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International audienceWe study a spectral problem related to a reaction-diffusion model where the preys and the predators do not live on the same area. We are interested in the optimal zone where the control should take place. First we prove existence of an optimal domain in a natural class. Then, it seems plausible that the optimal domain is localized in the intersection of the living areas of the two species. We prove this fact in one dimension for small size of domains

On some spectral problems arising in dynamic populations