【作者】 朱丽娜；

【导师】 王智诚；

【作者基本信息】 兰州大学 ， 数学·应用数学， 2024， 硕士

【摘要】 In recent years,the reaction-diffusion equation has been widely used to describe the dynamic behavior of populations because of its rich propagation dynamics.At present,the dynamics study of single species is relatively complete.For multiple species,because of the need to consider the complex interactions between them,the model is more complex,has more rich dynamic behavior,and brings greater difficulties to the research work.In this paper,we study the propagation dynamics of a threespecies predator-prey reaction-diffusion system with Beddington-DeAngelis type functional response.The main contents are the asymptotic spreading speeds and traveling wave solution.Firstly,considering that two predators have an interspecific competition relationship,we study the asymptotic propagation behavior of two predators invading prey when the prey has reached its equilibrium state.By using classical parabolic estimation to obtain priori estimates of the solution,then constructing suitable Lyapunov function and using a persistence lemma combined with the relevant conclusions of scalar equations and the method of upper and lower solutions,we can prove the persistence of species in some intermediate moving frames.The results show that if two predators travel at the same speed,two regions can be observed:the front(predator-free region)and the final region(three species co-exist);If two predators spread at different speeds,three regions can be observed:the frontier(predator-free region),the intermediate region(faster predator and prey co-exist),and the final region(three species co-exist).Secondly,simplifying the system to take into account the absence of interspecific competition between the two predators,we investigate the existence of travelling wave solutions connecting the semi-equilibrium and co-existence equilibrium.We consider that the environment is initially inhabited by prey and one of two predators,and eventually the three species co-exist when the other predator invades the environment.The proof of the traveling wave solution connecting the semi-equilibrium at negative infinity is based on the construction of generalized upper and lower solutions and using the Schauder fixed point theorem.Then,under the assumption that the co-existence equilibrium exists,we use the theory of stable manifold theroy to give the positive lower bound of the solution,and then construct a suitable Lyapunov function to prove the traveling wave solution connecting co-existence equilibrium points at positive infinity.更多还原

【Abstract】 In recent years,the reaction-diffusion equation has been widely used to describe the dynamic behavior of populations because of its rich propagation dynamics.At present,the dynamics study of single species is relatively complete.For multiple species,because of the need to consider the complex interactions between them,the model is more complex,has more rich dynamic behavior,and brings greater difficulties to the research work.In this paper,we study the propagation dynamics of a threespecies predator-prey reaction-diffusion system with Beddington-DeAngelis type functional response.The main contents are the asymptotic spreading speeds and traveling wave solution.Firstly,considering that two predators have an interspecific competition relationship,we study the asymptotic propagation behavior of two predators invading prey when the prey has reached its equilibrium state.By using classical parabolic estimation to obtain priori estimates of the solution,then constructing suitable Lyapunov function and using a persistence lemma combined with the relevant conclusions of scalar equations and the method of upper and lower solutions,we can prove the persistence of species in some intermediate moving frames.The results show that if two predators travel at the same speed,two regions can be observed:the front(predator-free region)and the final region(three species co-exist);If two predators spread at different speeds,three regions can be observed:the frontier(predator-free region),the intermediate region(faster predator and prey co-exist),and the final region(three species co-exist).Secondly,simplifying the system to take into account the absence of interspecific competition between the two predators,we investigate the existence of travelling wave solutions connecting the semi-equilibrium and co-existence equilibrium.We consider that the environment is initially inhabited by prey and one of two predators,and eventually the three species co-exist when the other predator invades the environment.The proof of the traveling wave solution connecting the semi-equilibrium at negative infinity is based on the construction of generalized upper and lower solutions and using the Schauder fixed point theorem.Then,under the assumption that the co-existence equilibrium exists,we use the theory of stable manifold theroy to give the positive lower bound of the solution,and then construct a suitable Lyapunov function to prove the traveling wave solution connecting co-existence equilibrium points at positive infinity.更多还原