| (i) The interactions between diffusion and spatial
heterogeneity could create interesting dynamics for a spatial
population model. We show that for two-component Lotka-Volterra
competition model with spatially heterogeneous diffusion
coefficients, intrinsic growth rates and competition rates, a
spatially heterogeneous positive equilibrium solution is globally
asymptotically stable when it exists. A similar result is also
proved for multi-component Lotka-Volterra competition model with
spatially heterogeneous intrinsic growth rates. We prove the result
by using monotone dynamical theory, upper and lower solution
methods, and a new Lyapunov functional method. This is a joint work
with Wenjie Ni and Mingxin Wang. (ii) The global asymptotic behavior
of the classical diffusive Lotka-Volterra competition model with
stage structure is studied. A complete classification of the global
dynamics is given for the weak competition case. It is shown that
under otherwise same conditions, the species with shorter maturation
time prevails. The method is also applied to the global dynamics of
another delayed competition models. This is a joint work with
Shanshan Chen. |