Diffusion Models And Their Risk Characterizations In The Changing Domains

Diffusion phenomenon is ubiquitous,and it is widely used in ecology,epidemiology,physics and other disciplines.At present,the study of diffusion has become a hot spot in ecological research,which is closely related to population diffusion(especially the invasion of alien species)and the prevention and control of infectious diseases.Taking population diffusion and infectious disease diffusion mechanism as the research object,and taking different diffusion regions and different diffusion ways as the main line,the influence of evolutionary region,free boundary,spatial heterogeneity and seasonal succession on the asymptotic behaviors of the solution are considered.The discussion includes the following aspects:Chapter 1 mainly introduces some background knowledge and research status of ecological diffusion,diffusion in different regions and in different ways,and gives the main content of this paper.Chapter 2,Chapter 3 and Chapter 4 focus on the local diffusion in known regional,where Chapter 2 is concerned with the diffusion of species,while the diffusion of infectious diseases is considered in Chapter 3 and Chapter 4.Chapter 5 depicts the nonlocal diffusion of infectious diseases in unknown regions.Chapter 2 considers the diffusion of a single phytoplankton species that only depends on light to maintain the metabolism of life in a periodically evolving environment,where both the death rate and the light intensity rely on the depth of the water column triggered by seasonal variation.We give the net propagation rate R0 by defining the spectral radius of the operator and some corresponding properties.Based on net reproduction rate R0,sufficient conditions for the expansion or extinction of phytoplankton are established.In particular,the relationships between the evolution rate p(t),the vertical turbulent diffusion rate D,the buoyant or sinking rate ?,water column depth L0 and net propagation rate R0 are derived.Both our theoretical results and numerical simulations reveal that there exists negative correlation between evolution rate and survival of phytoplankton,moreover,big vertical diffusion rate have an adverse effect on survival of phytoplankton,and so does water column depth.Chapter 3 is concerned with a diffusive SIS epidemic model in a heterogeneous and periodically evolving domain are considered,which is a known regional change caused by periodic changes of natural environment.By assuming that the change is periodic and isotropic,the epidemic model in a evolving domain is converted to the reaction diffusion problem in a fixed domain.Based on operator semigroup theory and spectral theory,the basic reproduction number determined by spatial heterogeneity and evolving rate of the domain is defined.A threshold type result on the dynamics of the model is established by the upper and lower solutions method and using the principal eigenvalue,and a biological explanation of the impact of regional evolution on disease is also given.Both our theoretical results and numerical simulations show that small evolving rate have an beneficial effect on the control of the infectious disease.To understand the relationship of the growing rate and the transmission risk of West Nile virus(WNv),a WNv model on a growing domain in Chapter 4,which accounts for climate warming results in habitat expansion of mosquitoes.The basic reproduction number,which is related to the growing rate and diffusion rate,is introduced through spectral theory.We derive the conditions to determine whether the virus vanishes or spreads in accordance with the basic reproduction number.The obtained results reveal that there exists positive correlation between rate of domain growth and risk of infection,and it is negative to the prevention and control of WNv.To show a agreement with our analytical results on the long-time behavior of WNv,some numerical simulations are performed.Chapter 5 is devoted to dealing with a WNv nonlocal diffusion model with free boundary,where the nonlocal diffusion characterizes a long-range dispersal,the free boundary describes the spreading front,and seasonal succession accounts for the effect of the warm and cold seasons.The well-posedness of the model is established,and its long-term dynamical behaviours,which depend on the generalized principal eigenvalues of the corresponding linear operator,are investigated.For both spatially independent and nonlocal WNv models with seasonal successions,the generalized eigenvalues are determined.We develop the indexes to the case with the free boundary and apply these indexes to determine whether spreading or vanishing happens.Our new criteria extends previous results for the case with the nonlocal diffusion and the case with the free boundary.The generalized principal eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection.Moreover,the initial infection length,the initial infection scale and the spreading ability to the new area play an important role for the long time behavior of the solution.Finally,we briefly summarize the work of this paper in Chapter 6,and make plans for the future research work.