Research On Dynamic Behaviors Of Mussel-algae Reaction-diffusion Model

Mussels that live on soft sediments feed mainly on algae,and together,they form the main part of the mussel bed ecosystem.Besides the high economic and nutritional value,mussels are particularly suited for the study of pattern formation.Therefore,establishing relevant mathematical models and studying their dynamics are necessary,which can provide theoretical support for preventing the collapse of the mussel bed ecosystem,and has very important practical significance.In this dissertation,we mainly study the dynamics of two mussel-algal reactiondiffusion models with different mussel mortality.For the first model,we are concerned with the Hopf bifurcation and Turing-Hopf bifurcation.For the second model,we mainly focus on the existence and non-existence of nonconstant steady states.Details are as follows:(1)We first consider the mussel-algae model with time-delay effect.Considering that mussels need some time to digest algae before they can convert them into their own biomass,we add the time delay into the model and study its effect on the dynamics of the system.When the delay is zero,the existence and uniqueness of the non-negative solution are given by using the upper and lower solution method,and a priori estimate is established according to the phase diagram analysis under certain conditions.The global stability of the semi-trivial steady state is obtained.Moreover,we study the local stability and Hopf bifurcation of the constant steady state in one-dimensional spatial space.When the delay is greater than zero,we investigate the Hopf bifurcation purely induced by time delay at the constant positive steady state.By using the central manifold theory and normal form method,we obtain the direction and stability of the bifurcating periodic solution.Finally,we show some numerical simulations to support our theoretical results.(2)The model with time-delay effect is further investigated in this part.Due to the aggregation,mussels present regular banded patterns at large spatial scales.We will study the potential mechanism of pattern formation from the perspective of bifurcation.We first obtain the existence and uniqueness of the nonnegative solutions by the upper and lower solution method for a mixed quasi-monotone system.According to the distribution ofroots of the characteristic equations,we study the stability of the constant positive steady state,Turing bifurcation and Turing-Hopf bifurcation,as well as the dynamic classification at the Turing-Hopf singularity.Under a given set of parameters,we find the coexistence of two stable states: spatially homogeneous periodic solution and spatially nonhomogeneous steady-state solution,and a further study show that the spatially nonhomogeneous periodic solution appearing in the evolution process is only a "transient".Although this state may last for some time,it will eventually converge to one of the two stable states mentioned above.Under the one-dimensional space,we give a mathematical explanation for the formation of the regular banded patterns of mussels.(3)We then consider the model in which mussel mortality has both positive and negative feedback effects.Based on the research of mussel-algae interaction on a small spatial scale,we modify the mussel mortality in the first model,the new mortality includes two feedback controls: one is a positive feedback relates to reduction of wave disturbance and predatory losses,and the other is a negative feedback relates to increment of intraspecific competition for food resources.The interaction of this two feedback can ensure that the model will not produce a unrealistically high mussel biomass,which addresses the problem that no prior bound of mussels can be obtained in the first model.Under the homogeneous Neumann boundary conditions,we discuss the global stability of the semitrivial steady state and constant positive steady state respectively.By using the elliptic estimate,we get a priori estimate for the nonnegative steady states,and then obtain the existence and non-existence of the nonconstant steady states.Finally,numerical simulations corresponding to the analytic theory are given,and the obtained images are very similar to the labyrinthine patterns of mussel beds on intertidal flats under wind-sheltered conditions.Our research results give a mathematical explanation for the pattern formation of the mussel bed ecosystem.