Two Types Of Pulse Infectious Disease Models And The Watt-type Functional Response Predator System

Bio-mathematics is a branch which is interacted between biology and mathematics.The main problem of bio-mathematics is studying the regular pattern in the infection disease and population ecology by using dynamics theory. In this paper, we do some research about two epidemic models with impulse and prey-predator model.First, we consider SIR epidemical models with pulse effects on the basis of standard incidence. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the model. By means of the Floquet theory and the compared principle, we prove the local and global stabilities of the infection-free periodic solution. The threshold is identified. The theoretical results are confirmed by numerical simulations.Second, we establish an SIR epidemic model with impulsive constant vaccination. By means of dynamical system and the basic theories of impulsive differential equation, we give out the basic reproduction number corresponding to the model. Further, the global stability of an infection-free periodic solution is analyzed. And the existence of a nontrivial periodic solution is considered by using bifurcation theory.Third, we consider the predator-prey system with reaction-diffusion and study how diffusion affects the stability of predator-prey coexistence equilibrium. The conditions for Hopf and Turing bifurcation is derived with symbolic calculation. Based on the bifurcation analysis, we give the spatial pattern formation for the evolution process of the system near the coexistence equilibrium point via numerical simulation. We also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations.