We consider an Allen-Cahn type equation of the form u t = u + ε −2 f ε (x, t, u), where ε is a small parameter and f ε (x, t, u) (x, t, u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u 0 that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε 2 | ln ε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where g ε ≡ 0.Next we consider systems of reaction-diffusion equations of the formwhich include the FitzHugh-Nagumo system as a special case. Given a rather general initial data (u 0 , v 0 ), we show that the component u develops a steep transition layer and that all the above-mentioned results remain true for the u-component of these systems.

In this note, we give a positive answer to a question addressed in [8]. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are "rapid") of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with fat tails.

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c * > 0, and prove the existence of waves when c ≥ c * and the nonexistence when 0 ≤ c < c * .

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. We introduce the climate shift due to Global Warming and discuss the dynamics of the population by studying the long time behavior of the solution of the Cauchy problem. We consider three sets of assumptions on the growth function. In the so-called confined case we determine a critical climate change speed for the extinction or survival of the population, the latter case taking place by "strictly following the climate shift". In the socalled environmental gradient case, or unconfined case, we additionally determine the propagation speed of the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider mixed scenarios, that are complex situations, where the growth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left.The main difficulty comes from the nonlocal competition term that prevents the use of classical methods based on comparison arguments. This difficulty is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality.

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We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.

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We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic, acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.