Study On Periodic Solution And Branch Of Several Mathematical Ecology Models

In this paper, we discuss the existence, stability of the periodic solutions and bifurca-tion of the two kinds of mathematical ecology models.For the first kind of model, we study the dynamical behaviors of a two-competitive metapopulation system with impulsive control and focus on the stable coexistence of the superior and inferior species. Poincare map is introduced to prove the existence of periodic solution and its stability. It is also shown that a stably positive periodic solution bifurcates from the semi-trivial periodic solution through a transcritical bifurcation.For the second kind of model, we study a eco-epidemiological predator-prey system where prey population have group defence and harvesting. Mathematical results like pos-itive invariance, boundedness, stability of equilibria and the permanence of system have been established. The dynamics of all equilibriums have been thoroughly investigated to find out conditions on the system parameters. We have also studied suitable conditions for non-existence of a periodic solution around the interior equilibrium. Number simulations have been carried out to illustrate different analytical findings.