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

Journal article

In this work we study a non-local version of the Fisher-KPP equation, (∂u ∂t = 1 2∆u + u(1 − φ ∗ u), t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R and its relation to a branching Brownian motion with decay of mass as introduced in [1], 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 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 φ). 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 φ and u0, the solutions converge to 1 behind the front and are globally bounded, improving recent results in [11]. 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.

Authors: Berestycki, J; Penington, S; Brunet, E

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: John Wiley & Sons
• Journal: Communications on Pure and Applied Mathematics
• Volume: 72
• Issue: 12
• Pages: 2487-2577
• Publication date: 2019-04-19
• Acceptance date: 2018-07-14
• DOI: 10.1002/cpa.21827
• EISSN: 1097-0312
• ISSN: 0010-3640
• Language: English