Spatially Structured Models And Distribution Patterns Of Populations

The size and spatial distribution of populations are the two major subjects in mathematical ecology. Classical methods from mathematics have played an important role in studying the problems about population size. Generally, the temporal dynamics of the populations are modeled by the dynamical equations, where many well-known achievements also have been obtained On the contray, few studies have contributed to the subject of spatial distributions of populations and related research is not thrived at all. This PhD thesis, in which four chapters are included, will mainly focus on the spatial dynamics of populations—spatial models and spatial distributions. In chapter one, we briefly reviewed the history of the development of population models from the view point of mathematical ecology. The transition from non-spatial models to spatial models is also included. Moreover, some newly developed methods which are used in modeling spatial dynamics of populations are introduced. All these methods are compared and categorized from the perspective of dynamics theory. In chapter two, two spatially-structured competitive systems are studied: The first one is an integro-differential model while the second one is cellular automata model: After a comprehensive analysis, the theory that spatial structure and non-transitive competition favor coexistence of species is verified. The importance of this research is providing some theoretical evidence and support to biodiversity. In chapter three, a individual-based model (IBM) is introduced and its spatial dynamics and patterns are approximated by a group of moment equations. Actually, the IBM is an interactive particle system, where two competitive species are placed on a plane. Some new results are obtained and a comparison with previous studies about non-spatial system is also conducted. It proves that the dispersal of individuals favor the coexistence of interspecific competitive species. In chapter four, we first introduced the application of spatial point process and spatial analysis in population pattern analysis. The first concern in this chapter is to compute aggregation parameter which reflects the aggregation level of the spatial point pattern. Two new methods, K-function and distance sampling, are proposed. The other concern in this chapter is to give an ideal-estimator of the density. Based on distance sampling (distance from individual to individual), we propose a maximum likelihood estimator of the density. All above methods are assessed by simulated point patterns and mapped point patterns from real forests. The meaning of this research is helping field workers to complete inventory and survey efficiently.