We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. We introduce theclimate shift due to Global Warming and discuss the dynamicsof the population by studying the long time behavior of thesolution of the Cauchy problem. We consider three sets ofassumptions on the growth function. In the so-called confinedcase we determine a critical climate change speed for theextinction or survival of the population, the latter case taking place by "strictly following the climate shift". In the so-called gradient case , or unconfined case , we additionally determine the propagation speedof 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 thegrowth 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.
The main difficulty in the mathematical analysis of is to handle the nonlocal competition term. When the competition term is replaced by a local (in @math and @math ) density regulation, many techniques based on the comparison principle --- such as some monotone iterative schemes or the sliding method --- can be used to get, among other things, monotonicity properties of the solution. Since integro-differential equations with a nonlocal competition term do not satisfy the comparison principle, it is unlikely that such techniques apply here. Problem shares this difficulty with the nonlocal Fisher-KPP equation which describes a population structured by a spatial variable only, and submitted to nonlocal competition modelized by the kernel @math . As far as equation is concerned, let us mention the possible destabilization of the steady state @math by some kernels @cite_17 , the construction of travelling waves @cite_12 , additional properties of these waves @cite_2 , @cite_1 , and a spreading speed result @cite_6 . We also refer to @cite_15 , @cite_9 for the construction of travelling waves for a bistable nonlocal equation, for an epidemiological system with mutations respectively.
@cite_9 @cite_1 @cite_6 @cite_2 @cite_15 @cite_12 @cite_17
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A sufficient and necessary condition for the existence of monotone travelling waves in the nonlocal Fisher–KPP equation is established, and the uniqueness of travelling wavefronts (up to translation) is also proved. Abstract In this note, we give a positive answer to a question addressed in (2011) [7] . To be precise, we prove that, for any kernel and any slope at the origin, there exist traveling 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 . Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. In this article, we are interested in a non-monotone system of logistic reaction-diffusion equations. This system of equations models an epidemics where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of minimal speed travelling waves, that are usually non monotonic. We then provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed.