Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems

The current paper is devoted to the study of spreading speeds and linear determinacy of diffusive cooperative/competitive systems of the formut(t,x)=(Au)(t,x)+F(t,u(t,x)),x∈H, where H=RN or ZN, A is a random or nonlocal dispersal operator in the cases H=RN, and is a discrete dispersal operator in the case H=ZN, F(t,u) is cooperative or competitive in u and is recurrent in t, which includes F(t,u) is periodic and almost periodic in t as special cases. First, the notion of spreading speed intervals for diffusive cooperative systems is introduced via the natural features of spreading speeds and some basic spreading properties are established. Next, some principal Lyapunov exponent and principal Floquent bundle theory for linear cooperative systems of ordinary differential equations is developed, which plays an important role in the study of spreading speeds of diffusive cooperative/competitive systems with time dependence and is also of independent interest. In terms of the principal Lyapunov exponent and principal Floquent bundle theory, some upper and lower bounds for the spreading speed intervals for diffusive cooperative systems are then established. Under certain conditions, it is proved that a diffusive cooperative system has a single spreading speed and is linearly determinant. Finally, by transforming a diffusive competitive system to a diffusive cooperative system and applying established results for diffusive cooperative systems, upper and lower bounds for the spreading speed intervals for diffusive competitive systems are obtained and linearly determinacy for such systems is discussed.