Spatial Spread and Front Propagation Dynamics of Nonlocal Monostable Equations in Periodic Habitats

Abstract

This dissertation is concerned with spatial spread and front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. Such equations arise in modeling the population dynamics of many species which exhibit nonlocal internal interactions and live in spatially periodic habitats. The main results of the dissertation consist of the following four parts. Firstly, we establish a general principal eigenvalue theory for spatially periodic nonlocal dispersal operators. Some su cient conditions are provided for the existence of principal eigenvalue and its associated positive eigenvector for such dispersal operators. It shows that a spatially periodic nonlocal dispersal operator has a principal eigenvalue for the following three special but important cases: (i) the nonlocal dispersal is nearly local; (ii) the periodic habitat is nearly globally homogeneous or (iii) it is nearly homogeneous in a region where it is most conducive to population growth. It also provides an example which shows that in general, a spatially periodic nonlocal dispersal operator may not possess a principal eigenvalue, which reveals some essential di erence between random dispersal and nonlocal dispersal. The principal eigenvalue theory established in this dissertation provides an important tool for the study of the dynamics of nonlocal monostable equations and is of also great importance in its own. Secondly, applying the principal eigenvalue theory for nonlocal dispersal operators and comparison principle for sub- and super-solutions, we obtain one of the important features for monostable equations, that is, the existence, uniqueness, and global stability of spatially periodic positive stationary solutions to a general spatially periodic nonlocal monostable equation. In spite of the use of the principal eigenvalue theory for nonlocal dispersal operators in the proof, this feature is generic for nonlocal monostable equations in the sense it is iiindependent of the existence of the principal eigenvalue of the linearized nonlocal dispersal operator at the trivial solution of the monostable equation, which is of great biological importance. Thirdly, applying the principal eigenvalue theory for nonlocal dispersal operators and comparison principle for sub- and super-solutions, we obtain another important feature for monostable equations, that is, the existence of a spatial spreading speed of a general spatially periodic nonlocal equation in any given direction, which characterizes the speed at which a species invades into the region where there is no population initially in the given direction. It is also seen that this feature is generic for nonlocal monostable equations in the same sense as above. Moreover, it is shown that spatial variation of the habitat speeds up the spatial spread of the population. Finally, this dissertation also deals with front propagation feature for monostable equa- tions with non-local dispersal in spatially periodic habitats. It is shown that a spatially periodic nonlocal monostable equation has in any given direction a unique stable spatially periodic traveling wave solution connecting its unique positive stationary solution and the trivial solution with all propagating speeds greater than the spreading speed in that direc- tion for the special but important cases mentioned above, that is, (i) the nonlocal dispersal is nearly local; (ii) the periodic habitat is nearly globally homogeneous or (iii) it is nearly homogeneous in a region where it is most conducive to population growth. It remains open whether this feature is generic or not for spatially periodic nonlocal monostable equations.