Spatially heterogeneous competitions

Under these assumptions, (10.1) provides us with a model for the evolution of two competing species with densities u and v in the habitat Ω. In the absence of v, the species u grows according to the (non-classical) generalized logistic parabolic problem

∂u ∂t − d1∆u = λu− a(x)f1(x, u)u in Ω× (0,∞), u = 0 on ∂Ω× (0,∞), u(·, 0) = u0 ≥ 0 in Ω,

(10.2)

whose dynamics were described in Chapter 5. Similarly, in the absence of u, the species v grows according to the (classical) generalized logistic problem

∂v ∂t − d2∆v = µv − d(x)f2(x, v)v in Ω× (0,∞), v = 0 on ∂Ω× (0,∞), v(·, 0) = v0 ≥ 0 in Ω.