Spatio-temporal Dynamics In Ecosystems

Ecosystem is a typical complex adaptive system far from thermody-namic equilibrium. Owing to its self-organization, ecosystem can produce complex spatio-temporal patterns. The spatio-temporal patterns possess special physical properties. The study on those patterns is of importance for understanding of biodiversity.In this thesis, we study the phenomenon of self-organization in ecosys-tems with Monte Carlo simulations and mean field models based on cyclic dynamics. One of our focus lies on the understanding of the influence of long-range interactions between species on the pattern formation and bio-diversity. The others are the understanding of the plant-animal mutualistic networks on the pattern formation and biodiversity.Firstly, the pattern formation of spiral waves in inhomogeneous ex-citable medium is investigated. The normal distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the formation of spiral pattern. For a larger variance of the parameter fluctuations, the spiral waves are rough. In the case of the uniform distributions for two param-eters, spiral wave cannot be observed for the larger range of fluctuations. It is conjectured that these results are induced by the rotating frequency of spiral wave for different parameters. For the larger rotating frequency, spiral wave is crowded, but is sparse for small frequency. Furthermore, we studied synchronization behaviours of spiral waves in a two-layer coupled inhomogeneous excitable system. It was found that phase synchronization can be observed under weak coupling strength. By increasing the coupling strength, the synchronization is broken down. With the further increase of the coupling strength, complete synchronization and phase synchronization occur again. We also found that the inhomogeneity in excitable systems is helpful to the synchronization.Secondly, We study the cyclic dominance of three species in two-dimensional constrained Newman-Watts networks with a four-state vari-ant of the rock-paper-scissors game. By limiting the maximal connection distance Rmax in Newman-Watts networks with the long-range connection probability p, we depict more realistically the stochastic interactions among species within ecosystems. When we fix mobility and vary the value of p or Rmax, the Monte Carlo simulations show that the spiral waves grow in size, and the system becomes unstable and biodiversity is lost with increasing p or Rmax.We compared extinctions with or without long-range connec-tions and computed spatial correlation functions and correlation length. We conclude that long-range connections could improve the mobility of species, drastically changing their crossover to extinction and making the system more unstable.Thirdly, we study the effects of mutualistic network structure of plants and their animal pollinators on the self-organized patterns and biodiversity with two-layer coupled lattices. In this model, plants and animals compete for resources within their respective groups. The animals exhibit cyclic dominance and the plants compete all to all. At the same time, the animals may move randomly. Through comparing the effects of three kinds of plant-animal mutualistic networks on pattern formation and biodiversity, we found that nested mutualism would lead to instability.At last, we make a conclusion and outlook of this thesis.