Qualitative analysis of a korean pine forest model with impulsive thinning measure

A korean pine forest model with impulsive thinning measure is presented by using impulsive state feedback system to investigate the periodicity of the regeneration process of the forest. Based on the qualitative properties of the corresponding continuous system, the existences of order-1 periodic solutions are discussed. If the positive equilibrium of the continuous system is globally stable, then the impulsive state feedback system has an order-1 periodic solution and no order-k(k⩾2) periodic solution. The conditions for the orbitally asymptotical stability of order-1 periodic solution are given and discussed by the analogue of the Poincaré criterion. For the case that the continuous system has a stable limit cycle, the existence of order-1 periodic solution of the impulsive state feedback system are also discussed, the results show that there are three kinds of order-1 periodic solutions. Finally, the mathematical results are verified by the numerical simulations. Moreover, the numerical results show that the impulsive state feedback system has order k(k⩾1) periodic solutions in the interior of the limit cycle of the continuous system for some parameters.