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This paper is concerned with the limit, as the interspecific competition rategoes to infinity, of pulsating front solutions in space-periodic media for abistable two-species competition--diffusion Lotka--Volterra system. Wedistinguish two important cases: null asymptotic speed and non-null as-ymptoticspeed. In the former case, we show the existence of a segregated stationaryequilibrium. In the latter case, we are able to uniquely characterize thesegregated pulsating front, and thus full convergence is proved. The segregatedpulsating front solves an interesting free boundary problem. We alsoinvestigate the sign of the speed as a function of the parameters of thecompetitive system. We are able to determine it in full generality, withexplicit conditions depending on the various parameters of the problem. Inparticular, if one species is sufficiently more motile or competitive than theother, then it is the invader. This is an extension of our previous work inspace-homogeneous media.

Competition in periodic media: II -- Segregative limit of pulsating fronts and "Unity is not Strength"-type result