Research On Pattern Dynamics Of Plankton And Fish System

The study on complex dynamic behaviors of aquatic ecosystems is a hot topic in population ecology.In this thesis,according to the interactive relationship between plankton and fish in aquatic ecosystems,reaction-diffusion models were constructed describing by partial differential equations.Through stability analysis,Turing bifurcation analysis,Hopf bifurcation analysis,weakly nonlinear analysis and numerical simulations,the complex dynamic behavior of the plankton and fish system and the formation,evolution and transition of patterns were studied.The main research results are as follows:(1)In the toxic phytoplankton-plankton system with the Allee effect,based on the occurrence of stable homogeneous stationary states,the parameter conditions of Turing bifurcation and Hopf bifurcation were determined,and the standard multi-scale analysis method was used to derive amplitude equations.The numerical simulations show that cross-diffusion is the key mechanism for the formation of spatial Turing patterns.Without cross-diffusion,Turing instability will not occur in the system,but Hopf bifurcation can induce the formation of spatiotemporal patterns of phytoplankton,such as spiral patterns.When the cross-diffusion rate is higher than its critical value,the distribution of all plankton species in space becomes heterogenuous,and different types of Turing patterns are generated:spots,spot-strip mixed,and striped patterns.Besides,the system can produce the transition between Turing patterns under different Allee effect thresholds and toxin production rates.When the toxin release is too high,a large number of cold spots can occur,indicating that high-density phytoplankton gradually occupy most of the space.(2)In the food chain model of plankton and fish,the conditions for the formation of Turing pattern were obtained based on Turing instability theory,and the amplitude equations were deduced by the weak nonlinear analysis.The conditions for the appearance and stability of various Turing patterns near the Turing bifurcation boundary were obtained.Numerical simulations showed that the growth rate of fish plays an important role in the occurrence of dynamic behaviors of the plankton and fish food chain system.As the fish growth rate changes,the spatiotemporal patterns of phytoplankton,zooplankton and fish show a transitional change from spots,to spot-strip mixed,and then to strips.The population densities of fish and phytoplankton change positively with the increase of fish growth rate.(3)Studies on the plankton and fish food chain system with linearly fishing showed that the system can produce complex dynamic behaviors such as Hopf bifurcation and Turing bifurcation.Hopf bifurcation can lead to periodic oscillations of the system.With the increase of fishing coefficient,patterns consisting of cold spots,cold spot-strip mixtures,strips,hot spot-strip mixtures,and hot spots were observed in the phytoplankton populations.Areas with high phytoplankton density correspond to low densities of zooplankton and fish,and vice versa.In addition,proper harvesting of fish can increase the population density of fish in the entire sy stem in a short period of time.From a long-term perspective,it has no effect on the stability of the system,while sustained overfishing is likely to lead to a gradual decline or even extinction of fish density.In this thesis,through the study of the pattern dynamics of the plankton and fish system,the formation and stabilization mechanism of the plankton patterns were revealed,the evolution and transition among different patterns were explored,and the understanding of complex dynamics such as bifurcation and chaos in the plankton and fish system was promoted.The research provides a theoretical basis for regulating the density and spatial distribution of plankton and fish in aquatic ecosystems by harvesting control.