Abstract
We study a system of branching Brownian motions on $\mathbb{R} $ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. The case of strictly local creations (the new particle is put exactly at the same position of the branching particle) was studied in [10]. In [11] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [10] and non local branching as in [11] and prove convergence in the continuum limit (when the number $N$ of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions. We use in the convergence a stronger topology than in [10] and [11] and have explicit bounds on the rate of convergence.
Highlights
The system considered in this paper fits in a class of models proposed by Brunet and Derrida in [3] to study selection mechanisms in biological systems and continues a line of research initiated by Durrett and Remenik in [11].Durrett and Remenik have studied a model of particles on R which independently at rate 1 branch creating a new particle whose position is chosen randomly with probability p(x, y)dy, p(x, y) = p(0, y − x), if x is the position of the generating particle
Even if the duplication rule is regressive, i.e. p(x, y) has support on y < x, the population fitness improves and if p(0, x) > 0 for x ∈ (−a, 0), a > 0, as time diverges the whole population concentrates around the position of the initially best fitted individual
Similar formulas hold as well in the case considered in this paper where we study a natural extension of the Durrett-Remenik model where particles move as independent Brownian motions in between branching times: biologically this means that the individuals fitness changes randomly in time
Summary
The system considered in this paper fits in a class of models proposed by Brunet and Derrida in [3] to study selection mechanisms in biological systems and continues a line of research initiated by Durrett and Remenik in [11]. Durrett and Remenik have studied the case where p(x, y) is symmetric and discussed the occurrence of traveling - wave solutions which describe a steady improvement of the population fitness, see [15] and Brunet, Derrida [1], [3] for the analysis of traveling waves in a large class of systems. Similar formulas hold as well in the case considered in this paper where we study a natural extension of the Durrett-Remenik model where particles move as independent Brownian motions in between branching times: biologically this means that the individuals fitness changes randomly in time. In the two sections we make precise the model and state the main results, an outline of how the paper is organized is given at the end of Section 3 We conclude this introduction by mentioning that there have been several papers about particles processes which in the continuum limit are described by free boundary problems. Some of them will be mentioned in the sequel, for a list we refer to a survey on the subject, [6]
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