Spreading Speeds of Time-Dependent Partially Degenerate Reaction-Diffusion Systems?

Jia LIU

1School of Science,Chang’an University,Xi’an 710064,China.E-mail:liujia@chd.edu.cn

Abstract This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity.By using the principal Lyapunov exponent theory,the author first proves the existence,uniqueness and stability of spatially homogeneous entire positive solution for time dependent partially degenerate reaction-diffusion system.Then the author shows that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions.The author also applies these results to a time dependent partially degenerate epidemic model.

Keywords Partially degenerate,Reaction-diffusion system,Time dependent,Spreading speed

1 Introduction

Reaction-diffusion models have been widely used to describe the spatial dynamics of populations in biology and ecology,for example,see the benthic-pelgic population model in[14]and man-environment-man epidemics model in[6–7]and so on.In these two models,at least one diffusion coefficients of population are zero and we usually call such models as partially degenerate reaction-diffusion system,in which some diffusion coefficients are zero.Spreading speeds and traveling waves of partially degenerate reaction-diffusion system are two important issues in the study of biological invasions and disease spread and have attracted a lot of attention,see[2,7–8,11–12,26,28–32]and references therein.

Due to the presence of various temporal and spatial in many biological models,spreading speeds and traveling wave solution for general time andor space dependent reaction-diffusion models have been widely investigated in many works.For example,see[1,4–5,13,16–17,19–24,27]and references therein.Recently,Bao et al.[4]have introduced the notion of spreading speeds for general time dependent cooperative systems with different dispersal types and see[3]for the existence and stability of generalized transition waves for time-dependent reactiondiffusion systems.However,in general time dependent case,there is little understanding about the spatial spread and front propagation dynamics for partially degenerate reaction-diffusion systems.The objective of the current paper is to study the spatial spreading speed of partially degenerate reaction-diffusion systems with general time dependent coefficients.

Consider the following time dependent partially degenerate reaction-diffusion system:

We point out that the spreading speed and traveling wave solution of(1.1)with time periodic have been studied in[12]for monostable nonlinearity case and in[30]for bistable nonlinearity case.For system(1.1)with monostable nonlinearity in space periodic habitat,Wu et al.[29]has established the existence of spreading speed and Wang and Zhao[28]proved the existence and stability of pulsating wave to such partially degenerate reaction-diffusion system.Here,we establish the existence of spreading speed of time dependent partially degenerate reactiondiffusion system(1.1)with monostable nonlinearity and apply the results to a time dependent epidemic model.The existence and stability of generalized transition wave solution of(1.1)will be studied somewhere else.

The rest of this paper is organized as follows.In Section 2,we will present the Lyapunov exponent theory and prove the existence of entire solution for system(1.1).In Section 3,we will establish the existence of the spreading speed and prove some important spreading features.We apply the results to a time dependent epidemic model in Section 4.

2 The Lyapunov Exponent Theory and Entire Solution

In this section,we present the principle Lyapunov exponent theory for time dependent linear system and study entire positive solution of(1.1).

3 Spreading Speed

4 A Time-Dependent Epidemic Model

AcknowledgementThe author would like to thank the referee for valuable comments and suggestions which improved the presentation of this manuscript.