Research On Several Time-delay Slow-fast Predator-prey Dynamics Models

Predator-prey relationship is an important topic in the field of mathematics and ecology.It is of great theoretical significance and application value to study the predator-prey interaction relationship.In this paper,several kinds of time-delay slow-fast predator-prey dynamics models are studied according to the smoothness of the response function.We study two kinds of time-delay slow-fast modified Leslie-Gower models through the geometric singular perturbation theory and entry-exit function,in which the response function is smooth.These two kinds of models are called hourly delay model and con-stant delay model.For the hourly delay model,we obtain the approximate system by Taylor's formula.By studying the equilibrium of the approximate system,we analyze the bifurcation of the system at the unique positive equilibrium and conclude the conditions for Hopf bifurcation.Then we consider the dynamics of the limit systems.Based on the above analysis,we construct a singular slow-fast cycle and give a theorem to prove the existence and uniqueness of relaxation oscillation cycles.For the constant delay model,we analyze the linearization system of the constant delay model and conclude the con-ditions for Hopf bifurcation.We find that there are two states of asymptotically stable equilibrium and periodic solution in the system.Then we thought about the limit sys-tems.Through numerical simulation,we get the existence and uniqueness of relaxation oscillation cycles of the model.We conjecture that the relaxation oscillation ring of the model is unique.Finally,we consider a class of predator-prey model with time delay and piecewise smooth response function.We prove the existence and uniqueness of the stable relaxation oscillation cycles of the model.The numerical simulation verifies our theoretical results and shows that our method to deal with time delay is effective to a certain extent.