Modeling And Analysis Of Coral Reef Ecosystems

In this paper, we are mainly concerned with modelling on coral reef ecological sys-tem consisting of coral, algae, herbivorous fish and carnivorous fish. Hereby three models are given to illustrate the relationships among these species. The first one focuses on the relationship between herbivorous fish (parrotfish) and carnivorous fish (groupers), fishing impact on the two groups of fishes is also considered in the model, what is new is that we divide parrotfish into two groups in terms of there ages or volumes, say, juveniles and adults. More important, our results coincide very well with the data presented in ref [40]by P. Mumby, etc. The second model is concerned with the relationships among two groups of herbivorous fishes (Damselfish and Parrotfish) and algae. A lot of empirical data shows that Damselfish plays a, critical role in maintaining the diversity of algae and gains the name" algae gardener", while there is no model take its importance seriously to the best of our knowledge. So basically it is the first time to consider both Damselfish and Parrotfish in one model. To be more precisely, Damselfish and parrotfish are competitive on gazing algae, while Damselfish and algae are cooperative to some extent from the empirical data.So our model works very well in representing the real world. Consequently, we carry out mathematical analysis on the positive invariant of the system, stability and bifurcation of equilibrium solutions. The third model deal with Algae-Damselfish-Parrotfish as well.Nevertheless, we use the cooperative function introduced by May in 1976 to reflect the relationship between Damselfish and algae. Dynamical behavior of the system are thor-oughly studied and corresponding simulation is also given. The main mathematical tools employed in this paper are Hartman-Grobman's Theorem?Routh-Hurwitz's Theorem and Sotomayor's Bifurcation Theorems, etc.