the system. Numeric simulations are carried out to show the feasibility of our results.
Highlights
Cooperation, one of the basic relationship between the species, has been studied by many scholars during the last decades, see [2]-[35] and the references cited therein
By using the comparison theorem of the differential equation, sufficient conditions which ensure the global attractivity of the boundary equilibria are obtained
Wei and Li[2] had considered the influence of the harvesting to the May cooperative system, they only considered the harvesting of the first species
Summary
Cooperation, one of the basic relationship between the species, has been studied by many scholars during the last decades, see [2]-[35] and the references cited therein. − q1Emx, dy dt = y(a2 − b2y) − q2Emy, where ai, bi, qi, i = 1, 2 c1, E, m(0 < m < 1) and d1 are all positive constants, where E is the combined fishing effort used to harvest and m(0 < m < 1) is the fraction of the stock available for harvesting His studied shows that depending on the range of the parameter m, the system may be collapse, or partial survival, or the two species could be coexist in a stable state. Stimulated by the works of [2]-[4], [36]-[38], in this paper, we will study the dynamic behaviors of the following non-selective harvesting May cooperative system incorporating partial closure for the populations x a1x + k1.
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