We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved by B. Perthame and G. Barles for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.
Lotka-Volterra competition models on finite graphs
本文研究了有限连通图上三个两种群竞争的Lotka-Volterra竞争模型，分别具有Dirichlet、Neumann和无边界条件.在Neumann边界条件和无边界条件下，当时间趋于无穷大时，种群中的一个种群灭绝而另一个种群生存，或者两个竞争种群共存，这主要取决于种群竞争力的强弱和种群初始规模.我们的一个结果部分回答了斯拉夫·勒克在[SIAM J. Appl. Dyn.系统：19（2020）]。证明主要结果的关键技术是上、下解方法，本文针对有限图上的弱耦合抛物型方程组发展了上、下解方法。
Propagation Direction of Bistable Traveling Fronts in the Lotka--Volterra Competition--Diffusion System
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）
Sharp Thresholds for Linear Determinacy and Zero Wave Speed in Lotka-Volterra Competition-Diffusion Systems
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然博物馆）；Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）
Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays
