ABSTRACT
The main goal of this chapter is ascertaining the asymptotic behavior of the solutions of (1.1) as t ↑ ∞ according to the each of the different ranges of values of the parameter λ ∈ R. From the point of view of population dynamics, these behaviors can be briefly sketched as follows:
• If λ ≤ dσ0, then the inhabiting region Ω cannot support the species u. • If dσ0 < λ < dσ1, then the species u exhibits logistic growth in Ω. • If q0 ≥ 2 and dσj ≤ λ < dσj+1 for some j ∈ {1, ..., q0 − 1}, then u has
Malthusian growth in ( Ω¯0,1 ∪ · · · ∪ Ω¯0,j
) \ ∂Ω, and logistic growth in the complement
Ωj := Ω \ ( Ω¯0,1 ∪ · · · ∪ Ω¯0,j
) .
