The convective diffusion equation with drift b(x) and indefinite weight r(x),
∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]+λr(x)ϕ, (1)
is introduced as a model for population dispersal. Strong connections between
Equation (1) and the forced Burgers equation with positive frequency (m≥0), ∂u∂t=∂2u∂x2−u∂u∂x+mu+k(x), (2)
are established through the Hopf-Cole transformation. Equation (2) is a prime
prototype of the large class of quasilinear parabolic equations given by
∂u∂t=∂2u∂x2+∂(f(v))∂x+g(v)+h(x). (3)
A compact attractor and an inertial manifold for the forced Burgers equation are
shown to exist via the Kwak transformation. Consequently, existence of an
inertial manifold for the convective diffusion equation is guaranteed. Equation (2)
can be interpreted as the velocity field precursed by Equation (1). Therefore, the
dynamics exhibited by the population density in Equation (1) show their effects on
the velocity expressed in Equation (2). Numerical results illustrating some aspects
of the previous connections are obtained by using a pseudospectral method.
