Dynamical Modeling Of Coral Reef Ecosystem

Coral reef ecosystems have rich biodiversity,high primary productivity,and effective material cycles,which are essential for the construction of marine ecological civilization and the sustainable development of the marine economy.However,coral reefs around the world are facing existential threats under the pressure of climate change and human activities.It is acknowledged that the outbreak of crown-of-thorns starfish(Co TS),the natural enemy of coral,is one of the factors of coral reef degradation.Meanwhile,the increase of macroalgae cover in reef area is one of the manifestations of coral reef degradation.In this thesis,three kinds of dynamical models of coral reef ecosystems are established from the perspective of controlling the outbreak of Co TS and preventing the excessive growth of macroalgae.The effect of the environmental noises on coral reefs are also considered to explore the mechanism of coral reef degradation and provide the theoretical basis for the protection of coral reefs.The reasons for the outbreak of Co TS are complex.A series of trophic cascades trigger when triton(the natural enemy of Co TS)is over-harvesting,which will reduce the control effect of triton on Co TS and cause the outbreak of Co TS.Firstly,a tri-trophic food chain model consisting of coral,Co TS,and triton is proposed to investigate the effect of harvesting in this thesis.The existence of possible steady states along with their stability are rigorously discussed in the theoretical analysis.From the economic perspective,we examine the existence of the bionomic equilibrium and establish the optimal harvesting policy.Subsequently,the deterministic model is extended to a stochastic model to study the effect of the environmental noises.The existence and uniqueness of the positive global solution and the long-time behaviors of the stochastic model are theoretically investigated.Numerical simulations show that over-harvesting of triton is not beneficial to coral reefs and modest harvesting of Co TS may promote sustainable growth in coral reefs.In addition,the presence of strong noises is harmful to coral reefs and can lead to population extinction.When terrestrial nutrients flow into the ocean,the food resources of Co TS larvae feeding on the phytoplankton increase,which greatly improve the survival rate of larvae and eventually leads to the outbreak of adult Co TS.As a consequence,effective measures for managing Co TS outbreak require a complete knowledge of the life cycle of Co TS.Then,a stochastic model containing coral,immature Co TS,and mature Co TS is formulated.We first prove that there exists a unique global positive solution with any given initial value.Sufficient conditions for the extinction of Co TS are investigated.The Markov semigroup theory is used to obtain the existence of a unique stable stationary distribution.By solving the Fokker-Planck equation,the approximate expression of the probability density function of the distribution around its quasipositive equilibrium is described.Our findings indicate that the survival rate and mortality of Co TS larvae are the key mechanisms that affect Co TS outbreak.Moreover,the qualitative analyses and numerical simulations show that coral is more vulnerable and easily go to extinction in the presence of strong noises,while the weak white noises can guarantee the existence of a stable stationary distribution which implies the long-term persistence of coral and Co TS.Overfishing and eutrophication promote the growth of macroalgae and give it a competitive advantage over coral.Sea urchins can control macroalgae biomass through predation and protect the health of coral reef ecosystems.However,both herbivorous fish competes with sea urchins for macroalgae and large predatory fish preys on sea urchins can directly influence the density of sea urchins,thereby affecting the benthic ecology of coral reefs.Finally,a competitive system containing sea urchins,herbivorous fish,and large predatory fish is established to explore the impact of the fear effect generated by large predatory fish on sea urchins and herbivorous fish.The existence and global stability of the equilibria are analyzed theoretically.Then,we extend the system from a deterministic framework to a stochastic one which is achieved by nonlinear perturbation.The long-time behaviors of the system and the approximate expression of the probability density function of the distribution around its quasi-positive equilibrium are investigated in detail.The results show that the fear effect makes the growth of sea urchins slow,which may reduce foraging activity and thus increase the feeding of herbivorous fish by large predatory fish.Furthermore,the presence of strong noises are not conducive to the longterm survival of the population,while the weak white noises can guarantee the existence of stationary distribution.