Gause's principle of competition between two species is studied when one of them is sterile. We study the condition for total extinction in the niche, namely, when the sterile population exterminates the native one by an optimal use of resources. A mathematical Lotka-Volterra non linear model of interaction between a native and sterile species is proposed. The condition for total extinction is related to the initial number M_o of sterile individuals released in the niche. In fact, the existence

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... of a critical sterile-population value M_c is conjectured from numerical analysis and an analytical estimation is found. When spatial diffusion (migration) is considered a critical size territory is found and, for small territory, total extinction exist in any case. This work is motived by the extermination agriculture problem of fruit flies in our region.