ncts

In this talk, we discuss the dynamics of the 3D Lotka-Volterra competition model of differential equations with seasonal succession, which exhibits that populations experience an external periodically forced environment. We first establish an index formula on the carrying simplex for the associated Poincaré map of the model. We then classify the dynamics of the discrete dynamical systems induced by all 3D Poincaré maps into 33 classes n terms of inequalities on parameters. There is no positive fixed point in classes 1–18 so that every orbit tends to certain boundary fixed point, while there exists a positive fixed point (but its uniqueness is unknown) for classes 19–33. We further present necessary and sufficient conditions for the (non)uniqueness of the positive fixed point when the model has identical intrinsic growth rate and death rate respectively, by which we completely classify the global dynamics into 37 dynamical classes. As a by-product, we obtain in which classes the positive fixed point is (non)unique. Especially, we find new phenomena and cases that possess a family of invariant closed curves on which all orbits are positive fixed points, or periodic orbits, or dense orbits, which implies the existence of infinitely many positive harmonic solutions, or subharmonic solutions, or quasiperiodic solutions, respectively.