Dynamic Analysis Of A Coral Reef Model

As the impact of global climate change and human activities,many sea areas often experience coral depletion and excessive growth of macroalgae in recent years.In response to this situation,mathematical models are built for coral reef ecosystems to explore the mechanism.We establish and analyze a coral reef model with functional responses based on the predation of parrotfish.Firstly,we consider the coral reef model with Holling I functional response,prove the positive invariance of the model,and discuss the conditions for the existence of equilibria and the local asymptotic stability.Through sensitivity analysis,we obtain the two parameters b₁ and f that have a significant impact on the model and perform bifurcation analysis.The analysis results indicate that under certain conditions,as ₁b increases,the final stable state of the system experiences stable coexistence of only macroalgae,stable coexistence of macroalgae,algal turfs,and parrotfish;and stable coexistence of four populations.As f increases,the final stable state of the system undergoes stable coexistence of coral,algal turfs,and parrotfish,stable coexistence of four populations,stable coexistence of macroalgae,algal turfs,and parrotfish,and only macroalgae existing.Increasing ₁b or decreasing f is beneficial for the coral population recovery.Secondly,a coral reef model with Holling II functional response is studied.We demonstrate its positive invariance and analyze the existence and local asymptotic stability of equilibria.The influence of parameter f on the dynamical behaviors is explored in the numerical simulations.Bifurcation diagrams show that as f increases,the system exhibits complex dynamic behaviors,experiencing two Hopf bifurcations,BP bifurcation and LP bifurcation.The final state of the system undergoes stable coexistence of coral,algal turfs,and parrotfish,four population cycles,four population stable coexistence,stable coexistence of macroalgae,small algae and parrotfish,and only macroalgae existing.Decreasing f is also beneficial for the coral population recovery.It is concluded that reducing the loss rate of parrotfish is beneficial for coral population recovery in coral reef ecosystems and increasing the proportion of corals.Our work can provid a theoretical basis for the restoration of coral reef systems.