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Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. (English) Zbl 1281.92069
Summary: In this paper, a predator-prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order $$k$$ $$(k\geq 2)$$ periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.

##### MSC:
 92D40 Ecology 34K13 Periodic solutions to functional-differential equations 34C23 Bifurcation theory for ordinary differential equations
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