Study On Nonlinear Dynamics And Grazing Models Of Zooplankton

In the ecological research field, selective grazing behavior has got ecologists’attention for a long time, of which their focuses vary from the details of individualbehavior to predator-prey system of dynamics. Researchers have found that somepredators can distinguish different types of prey and avoid the ingestion of somecertain prey, which is called selective grazing. In the plankton ecosystem, zooplanktonhas shown the alternatives of grazing when they distinguish toxic algae. Some causesand mechanisms that lead to selective grazing are known, such as morphology (size,shape, colony formation), viable gut passage, and toxic chemical production. Butsome reasons are still under discovery. The selective grazing of zooplankton has asignificant impact on the coexistence of plankton ecosystem and the rapid evolutionof phytoplankton populations.In the plankton ecosystem with three interacting species, including nontoxic andtoxic phytoplankton (NTP and TTP) as well as zooplankton, zooplankton can reducethe grazing rate on toxic phytoplankton to avoid the toxic chemical production andprefer to graze the nontoxic one. In this work, we gave a series of selective grazingfunctions of zooplankton on toxic plankton in the presence of nontoxic one todescribe the avoidance of toxic phytoplankton and preference of non-toxic one byzooplankton using the grazing rate data derived from the result of barbed tail spindleDaphnia on Alexandrium tamarense in the mixture of Alexandrium tamarense andplatymonas subcordiformis. We introduced two parameters in the double-prey grazingmodels. The parameter c is defined as “preference coefficient” and measures thelevel of zooplankton preference of NTP. The parameter d is defined as “avoidancecoefficient” and measures the level of zooplankton avoidance of TTP.Next, we established a dynamical model in the nontoxic phytoplankton-toxicphytoplankton-zooplankton system based on the selective zooplankton grazingfunctions. The objective of our study was to investigate the role of avoidance andpreference in zooplankton grazing behavior for species coexistence and system stability. We analyzed this dynamical system in three aspect:(i) Ultimatelyboundedness of the system;(ii) Equilibria and their existence condition;(iii)Eigenvalue analysis to establish local asymptotic by Lyapunuv stability theory;(iv)Limit cycle and Hopf-bifurcation around the positive equilibrium;(v) Numericalanalysis;(vi) Model application in microalgae-rotifer system.The grazing rate of rotifers is closely related to the growth rate and characteristic ofmicroalgae such as morphology (size, shape, colony formation), viable gut passage,and toxic chemical production.We study on the microalgae-rotifer system and give abiological background to learn the character and mechanism of the selective grazingbehavior. Predictions of the microalgae and rotifer biomass in Chlorella sp.MRA-1-Rotifer system and Didymogenes sp. HN-4-Chlorella sp. MRA-1-Rotifersystem are given by using the NTP-TTP-Z dynamical model. The predicted resultsexplain the experiment well, and give the theoretic foundation for the initial densitiesof microalgae an rotifer in experiment.From stability analysis, numerical analysis and the application of the NTP-TTP-Zmodel, the following conclusions were made:(1) The avoidance and the preference as two characteristics displayed inzooplankton grazing behavior play an important role in maintaining the coexistence ofall population. The system has a stable interior equilibrium point when the avoidancelevel is suitable while the zooplankton concentration remains at a very low level; thenthe system generates a periodic oscillations as the avoidance and preference level rises,describing the periodic algae blooms. When the avoidance coefficient and preferencecoefficient increase to cross a critical value, zooplankton population will extinct whilenontoxic phytoplankton and toxic phytoplankton coexist in the system depicting algalblooms.(2) The preference of zooplankton grazing on nontoxic phytoplankton plays animportant role in maintaining the system stable when the toxin liberation by toxicphytoplankton population is very low. All population co-exist with the suitablepreference level, but zooplankton extincts when the preference level is too high.When the toxin liberation is high, the avoidance level controls the system stability. Zooplankton will extinct in too high or too low avoidance level.(3) There exist the critical values. When the avoidance coefficient and preferencecoefficient cross the critical value the system enters into Hopf-bifurcation around thepositive equilibrium and that induces oscillations of the population. The co-existenceof all population at stable equilibriums changes into co-existence though oscillationson limit cycle.Finally based on the analysis of anti-rotifer robustness of microalgal strains wediscuss the NTP-TTP-Z model in three aspects: growth rate, ratio of biomassconsumed by zooplankton, and competition in phytoplankton population. The resultsshow that the system maintain stable easily by introducing the growth rate functionsin model which depict the effect of grazing on growth rate of toxic phytoplankton andzooplankton. If the ratio of biomass consumed by zooplankton on toxic phytoplanktonis positive, the nontoxic phytoplankton population will extinct while toxicphytoplankton and zooplankton co-exist in system with the increasing of the value ofthe ratio. When the competition in phytoplankton population crosses a critical value,we observe a strange attractor in system which depicts chaotic bloom in nature.The selective grazing behavior of zooplankton plays an important role inmaintaining the stability of the system. When the avoidance and preference levelchange, the system enters a periodic oscillation depicting blooms phenomenon.