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Global existence for a free boundary problem of Fisher–KPP type

IOP Publishing
Nonlinearity
Authors:
Julien Berestycki at University of Oxford
Julien Berestycki
Éric Brunet
Éric Brunet
Éric Brunet
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Sarah Penington at University of Oxford
Sarah Penington
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Abstract

Motivated by the study of branching particle systems with selection, we establish global existence for the solution of the free boundary problem when the initial condition is non-increasing with as and as . We construct the solution as the limit of a sequence , where each un is the solution of a Fisher–KPP equation with the same initial condition, but with a different nonlinear term. Recent results of De Masi A et al (2017 (arXiv:1707.00799)) show that this global solution can be identified with the hydrodynamic limit of the so-called N-BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by removing the leftmost particle at each branching event.
... (1.6) Basic properties of solutions of (1.6) are given by [BBP19]. We also have the following. ...
... It follows from the definitions that κ : P(R) → D is a homeomorphism. Existence and uniqueness of solutions to (1.6) was established in [BBP19] for initial conditions U 0 belonging to D. Furthermore, it is also shown in [BBP19] that (u, L) is a solution of (1.4) with initial condition µ 0 ∈ P(R) if and only if (U, L) is a solution of (1.6) with initial condition κ(µ 0 ), where U t = κ(u t ) for t > 0 and the free boundary L t is unchanged (here we identify u t with the measure having density u t ). ...
... It follows from the definitions that κ : P(R) → D is a homeomorphism. Existence and uniqueness of solutions to (1.6) was established in [BBP19] for initial conditions U 0 belonging to D. Furthermore, it is also shown in [BBP19] that (u, L) is a solution of (1.4) with initial condition µ 0 ∈ P(R) if and only if (U, L) is a solution of (1.6) with initial condition κ(µ 0 ), where U t = κ(u t ) for t > 0 and the free boundary L t is unchanged (here we identify u t with the measure having density u t ). ...
Selection principle for the N-BBM
Preprint
  • Jul 2024
The N-branching Brownian motion with selection (N-BBM) is a particle system consisting of N independent particles that diffuse as Brownian motions in R\mathbb{R}, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We establish the following selection principle: as N→∞N \rightarrow \infty the stationary empirical measure of the N-particle system converges to the minimal travelling wave of the associated free boundary PDE. This resolves an open question going back at least to \cite[p.19]{Maillard2012} and \cite{GroismanJonckheer}, and follows a recent related result by the second author establishing a similar selection principle for the so-called Fleming-Viot particle system \cite{Tough23}.
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... In practice, for given values of ∆, τ, P and Γ, we will use the approximation D = PΔ 2 /(4τ) and γ = Γ /τ The evolution equation for s, equation (2.7), arises directly from our discrete model where we assume that each lattice site can occupy a maximum amount of substrate. This leads to a mechanism that is very similar to an approach that has been recently adopted to study a generalization of the well-known Fisher-KPP model where the nonlinear logistic source term is replaced with a linear saturation mechanism [73][74][75]. Solutions of these saturation-type models of invasion involve moving boundaries that form as a result of the saturation mechanism since this provides a natural moving boundary between regions where s = 1 and s < 1. Later in § §2.3 and 2.4, we will show that equations (2.6) and (2.7) can also be interpreted as moving boundary problems in exactly the same way as [73][74][75]. ...
... Solutions of these saturation-type models of invasion involve moving boundaries that form as a result of the saturation mechanism since this provides a natural moving boundary between regions where s = 1 and s < 1. Later in § §2.3 and 2.4, we will show that equations (2.6) and (2.7) can also be interpreted as moving boundary problems in exactly the same way as [73][74][75]. ...
... As discussed in §2.2, the moving boundary at x = η(t) arises because of the saturation mechanism governing the dynamics of s in equation (2.7). This kind of moving boundary problem has been previously studied in the case of a generalized Fisher-KPP model [73][74][75], except that these previous investigations have not involved any discrete stochastic models, or any kind of coarse-graining to arrive at an approximate PDE model. ...
Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion
Article
Full-text available
  • May 2024
Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction–diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction–diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.
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... A modification of the problem seeing the inverse unkown in the random starting position has been studied in [JKZ09], [JKV09b], [Abu13b] and [JKV14]. The branch of research of [De +19a], [BBP19], [Bec19], [Lee20], [Ber+20] and [Ber+21], studying certain branching particle systems with selection and corresponding free boundary problems, is related to the particular case of the inverse first-passage time problem where ξ has exponential distribution. A more detailed overview over the existent literature mentioned above is given further below. ...
... Note that in the special case of g 1 (t) = g 2 (t) = e −t a corresponding comparison principle has been established in the free boundary problem context in [BBP19] and [Ber+21]. ...
... The original free boundary problem in [BBP19] is formulated as down-crossing variant and omits the scalar 1 2 in front of the Laplacian, thus the problems can be equivalently transformed into each other by a scaling of the spatial variable by factor − √ 2. In the following we will explain the connection to the more general free boundary problem (1.14) and therefore a connection to the inverse first-passage time problem. For v 0 (x) = P (X 0 < x) and if b is the solution to the inverse first-passage time problem for the standard exponential distribution, i.e. ...
The classical and the soft-killing Inverse First-Passage Time Problem: a stochastic order approach
Thesis
  • Sep 2022
We study the classical and the soft-killing variant of the inverse first- passage time problem for Brownian motion. Given a distribution on the positive real line, the (soft-killing) inverse first-passage time problem consists of asking for an unknown function, such that the (soft-killing) first-passage time of this function has the given distribution. By the use of stochastic order relations we provide new probabilistic and more elementary approaches to these problems, which were hitherto mostly tackled in the context of partial differential equations. In the classical problem, at the one hand we obtain the known uniqueness result for solutions, but on the other hand establish new results, such as a comparison principle and sufficient conditions for monotonicity and Lipschitz continuity. Using these results we study the special case of the exponential distribution and other examples. Further, given a distribution, we study an interacting particle system, whose hydrodynamic limit finds the solution of the inverse first-passage time problem. In the soft-killing problem we show a stronger version of the known existence and uniqueness result for continuous solutions, assuming only the necessary condition for existence, and extend the result to a more general class of Markov processes.
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... For Brownian motion the modification of the problem to fix both b and ξ and to ask whether X 0 can be randomly distributed such that τ b has distribution according to ξ has been studied in [28], [30], [2], [31] and is naturally related to our comparison principle. Moreover, the inverse first-passage time problem for the case of Brownian motion and exponentially distributed ξ is related to [18], [9], [38], [8] and [7], where hydrodynamic limits of certain particle systems and corresponding free boundary problems are studied. For general ξ a related particle system whose hydrodynamic limit is characterized by the inverse first-passage time problem for reflected Brownian motion has been constructed in [34]. ...
... • For Brownian motion relations to free boundary problems [13], [9], optimal stopping problems [22], [6], integral equations [29], [39] and particle systems [18], [34] are known. Hence the question arises whether such relations extend to other stochastic processes, • By the comparison principle, the inverse first-passage time problem is related to the firstpassage time problem and the modified problems studied in [28] or [17], which therefore gain interest in this general setting. ...
CONDITIONS FOR EXISTENCE AND UNIQUENESS OF THE INVERSE FIRST-PASSAGE TIME PROBLEM APPLICABLE FOR LÉVY PROCESSES AND DIFFUSIONS
Article
Full-text available
  • Dec 2025
  • ANN APPL PROBAB
For a real-valued stochastic process (X t) t≥0 we establish conditions under which the inverse first-passage time problem has a solution for any random variable ξ > 0. For Markov processes we give additional conditions under which the solutions are unique and solutions corresponding to ordered initial states fulfill a comparison principle. As examples we show that these conditions include Lévy processes with infinite activity or unbounded variation and diffusions on an interval with appropriate behavior at the boundaries. Our methods are based on the techniques used in the case of Brownian motion and rely on discrete approximations of solutions via Γ-convergence from [3] and [13] combined with stochastic ordering arguments adapted from [35].
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... For Brownian motion the modification of the problem to fix both b and ξ and to ask whether X 0 can be randomly distributed such that τ b has distribution according to ξ has been studied in [27], [30], [2], [28] and is naturally related to our comparison principle. Moreover, the inverse first-passage time problem for the case of Brownian motion and exponentially distributed ξ is related to [17], [9], [37], [8] and [7], where hydrodynamic limits of certain particle systems and corresponding free boundary problems are studied. For general ξ a related particle system whose hydrodynamic limit is characterized by the inverse first-passage time problem for reflected Brownian motion has been constructed in [33]. ...
... • For Brownian motion relations to free boundary problems [12], [9], optimal stopping problems [21], [6], integral equations [29], [38] and particle systems [17], [33] are known. The question arises whether such relations extend to other stochastic processes. ...
Conditions for existence and uniqueness of the inverse first-passage time problem applicable for Lévy processes and diffusions
Preprint
Full-text available
  • May 2023
For a stochastic process (X_t) t≥0 we establish conditions under which the inverse first-passage time problem has a solution for any random variable ξ > 0. For Markov processes we give additional conditions under which the solutions are unique and solutions corresponding to ordered initial states fulfill a comparison principle. As examples we show that these conditions include Lévy processes with infinite activity or unbounded variation and diffusions on an interval with appropriate behavior at the boundaries. Our methods are based on the techniques used in the case of Brownian motion and rely on discrete approximations of solutions via Γ-convergence from [3] and [12] combined with stochastic ordering arguments adapted from [34].
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... When the search strategy of each agent does depend on the locations of the other agents, the learning process described above has a similarity with a branching process known as N -BBM. This is a version of a branching Brownian motion on the real line, introduced by Brunet, Derrida, Mueller and Munier in [17,18] and later studied in [10,11,12,13,14,23]. That process also consists of N particles that may diffuse and have exponential clocks attached to them. ...
Diffusion of knowledge and the lottery society
Preprint
Full-text available
  • Sep 2024
The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~\cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.
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Branching random walk with non‐local competition
Article
  • May 2024
We study the Bolker–Pacala–Dieckmann–Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non‐local competition . This makes the particle system non‐monotone and of infinite‐range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands‐on approach. Some ideas in the proof are inspired by works on the non‐local Fisher‐KPP equation, but the stochasticity of the model creates new difficulties.
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Ahead of the Fisher–KPP front
Article
Full-text available
The solution h to the Fisher–KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h ( c t , t ) of the solution ahead of the front, at time t and position ct , with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c , the initial condition and the nonlinearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly explicit and irregular. The main tool of this paper is the so-called ‘magical expression’ which relates the position of the front, the initial condition, and the quantity we investigate.
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Ahead of the Fisher-KPP front
Preprint
  • Feb 2023
The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly explicit and irregular. The main tool of this paper is the so-called "magical expression" which relates the position of the front, the initial condition, and the quantity we investigate.
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Precise asymptotics for Fisher-KPP fronts
Article
Full-text available
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution u converges at long time to a traveling wave ϕ\phi at a position σ~(t)=2t−(3/2)log⁡t+α0−3π/t\tilde \sigma(t) = 2t - (3/2)\log t + \alpha_0- 3\sqrt{\pi}/\sqrt{t}, with error O(tγ−1)O(t^{\gamma-1}) for any γ>0\gamma>0. With their methods, we find a refined shift σ(t)=σ~(t)+μ∗(log⁡t)/t+α1/t\sigma(t) = \tilde \sigma(t) + \mu_* (\log t)/t + \alpha_1/t such that in the frame moving with σ\sigma, the solution u satisfies u(t,x)=ϕ(x)+ψ(x)/t+O(tγ−3/2)u(t,x) = \phi (x) + \psi(x)/t + O(t^{\gamma-3/2}) for a certain profile ψ\psi independent of initial data. The coefficient α1\alpha_1 depends on initial data, but μ∗=9(5−6log⁡2)/8\mu_* = 9(5-6\log 2)/8 is universal, and agrees with a finding of Berestycki, Brunet, and Derrida in a closely-related problem. Furthermore, we predict the asymptotic forms of σ\sigma and u to arbitrarily high order.
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Exact solution and precise asymptotics of a Fisher-KPP type front
Article
Full-text available
The present work concerns a version of the Fisher-KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in [BrunetDerrida.2015], we show that the Laplace transform of the initial condition is directly related to some functional of the front position μt\mu_t. We then obtain precise asymptotics of the front position by means of singularity analysis. In particular, we recover the so-called Ebert and van Saarloos correction [EbertvanSaarloos.2000], we obtain an additional term of order log⁡t/t\log t /t in this expansion, and we give precise conditions on the initial condition for those terms to be present.
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Free boundary problems and biological systems with selection rules
Article
  • Jul 2019
We prove the local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida (Phys Rev E 3(56):2597–2604, 1997). The problem we consider describes the limit evolution of branching Brownian particles on the line with death of the leftmost particle at each creation time as studied in De Masi et al. (Hydrodynamics of the N-BBM process, arXiv:1705.01825, 2017). We extensively use results in Cannon (The one-dimensional heat equation, Addison-Wesley Publishing Company, Boston 1984) and Fasano (Mathematical models of some diffusive processes with free boundaries, SIMAI e-Lecture Notes, 2008).
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Hydrodynamics of the N-BBM Process
Chapter
  • Jul 2019
The Branching Brownian Motion (BBM) process consists of particles performing independent Brownian motions in R\mathbb R, and each particle creating a new one at rate 1 at its current position. The newborn particles’ increments and branchings are independent of the other particles. The N-BBM process starts with N particles and, at each branching time, the left-most particle is removed so that the total number of particles is N for all times. The N-BBM process has been originally proposed by Maillard, and belongs to a family of processes introduced by Brunet and Derrida. We fix a density ρ\rho with a left boundary sup⁡{r∈R:∫r∞ρ(x)dx=1}>−∞\sup \{r\in \mathbb R: \int _r^\infty \rho (x) d x=1\}>-\infty , and let the initial particles’ positions be iid continuous random variables with density ρ\rho . We show that the empirical measure associated to the particle positions at a fixed time t converges to an absolutely continuous measure with density ψ(⋅,t)\psi (\cdot ,t) as N→∞N\rightarrow \infty . The limit ψ\psi is solution of a free boundary problem (FBP). Existence of solutions of this FBP was proved for finite time-intervals by Lee in 2016 and, after submitting this manuscript, Berestycki, Brunet and Penington completed the setting by proving global existence. KeywordsHydrodynamic limitFree boundary problemsBranching Brownian MotionBrunet-Derrida systems
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Non local branching Brownians with annihilation and free boundary problems
Article
  • Nov 2017
  • ELECTRON J PROBAB
We study a system of branching Brownian motions on R\mathbb R with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In \cite{DFPS} it has been studied the case of strictly local creations (the new particle is put exactly at the same position of the branching particle), in \cite{DR11} 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 \cite{DFPS} and non local branching as in \cite{DR11} 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, local in time existence of classical solution has been proved recently in \cite{JM2}. We use in the convergence a stronger topology than in \cite{DFPS} and \cite{DR11} and have explicit bounds on the rate of convergence.
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Refined long time asymptotics for Fisher-KPP fronts
Article
  • Jul 2016
  • COMMUN CONTEMP MATH
We study the one-dimensional Fisher-KPP equation, with an initial condition u0(x)u_0(x) that coincides with the step function except on a compact set. A well-known result of M. Bramson states that, as t→+∞t\to+\infty, the solution converges to a traveling wave located at the position X(t)=2t−(3/2)log⁡t+x0+o(1)X(t)=2t-(3/2)\log t+x_0+o(1), with the shift x0x_0 that depends on u0u_0. U. Ebert and W. Van Saarloos have formally derived a correction to the Bramson shift, arguing that X(t)=2t−(3/2)log⁡t+x0−3π/t+O(1/t)X(t)=2t-(3/2)\log t+x_0-3\sqrt{\pi}/\sqrt{t}+O(1/t). Here, we prove that this result does hold, with an error term of the size O(1/t1−γ)O(1/t^{1-\gamma}), for any γ>0\gamma>0. The interesting aspect of this asymptotics is that the coefficient in front of the 1/t1/\sqrt{t}-term does not depend on u0u_0.
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An Exactly Solvable Travelling Wave Equation in the Fisher–KPP Class
Article
  • Jun 2015
For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times tnt_n at which the travelling wave reaches the positions n, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher-KPP class can be recovered, in particular Bramson's shifts of the positions. We also recover and generalize Ebert-van Saarloos' corrections depending on the initial condition.
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