Global Asymptotic Behaviour of Lotka-Volterra competition systems with diffusion and time delays

Abstract

The Lotka-Volterra competition systems are mathematical models describing the evolution of the density of population (or the number of individuals) of multiple living species, competing with one another for the life resources. In this thesis we present the work of C.V. Pao [3] on the asymptotic behaviour of such populations in the long run. The Lotka-Volterra model of N competing species is given in the form    ∂tui(t, x) − Liui(t, x) = aiui(t, x)

1 − ui(t, x) − X N j6=i bijuj (t, x) − X N j=1 cijuj (t − τj , x) ! , t > 0, x ∈ Ω ∂ui ∂ν (t, x) = 0, t > 0, x ∈ ∂Ω ui(t, x) = ηi(t, x), −τi 6 t 6 0, x ∈ Ω, (1) Ω represents the enviroment (in R, R 2 , or R 3 ) inside which occures all the interactions between populations, ui(t, x) is the density of population i at time t ≥ 0 and in the position x ∈ Ω. The parameters are nonnegative constants where ai 6= 0 is the self-growth rate of population i ; bij is relative rate of the effect of populations j on population i and cij is same as bij execept that the effect between populations is delayed with a delay of τj , and both are called competition rates. ∂ui/∂ν(t, x) = 0 stats that no flux of all populations occures across ∂Ω the boundary of Ω. Li = Di(x)∆ + σ(x) · ∇ is a diffusion-convection operator (Di(x) > 0) introduced to take into consideration the dispersion effect if exists for some or all populations, othewise Li is allowed to be zero if the population shows no diffusion, hence the model is a coupled ordinary and parabolic system . As mentioned earlier the aime is to study the asymptotic behaviour of the solution of (1) more precisely, in [3] the interst is given to the investigation of the conditions on the competition rates (bij and cij ) underwhich the system has constant (independent of x) 5 asymptotic behaviour. The work is devides into three chapter, the first is preliminary, where all the necessary tools are set up such as elliptic maximum principle and results on semilinear parabolic systems. The second chapter is dedicated to steady state systems corresponding to time depending problems in the form    ∂tui − Liui = uifi(u, uτ ),(t > 0; x ∈ Ω), ∂ui ∂ν = 0, on ∂Ω ui(t, x) = ηi(t, x), (−τi 6 t 6 0, x ∈ Ω), where uτ (t, x) = (u1(tτ1 , x), · · · , uN (tτN , x)) In this case the steady state problem is    −Liui = uifi(u, u), in Ω, ∂ui ∂ν = 0, on ∂Ω the existence of constant quasisolutions and solutions is studied using upper and lower solutions method [3,4], pairs of quasisolutions (to be defined later) are important since they constitute attarcting sectors of solutions of the corresponding time depending problem as t tends to +∞ for suitable set for initial functions ηi [4] liying between upper and lower solutions. Finaly the third chapter is devoted to the study of possible constant asymptotic behaviour of the solutions of (1) under conditions given on the competition rates only. Asymptotic behaviour is said to be global if it is proved to be the limit of u(t, x), as t tends to infinity, for all nonnegative, non identically zero initial functions ηi .