From indirect  to direct taxis by fast reaction limit.

Many ecological population models consider taxis as the directed movement of animals in reaction to a stimulus which we name as direct (if it is guided by the density gradient of some other population), or indirect (if it is guided by the density of a chemical secreted by the individuals of the other population). We prove that in the so-called fast reaction limit a solution to the  Lotka -Volterra competition model with indirect taxis converges to the solution of the competition model with direct taxis.  Our result concerning the reaction-diffusion-advection systems refers to the small parameter method and the classical Tikhonov theorem known for systems of ordinary differential equations, in which a singular limit reduces the number of equations of the system in the so-called quasi-stationary approximation.