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In this work we study a non-local version of the Fisher-KPP equation,\begin{equation*} \begin{cases} \frac{\partial u}{\partialt}=\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 itsrelation to $\textit{branching Brownian motion with decay of mass}$ asintroduced by Addario-Berry and Penington (2017), i.e. a particle systemconsisting of a standard branching Brownian motion (BBM) with a competitiveinteraction between nearby particles. Particles in the BBM with decay of masshave a position in $\mathbb{R}$ and a mass, branch at rate 1 into two daughterparticles of the same mass and position, and move independently as Brownianmotions. Particles lose mass at a rate proportional to the mass in aneighbourhood around them (as measured by the function $\phi$).  We obtain two types of results. First, we study the behaviour of solutions tothe partial differential equation above. We show that, under suitableconditions on $\phi$ and $u_0$, the solutions converge to 1 behind the frontand are globally bounded, improving recent results of Hamel and Ryzhik(arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBMwith decay of mass is the solution of the non-local Fisher-KPP equation. Wethen harness this to obtain several new results concerning the behaviour of theparticle system.

Branching Brownian motion with decay of mass and the non-local Fisher-KPP equation