Modeling,Dynamics And Numerical Simulations Of Environmental Impact On Algal Bloom In Shallow Lakes

Shallow lake is one of the most important parts of human living environment,which plays a significant role in sustainable development of society and economy. In recent years, as a result of global climate change, rapid industrialization, the growing use of fertilizers and the increasing development of livestock agriculture in the sur-rounding region of lake, many shallow lakes on earth have been seriously polluted,which makes the eutrophication deeper and then the outbreak of algae has become more frequent, which has a crippling effect on the development of the ecological en-vironment and the lives of the people of the lake area. Due to its frequency and harmfulness, algal bloom in shallow lakes has received much more attention recently.However, the complex mechanisms of algal bloom are still far less than understood because of the influences of numerous factors.In this thesis, our aim is to further explore the mechanism of the formation and termination of algal bloom in shallow lakes by considering some factors (mainly in-cluding nutrient, toxin, the refuge for phytoplankton, dissolved oxygen, etc.) which have been paid more attention in recent studies. In this process, mathematical mod-elling and the corresponding theoretical methods of dynamical systems, especially the application and relative calculation of the bifurcation theory of nonlinear differ-ential equations, are very useful to study the complexity of lake ecosystem and give plausible biological explanations for the observed phenomena in reality.Just taking one factor, nutrient input or toxin released by phytoplankton, into consideration, earlier findings show that rich nutrient input can increase the incidence of the bloom-forming but the toxin has the opposite effect. In Chapter 1, we give a general framework for both nutrient uptake by phytoplankton and zooplankton grazing response due to the toxin,using a new nutrient-phytoplankton-zooplankton model to study the combined effect of nutrient input and toxin on algal blooms. We present some of the dynamical analysis of the existence, stability and bifurcations to explore the complex dynamics of the model equations. The dynamics include Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2 or higher. And then we illustrate our findings by taking specific forms of nutrient uptake and zooplankton grazing with numerical simulations. Except what were discovered in available studies, we also have some more interesting findings on the period of algal blooming and level of toxin, and nutrient input. That is, the duration of the bloom when occurs can be lengthened with the lower toxin level or the higher nutrient input.Besides toxin, another fascinating factor, the refuge provided for phytoplankton,is worthy of attention since it might stabilize the biomass of plankton by prevent-ing extinction of phytoplankton from the predation of zooplankton temporarily. To study the impacts of toxin produced by phytoplankton and refuges provided for phy-toplankton on phytoplankton-zooplankton interactions in lakes,we establish a simple phytoplankton-zooplankton system with Holling type II response function in Chapter 2. The existence and stability of positive equilibria are discussed. Bifurcation analy-ses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurca-tion, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the ob-served biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have significant impacts on the occurring and terminating of algal blooms in freshwater lakes. To some extent, they may act as not only a biological trigger but also a biological control for algal bloom.Dissolved oxygen, as one of most important environmental factors, plays a sig-nificant role in the interactions between phytoplankton and zooplankton and also in triggering outbreaks of algal blooms. In Chapter 3, considering the effect of dissolved oxygen concentration on the filtering rate of zooplankton, we propose a phytoplankton-zooplankton-dissolved oxygen model to study how dissolved oxygen enriches the dynamical behaviors of plankton ecosystem and triggers algal bloom. We discuss the stability of boundary equilibria, the existence of multiple positive equilib-ria and analyze the local dynamics by studying the bifurcations of a triple equilibrium on a manifold. The system exhibits complex dynamics, including saddle-node bifur-cations, Hopf bifurcation, and transcritical bifurcation. Our theoretical findings are illustrated by performing some numerical simulations with suitable bifurcation pa-rameters, and the corresponding biological interpretations are given for the interesting phenomena from the modeling. With other conditions in suitable range, the maximum relative effect rate of DO on the predation of zooplankton can increase the instabili-ty of ecosystem and the risk of algal bloom. When DO concentration is in a suitable range, the large death of zooplankton can also increase the instability of ecosystem and the risk of algal bloom. Compared with some previous phytoplankton-zooplankton models,besides Hopf bifurcation,the bistable phenomena used in mathematics to de-scribe the dramatic sharp growth and then sudden collapse of phytoplankton during the bloom can appear for suitable parameters. And this bistable phenomena also shows that the coexistence and the final state of phytoplankton and zooplankton depend on initial status of dissolved oxygen concentration.