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The existence and stability of order-1 periodic solutions for a state-dependent Kolmogorov predator-prey model with non-selective harvesting
Huilan Wang,Chunhua Ou,Binxiang Dai
Keywords:State dependent impulse; Periodic solution; Phase analysis; Poincar$\acute{e}$ map; Successor function; Bendixson domain
      In this paper, we focus on a general state-dependent Kolmogorov predator-prey model subject to non-selective harvesting along with delivery. Certain criteria are established for the existence, non-existence and multiplicity of order-1 impulsive periodic solutions to the system. Based on the geometric phase analysis and the method of Poincar$\acute{e}$ map or successor function with Bendixson domain theory, three typical types of Bendixson domains (i.e., Parallel Domain, Sub-parallel Domain and Semi-ring Domain) are introduced to deal with the discontinuity of the Poincar$\acute{e}$ map or successor function. We incorporate two discriminants $\Delta_1$ and $\Delta_2$ to link with the existence, non-existence and multiplicity as well as the stability of order-1 periodic solutions. At the same time, we locate the order-1 periodic solutions with the help of three characteristic points and the parameters ratio of delivery over harvesting. The results show that there must exist at least one order-1 periodic solution when the trajectory, tangent to the mapping line, can hit the impulsive line. While the trajectory, tangent to the mapping line, cannot hit the impulsive line, there is not necessary the existence of an order-1 periodic solution, which means the impulsive control may be invalid after finite times stimulation or suppression. In conclusion, we reveal that the delivery can prevent the predator from extinction and stabilize the order-1 periodic solution.