Physica A: Statistical Mechanics and its Applications
Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality
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 iswith 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 A⇌A+A reaction process and the stochastic logistic Itô equationwhere 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:Specifically, the pair coagulation probability rate 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 equationswhere the Wi(t) are independent Brownian motions at each site, i.e.,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 formwhere 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):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 A⇌A+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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