In this work we study a non-local version of the Fisher-KPP equation, \begin{ equation*} \begin{ cases} \frac{ \partial u}{ \partial t}=\tfrac{ 1}{ 2}\Delta u +u (1- \phi \ast u), \quad t>0, \quad x\in \mathbb{ R}, u(0,x)=u_0(x), \quad x\in \mathbb{ R} \end{ cases} \end{ equation*} and its relation to $\textit{ branching Brownian motion with decay of mass}$ as introduced by Addario-Berry and Penington (2017), i.e. a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in $\mathbb{ R}$ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighbourhood around them (as measured by the function $\phi$). We obtain two types of results. First, we study the behaviour of solutions to the partial differential equation above. We show that, under suitable conditions on $\phi$ and $u_0$, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik (arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the non-local Fisher-KPP equation. We then harness this to obtain several new results concerning the behaviour of the particle system.
Categories: math.PR math.AP