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

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  • Doering, Charles R.
  • Mueller, Carl
  • Smereka, Peter

Abstract

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is∂tU(x,t)=D∂xxU+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 A⇌A+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.

Suggested Citation

  • Doering, Charles R. & Mueller, Carl & Smereka, Peter, 2003. "Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 325(1), pages 243-259.
    • Handle: RePEc:eee:phsmap:v:325:y:2003:i:1:p:243-259
    DOI: 10.1016/S0378-4371(03)00203-6
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    Cited by:

    1. Lavrentovich, Maxim O. & Nelson, David R., 2015. "Survival probabilities at spherical frontiers," Theoretical Population Biology, Elsevier, vol. 102(C), pages 26-39.
    2. Pigolotti, S. & Benzi, R. & Perlekar, P. & Jensen, M.H. & Toschi, F. & Nelson, D.R., 2013. "Growth, competition and cooperation in spatial population genetics," Theoretical Population Biology, Elsevier, vol. 84(C), pages 72-86.
    3. Bryant, Adam S. & Lavrentovich, Maxim O., 2022. "Survival in branching cellular populations," Theoretical Population Biology, Elsevier, vol. 144(C), pages 13-23.
    4. Feng, Zhaosheng & Li, Yang, 2006. "Complex traveling wave solutions to the Fisher equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 115-123.
    5. Hallatschek, Oskar & Nelson, David R., 2008. "Gene surfing in expanding populations," Theoretical Population Biology, Elsevier, vol. 73(1), pages 158-170.

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