Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality

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Abstract

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation istU(x,t)=DxxU+γU(1−U)+εU(1−U)η(x,t)for 0⩽U⩽1 where η(x,t) is a Gaussian white noise process in space and time. Here D, γ and ε are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Itô equations. Solutions of this stochastic partial differential equation have an exact connection to the AA+A reaction–diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation.

Section snippets

Introduction and motivation

The Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation [1], [2] is one of the most fundamental models in mathematical biology and ecology [3]. It describes a population U(x,t) that evolves under the combined effects of spatial diffusion and local logistic growth and saturation. In one space dimension the FKPP equation istU=DxxU+γU(1−U),0⩽U⩽1,with diffusion coefficient D and low-density growth rate γ, and where the population has been normalized so that the stable saturation level is U=1.

Duality for the stochastic logistic equation

Here we present the details of the duality relation between the spatially homogeneous AA+A reaction process and the stochastic logistic Itô equationdU=γU(1−U)dt+σU(1−U)dW,where W(t) is the usual Brownian motion. These processes will be dual when the reaction coefficients in the birth–coagulation process correspond to the coefficients in the stochastic differential equation as follows:A→A+Aatrateγ,A+A→Aatrateσ2.Specifically, the pair coagulation probability rate χ/Ω from (6) is taken as the

Duality for the stochastic FKPP equation

We now establish the duality between the (spatially discretized) stochastic Fisher–Kolmogorov–Petrovsky–Piscounov partial differential equation and the spatially inhomogeneous birth–coagulation particle process. Discretize the x-axis with lattice spacing h and consider the coupled Itô equationsdUi(t)=DUi+1−2Ui+Ui−1h2+γUi(1−Ui)dt+σUi(1−Ui)dWi,where the Wi(t) are independent Brownian motions at each site, i.e.,dWi(t)dWj(t)=δijdt.With the identification U(x,t)=Ux/h(t), the solution of (33) gives

Wavefronts in the stochastic FKPP equation

The FKPP equation (1) describes the invasion of the stable saturated state (U=1) into regions of the determinstically unstable extinct phase (U=0). This invasion proceeds via propagation of a front at a constant velocity: to find these travelling waves we look for solutions of the formU(x,t)=w(x−ct),where the speed c is to be determined. Inserting this ansatz into the FKPP equation yields an ordinary differential equation for the front shape w(z):Dw″+cw′+γw(1−w)=0.The boundary conditions for

Summary and conclusions

In this paper we have presented a simple and complete derivation of the duality relation, first discovered by Shiga and Uchiyama [7], between solutions of the stochastic FKPP equation and the AA+A birth–coagulation reaction–diffusion interacting particle system. We then used duality to obtain an exact formula for the eventual extinction probability of any solution of the stochastic FKPP equation as a function of the initial configuration and the system parameters.

The dual particle process

Acknowledgements

C.R.D thanks Profs. Peter Jung (Ohio University), Joseph Keller (Stanford University) and Lutz Schimansky-Geier (Humboldt University) for thoughtful remarks. This research was supported in part by awards from the United States’ National Science Foundation and National Security Agency.

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