Propagation Dynamics Of A Three-species Predator-prey Model In A Shifting Environment

In recent years,due to the influence of temperature,climate and human factors,the living environment of species has changed,which directly affects the survival and spread of species,so it’s very necessary to study the spread of species in a shifting environment.At the same time,compared with the random diffusion equation,the nonlocal diffusion equation can more effectively describe the small-density and large-scale free movement process in ecology.In this paper,we study the propagation dynamics of a three-species predator-prey model(including two preys and one predator)with nonlocal diffusion in a shifting environment,mainly focusing on the study of forced waves and Spreading speeds.Firstly,the existence and nonexistence of forced traveling waves are considered for the above system.Suppose that before climate change,the environment is favorable for the preys,and over time,the environment deteriorates and eventually becomes unfavorable for the species.Under certain conditions,by constructing appropriate upper-lower solutions and applying the Schauder Fixed-point theorem,two different types of forced waves(front type and mixed front-pulse type)are obtained,they connect different non-trivial states to extinction state.At the same time,we can characterize the minimal shifting speed for each mixed front-pulse type forced waves.Secondly,the propagation theory of the system is studied.Specifically,we consider spreading speeds in the following three situations:(i)both prey spread faster than the predator;(ii)the predator spreads faster than both prey;and(iii)the predator spreads between two preys,we prove that the three species survival or extinction of species in different moving frames.In addition,under certain conditions,there are three types of coexistence: two prey coexist in the absence of the predator,the predator coexists with one of the prey and three species coexist.In this paper,the results are proved by using the relevant results of scalar equation and the method of upper-lower solutions.Finally,by using the prior estimate of the solution,we construct a suitable Lyapunov function to prove the persistence of species in some intermediate moving frames.