A Kind Of Prey And Predators Model Space Pattern Dynamics Research

In recent years, there have been considerable interests in spatial and temporal behaviorof interacting species in ecosystems. The dynamical behavior between predator and preyhas always been one of the dominant themes in ecosystems due to its universal existenceand importance. In this paper, we study the spatial dynamics of the Beddington-DeAngelispredator-prey model. More specifcally, we research the Turing pattern selection and noise-induced patterns.In chapter1, we give the background and signifcance of investigating predator-preymodel in ecosystems, development of this fled and the main work in this thesis.In chapter2, spatial dynamics of Beddington-DeAngelis predator-prey model is investi-gated. We analyse the linear stability and obtain the condition of Turing instability of thismodel. Moreover, we deduce the amplitude equations and determine the stability of diferentpatterns. In Turing space, we fnd that this model has coexistence of H0hexagon patternsand stripe patterns, Hπhexagon patterns and H0hexagon patterns. To better describe thereal ecosystem, we consider the ecosystem as an open system and take the environmentalnoise into account. It is found that noise can decrease the number of the patterns and makethe patterns more regular. What’s more, noise can induce two kinds of typical pattern tran-sitions. One is from the Hπhexagon patterns to the regular stripe patterns and the otherone is from the coexistence of H0hexagon patterns and stripe patterns to the regular stripepatterns.In chapter3, spatial dynamics in the Beddington-DeAngelis predator-prey model withself-difusion and cross-difusion is investigated. We analyse the linear stability and obtainthe condition of Turing instability of this model. We can deduce the amplitude equations anddetermine the stability of diferent patterns. Numerical simulations show that this systemhas three types of typical patterns. One is the coexistence of hexagon patterns and stripepatterns. The other two are hexagon patterns of diferent types: Hπhexagon patterns andH0hexagon patterns. Comparing to the model with only self-difusion, we can fnd that the predator-prey model with both self-difusion and cross-difusion has similar properties withthe predator-prey model only including self-difusion. But when cross-difusion is includedin the model, we observe that the Turing space becomes smaller than the case withoutcross-difusion. This implies that cross-difusion has certain efect on the stability of thepopulation.