Principal Eigenvalue Theory for Time Periodic Nonlocal Dispersal Operators and Applications

Abstract

The dissertation is concerned with the spectral theory, in particular, the principal eigenvalue theory for nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue.

In this dissertation, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. Among others, it is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. We also obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems.

Next, we discuss the applications of the established principal eigenvalue theory to the existence, uniqueness, and stability of time periodic positive solutions to Fisher or KPP type equations with nonlocal dispersal in periodic media. We prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution $u^+(t,x)$ which is globally asymptotically stable.

Finally, we discuss the application of the established principal eigenvalue theory to the spatial spreading and front propagation dynamics of KPP equations with nonlocal dispersal in periodic media. We show that such an equation has a spatial spreading speed $c^*(\xi)$ in the direction of any given unit vector $\xi$. A variational characterization of $c^*(\xi)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a periodic traveling wave solution connecting $u^+(\cdot,\cdot)$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^*(\xi)$.