Long-time behaviour of an advection-selection equation

We study the long-time behaviour of the advection-selection equation partial derivative(t)n(t, x)+del center dot (f(x)n(t,x)) = (r(x) - rho(t))n(t,x), t >= 0, x is an element of R-d, with rho(t)=integral(R)d n(t,x)dx an initial condition n(0. center dot) = n(0). In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, x bar right arrow n(t, x) represents the density of the population characterised by a phenotypic trait x, the advection term del center dot (f(x)n(t,x)) a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term (r(x) - rho(t))n(t,x) the growth of the population, which is of logistic type through the total population size rho(t)=integral(Rd) n(t,x)dx. In the one-dimensional case x is an element of R, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in L-1. Depending on the parameters n(0), f and r, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the L-1-function which is reached.