From indirect to direct taxis by fast reaction limit

Many population models formulated in terms of quasi-linear parabolic systems incorporate taxis, understood as directed movement of animals in response to some stimulus. Taxis is called direct if the animals follow the density gradient of another population, or indirect if they follow the density of a chemical secreted by individuals from another population. We prove that in the so-called fast reaction limit, i.e. when the ratio [Formula: see text] of the chemical signal diffusion to the signal production/degradation vanishes asymptotically, the solution of the model with indirect taxis converges to the solution of the model with direct taxis. It is justified then to replace the gradient of the chemical by the gradient in population density, reducing by one the number of equations in the indirect taxis model. The kinetic part of the considered models describes competition or predation, and the taxis term refers to the avoidance strategy assigned to one of the competitors or to a prey-taxis. Under certain compatibility condition satisfied by the initial data for the space dimension [Formula: see text], convergence occurs in a strong topology of function spaces, and its rate can be controlled by [Formula: see text]. Otherwise, the compactness argument only leads to convergence for a subsequence in the appropriate weak sense.