Research On The Cannibalism Model With Delay And COVID-19 Model With Diffusion

Biodynamics is an important branch of biomathematics,which studies how biomass and the morphology of organisms change over time.These include population dynamics and infectious disease dynamics.In this work,the population dynamics of a class of cannibalism model with double delay and stage-structure and the dynamics of an infectious disease of COVID-19 model with Beddington-DeAngelis incidence and diffusion are studied from these two aspects.In the first part,a class of cannibalism model with double delay and stage-structure is studied.One delay is to characterize the lag effect of negative feedback of the prey population,the other is used to describe the gestation period of an adult predator.First,we discuss the existence of all equilibrium points in the system,Secondly,sufficient conditions for the local stability of the system's triviality and boundary equilibrium point are established by analyzing the characteristic equations.Meanwhile,by choosing the delay as bifurcation parameter,criteria on the existence of Hopf bifurcation are established.Furthermore,by using the normal form theory and center manifold theorem,we derive the explicit formulas determining properties of periodic solutions.Finally,the theoretical results are illustrated by numerical simulations,by which we can see that the feedback time delay of prey is more sensitive to the stability of the system than the gestation time delay of predator;in the finite time delay interval,the predator's gestation time delay can make the chaotic phenomenon of the system disappear and make the system solution maintain periodic oscillation;a large feedback time delay of prey can make predators extinct and and make the prey form a periodic solution.In the second part,a SEIAR model with Beddington-DeAngelis incidence and reactiondiffusion is used to describe COVID-19 transmission with spatial transmission characteristics.First,the appropriateness of the system solution is studied.Second,the basic disease regeneration number R0 is derived as the threshold for determining whether COVID19 will spread.At the same time,considering the spreading dynamics of COVID-19 in a spatially homogeneous environment,it is obtained that when R0> 1,the disease-free steady-state is globally asymptotically stable;and when R0 <1,the endemic steadystate is globally asymptotically stable.In addition,according to the early official report data of COVID-19 in Wuhan,China,we estimated the actual value of R0,and found that compared with other types of incidence,the peak of the infected person estimated by Beddington-DeAngelis incidence model is closer real situation.Finally,it is observed through numerical simulations that the impact of spreading behavior on the spread of COVID-19 is mainly reflected in the heterogeneity of space,rather than the homogeneity of the environment.