Asymptotic phases in a cell differentiation model

T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet (x1) and GATA-3 (x2). The population density of the T cells, ϕ(t,x1,x2), satisfies a conservation law ∂ϕ/∂t+(∂/∂x1)(f1ϕ)+(∂/∂x2)(f2ϕ)=gϕ where fi depends on (t,x1,x2) and, in a nonlinear nonlocal way, on ϕ. It is proved that, as t→∞, ϕ(t,x1,x2) converges to a linear combination of 1, 2, or 4 Dirac measures. Numerical simulations and their biological implications are discussed.