  In this paper, we introduce and analyze an asymptotic-preserving scheme for
Lotka-Volterra parabolic equations. It is a class of nonlinear and nonlocal
stiff equations, which describes the evolution of a population structured with
phenotypic trait. In a regime of long time and small mutations, the population
concentrates at a set of dominant traits. The dynamics of this concentration is
described by a constrained Hamilton-Jacobi equation, which is a system coupling
a Hamilton-Jacobi equation with a Lagrange multiplier determined by a
constraint. This coupling makes the equation nonlocal. Moreover, the constraint
does not enjoy much regularity, since it can have jumps.
  The scheme we propose is convergent in all the regimes, and enjoys stability
in the long time and small mutations limit. Moreover, we prove that the
limiting scheme converges towards the viscosity solution of the constrained
Hamilton-Jacobi equation, despite the lack of regularity of the constraint. The
theoretical analysis of the schemes is illustrated and complemented with
numerical simulations.
