A non-local model for dispersal with continuous time and space is carefully justified and discussed. The necessary mathematical background is developed and we point out some interesting and challenging problems. While the basic model is not new, a 'spread' parameter (effectively the width of the dispersal kernel) has been introduced along with a conventional rate paramter, and we compare their competitive advantages and disadvantages in a spatially heterogeneous environment. We show that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal. However, fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results. For example, in a fairly general setting given two phenotypes with different, but small spread, the smaller spread is selected while in the case of large spread the larger spread is selected.

We study the travelling wave problem J * u - u - cu' + f(u) = 0 in R, u(-infinity) = 0, u(+infinity) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c not equal 0. For c = 0 we show examples of nonuniqueness

Abstract. Let J ∈ C(R), J ≥ 0, R J = 1 and consider the nonlocal diffusion operator M[u] = J ⋆ u − u. We study the equationwhere f is a KPP type non-linearity, periodic in x. We show that the principal eigenvalue of the linearization around zero is well defined and a that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, J is symmetric then the non-trivial solution is unique.

Artículo de publicación ISIWe examine whether the expectations of pre-service elementary school teachers about students' achievement, and their beliefs regarding student need for academic support, are influenced by future teachers' mathematics anxiety or by student gender and socioeconomic status. We found that mathematics anxiety can negatively influence pre-service teachers' expectations about students, and that future mathematics teachers' expectations of mathematics achievement are lower for girls than for boys. These effects are independent, as we did not find significant interaction effects between pre-service teacher's mathematics anxiety and student gender. Our results also suggest that mathematics anxiety could affect the capacity of pre-service teachers to develop inclusive learning environments in their classrooms.Fondecyt 1140834 PIA-Conicyt Basal Funds for Centers of Excellence Project BF0003 Fondef D09-I1023 Basal project CMM U. de Chile UMI 2807 CNR

Artículo de publicación ISIIn this paper we are interested in propagation phenomena for nonlocal reaction–diffusion equations of the type: ∂u ∂t = J ∗ u −u+ f (x,u) t ∈ R, x ∈ RN, where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution