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The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the(More)
On a compact Riemannian manifold (V m , g), we consider the second order positive operator L ǫ = ǫ∆ g + (b, ∇) + c, where −∆ g is the Laplace-Beltrami operator and b is a Morse-Smale (MS) field, ǫ a small parameter. We study the measures which are the limits of the normalized first eigenfunctions of L ǫ as ǫ goes to the zero. In the case of a general MS(More)
In this Note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse–Smale dynamical system on a compact Riemannian manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure µ concentrated on the hyperbolic sets of the field. The coefficients which appear in the(More)
We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant ǫ goes to zero. We assume that the first order term is given by a vector field b, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the(More)
We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant ǫ goes to zero. If the drift of the diffusion is given by a Morse-Smale vector field b, the limits of the eigenfunctions concentrate on the recurrent set of b. A blow-up analysis enables us to find(More)
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