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Journal articles on the topic 'Fisher-KPP equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

RODRIGO, M., and M. MIMURA. "Annihilation dynamics in the KPP-Fisher equation." European Journal of Applied Mathematics 13, no. 2 (April 2002): 195–204. http://dx.doi.org/10.1017/s0956792501004764.

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We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.

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2

Cai, Hong, Anna Ghazaryan, and Vahagn Manukian. "Fisher-KPP dynamics in diffusive Rosenzweig–MacArthur and Holling–Tanner models." Mathematical Modelling of Natural Phenomena 14, no. 4 (2019): 404. http://dx.doi.org/10.1051/mmnp/2019017.

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We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig–MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling–Tanner population model. In the second situation for the Rosenzweig–MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction–diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation.

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3

El-Hachem, Maud, Scott W. McCue, Wang Jin, Yihong Du, and Matthew J. Simpson. "Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2229 (September 2019): 20190378. http://dx.doi.org/10.1098/rspa.2019.0378.

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The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as the Fisher–KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher–KPP model cannot replicate, such as the extinction of invasive populations. The Fisher–Stefan model is an adaptation of the Fisher–KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher–Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher–Stefan model is that it is able to simulate population extinction, giving rise to a spreading–extinction dichotomy . In this work, we revisit travelling wave solutions of the Fisher–KPP model and show that these results provide new insight into travelling wave solutions of the Fisher–Stefan model and the spreading–extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher–Stefan model and often-neglected travelling wave solutions of the Fisher–KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher–Stefan model in the limit of slow travelling wave speeds, c ≪1.

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4

Angstmann, Christopher N., and Bruce I. Henry. "Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations." Entropy 22, no. 9 (September 16, 2020): 1035. http://dx.doi.org/10.3390/e22091035.

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A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

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5

Hamel, François, and Luca Rossi. "Transition fronts for the Fisher-KPP equation." Transactions of the American Mathematical Society 368, no. 12 (January 26, 2016): 8675–713. http://dx.doi.org/10.1090/tran/6609.

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6

Shapovalov, A. V., and A. Yu Trifonov. "An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850102. http://dx.doi.org/10.1142/s0219887818501025.

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A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

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7

Alfaro, Matthieu, and Arnaud Ducrot. "Sharp interface limit of the Fisher-KPP equation." Communications on Pure & Applied Analysis 11, no. 1 (2012): 1–18. http://dx.doi.org/10.3934/cpaa.2012.11.1.

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8

Fang, Jian, and Xiao-Qiang Zhao. "Monotone wavefronts of the nonlocal Fisher–KPP equation." Nonlinearity 24, no. 11 (September 23, 2011): 3043–54. http://dx.doi.org/10.1088/0951-7715/24/11/002.

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9

Stan, Diana, and Juan Luis Vázquez. "The Fisher-KPP Equation with Nonlinear Fractional Diffusion." SIAM Journal on Mathematical Analysis 46, no. 5 (January 2014): 3241–76. http://dx.doi.org/10.1137/130918289.

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10

Du, Yihong, and Zongming Guo. "The Stefan problem for the Fisher–KPP equation." Journal of Differential Equations 253, no. 3 (August 2012): 996–1035. http://dx.doi.org/10.1016/j.jde.2012.04.014.

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11

Danilov, V. G., and P. Yu Subochev. "Interaction of kinks in the KPP-Fisher equation." Mathematical Notes of the Academy of Sciences of the USSR 50, no. 3 (September 1991): 981–82. http://dx.doi.org/10.1007/bf01156147.

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12

李, 志君. "The Explicit Richardson Extrapolation Method for Two-Dimensional Fisher-KPP Equation." Pure Mathematics 11, no. 09 (2021): 1649–56. http://dx.doi.org/10.12677/pm.2021.119183.

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13

Murata, Mikio. "Tropical discretization: ultradiscrete Fisher–KPP equation and ultradiscrete Allen–Cahn equation." Journal of Difference Equations and Applications 19, no. 6 (June 2013): 1008–21. http://dx.doi.org/10.1080/10236198.2012.705834.

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14

Mansour, M. B. A. "Traveling wave solutions for the extended Fisher/KPP equation." Reports on Mathematical Physics 66, no. 3 (December 2010): 375–83. http://dx.doi.org/10.1016/s0034-4877(10)80009-6.

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15

Hasik, Karel, and Sergei Trofimchuk. "Slowly oscillating wavefronts of the KPP-Fisher delayed equation." Discrete & Continuous Dynamical Systems - A 34, no. 9 (2014): 3511–33. http://dx.doi.org/10.3934/dcds.2014.34.3511.

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16

Kuehn, Christian, and Pasha Tkachov. "Pattern formation in the doubly-nonlocal Fisher-KPP equation." Discrete & Continuous Dynamical Systems - A 39, no. 4 (2019): 2077–100. http://dx.doi.org/10.3934/dcds.2019087.

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17

Gu, Hong. "Fisher-KPP equation with advection on the half-line." Mathematical Methods in the Applied Sciences 39, no. 3 (April 21, 2015): 344–52. http://dx.doi.org/10.1002/mma.3485.

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18

King, John R., and Philip M. McCabe. "On the Fisher–KPP equation with fast nonlinear diffusion." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 459, no. 2038 (October 8, 2003): 2529–46. http://dx.doi.org/10.1098/rspa.2003.1134.

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19

Lou, Bendong, and Junfan Lu. "Spreading in a cone for the Fisher-KPP equation." Journal of Differential Equations 267, no. 12 (December 2019): 7064–84. http://dx.doi.org/10.1016/j.jde.2019.07.014.

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20

Gomez, Adrian, and Sergei Trofimchuk. "Monotone traveling wavefronts of the KPP-Fisher delayed equation." Journal of Differential Equations 250, no. 4 (February 2011): 1767–87. http://dx.doi.org/10.1016/j.jde.2010.11.011.

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21

Shapovalov, Alexander, and Andrey Trifonov. "Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type." Symmetry 11, no. 3 (March 12, 2019): 366. http://dx.doi.org/10.3390/sym11030366.

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We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

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22

Nolen, James, Jean-Michel Roquejoffre, and Lenya Ryzhik. "Refined long-time asymptotics for Fisher–KPP fronts." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850072. http://dx.doi.org/10.1142/s0219199718500724.

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We study the one-dimensional Fisher–KPP equation, with an initial condition [Formula: see text] that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531–581; Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (American Mathematical Society, Providence, RI, 1983)] states that, as [Formula: see text], the solution converges to a traveling wave located at the position [Formula: see text], with the shift [Formula: see text] that depends on [Formula: see text]. Ebert and Van Saarloos have formally derived in [Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000) 1–99; Front propagation into unstable states, Phys. Rep. 386 (2003) 29–222] a correction to the Bramson shift, arguing that [Formula: see text]. Here, we prove that this result does hold, with an error term of the size [Formula: see text], for any [Formula: see text]. The interesting aspect of this asymptotics is that the coefficient in front of the [Formula: see text]-term does not depend on [Formula: see text].

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23

Lagergren, John H., John T. Nardini, G. Michael Lavigne, Erica M. Rutter, and Kevin B. Flores. "Learning partial differential equations for biological transport models from noisy spatio-temporal data." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2234 (February 2020): 20190800. http://dx.doi.org/10.1098/rspa.2019.0800.

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We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.

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24

Li, Wan-Tong, Yu-Juan Sun, and Zhi-Cheng Wang. "Entire solutions in the Fisher-KPP equation with nonlocal dispersal." Nonlinear Analysis: Real World Applications 11, no. 4 (August 2010): 2302–13. http://dx.doi.org/10.1016/j.nonrwa.2009.07.005.

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25

Billingham, John. "Slow travelling wave solutions of the nonlocal Fisher-KPP equation." Nonlinearity 33, no. 5 (March 16, 2020): 2106–42. http://dx.doi.org/10.1088/1361-6544/ab6f4f.

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26

Jordan, P. M., and A. Puri. "Qualitative results for solutions of the steady fisher-KPP equation." Applied Mathematics Letters 15, no. 2 (February 2002): 239–50. http://dx.doi.org/10.1016/s0893-9659(01)00124-0.

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27

Nolen, James, Jean-Michel Roquejoffre, and Lenya Ryzhik. "Convergence to a single wave in the Fisher-KPP equation." Chinese Annals of Mathematics, Series B 38, no. 2 (March 2017): 629–46. http://dx.doi.org/10.1007/s11401-017-1087-4.

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28

Hamel, François, and Christopher Henderson. "Propagation in a Fisher-KPP equation with non-local advection." Journal of Functional Analysis 278, no. 7 (April 2020): 108426. http://dx.doi.org/10.1016/j.jfa.2019.108426.

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29

Bouin, Emeric, Christopher Henderson, and Lenya Ryzhik. "The Bramson delay in the non-local Fisher-KPP equation." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 37, no. 1 (January 2020): 51–77. http://dx.doi.org/10.1016/j.anihpc.2019.07.001.

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30

Tian, Ge, Haoyu Wang, and Zhicheng Wang. "Spreading Speed in the Fisher-KPP Equation with Nonlocal Delay." Acta Mathematica Scientia 41, no. 3 (April 19, 2021): 875–86. http://dx.doi.org/10.1007/s10473-021-0314-y.

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31

Tian, Ge, Zhi-Cheng Wang, and Guo-Bao Zhang. "Stability of traveling waves of the nonlocal Fisher–KPP equation." Nonlinear Analysis 211 (October 2021): 112399. http://dx.doi.org/10.1016/j.na.2021.112399.

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32

Defreitas, Colin L., and Steve J. Kane. "A Laplace transform finite difference scheme for the Fisher-KPP equation." Journal of Algorithms & Computational Technology 15 (January 2021): 174830262199958. http://dx.doi.org/10.1177/1748302621999582.

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This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

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33

DU, YIHONG, and LI MA. "LOGISTIC TYPE EQUATIONS ON ℝN BY A SQUEEZING METHOD INVOLVING BOUNDARY BLOW-UP SOLUTIONS." Journal of the London Mathematical Society 64, no. 1 (August 2001): 107–24. http://dx.doi.org/10.1017/s0024610701002289.

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We study, on the entire space ℝN(N [ges ] 1), the diffusive logistic equationand its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important role in understanding various population models and some other problems in applied mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and KPP equation, due to the pioneering works of Fisher [8] and Kolmogoroff, Petrovsky and Piscounoff [18].

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34

Bouin, Emeric, Vincent Calvez, and Grégoire Nadin. "Hyperbolic traveling waves driven by growth." Mathematical Models and Methods in Applied Sciences 24, no. 06 (March 28, 2014): 1165–95. http://dx.doi.org/10.1142/s0218202513500802.

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We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

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35

Yang, Gaoxiang. "Hopf bifurcation of traveling wave solutions of delayed Fisher-KPP equation." Applied Mathematics and Computation 220 (September 2013): 213–20. http://dx.doi.org/10.1016/j.amc.2013.06.051.

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36

LOU, BENDONG, JUNFAN LU, and YOSHIHISA MORITA. "Entire solutions of the Fisher–KPP equation on the half line." European Journal of Applied Mathematics 31, no. 3 (March 26, 2019): 407–22. http://dx.doi.org/10.1017/s0956792519000093.

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In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

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37

Achleitner, Franz, and Christian Kuehn. "On bounded positive stationary solutions for a nonlocal Fisher–KPP equation." Nonlinear Analysis: Theory, Methods & Applications 112 (January 2015): 15–29. http://dx.doi.org/10.1016/j.na.2014.09.004.

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38

Berestycki, Henri, Grégoire Nadin, Benoit Perthame, and Lenya Ryzhik. "The non-local Fisher–KPP equation: travelling waves and steady states." Nonlinearity 22, no. 12 (October 30, 2009): 2813–44. http://dx.doi.org/10.1088/0951-7715/22/12/002.

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39

Cristofol, Michel, and Lionel Roques. "Stable estimation of two coefficients in a nonlinear Fisher–KPP equation." Inverse Problems 29, no. 9 (August 15, 2013): 095007. http://dx.doi.org/10.1088/0266-5611/29/9/095007.

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40

Cao, Mei-Ling, and Wei-Jie Sheng. "Entire solutions of the Fisher-KPP equation in time periodic media." Dynamics of Partial Differential Equations 9, no. 2 (2012): 133–45. http://dx.doi.org/10.4310/dpde.2012.v9.n2.a3.

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41

Finkelshtein, Dmitri, Yuri Kondratiev, and Pasha Tkachov. "Doubly nonlocal Fisher–KPP equation: Speeds and uniqueness of traveling waves." Journal of Mathematical Analysis and Applications 475, no. 1 (July 2019): 94–122. http://dx.doi.org/10.1016/j.jmaa.2019.02.010.

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42

Brunet, Éric, and Bernard Derrida. "An Exactly Solvable Travelling Wave Equation in the Fisher–KPP Class." Journal of Statistical Physics 161, no. 4 (September 8, 2015): 801–20. http://dx.doi.org/10.1007/s10955-015-1350-6.

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43

Berestycki, Henri, and Jian Fang. "Forced waves of the Fisher–KPP equation in a shifting environment." Journal of Differential Equations 264, no. 3 (February 2018): 2157–83. http://dx.doi.org/10.1016/j.jde.2017.10.016.

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44

Ducrot, Arnaud, and Grégoire Nadin. "Asymptotic behaviour of travelling waves for the delayed Fisher–KPP equation." Journal of Differential Equations 256, no. 9 (May 2014): 3115–40. http://dx.doi.org/10.1016/j.jde.2014.01.033.

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45

McCue, Scott W., Maud El-Hachem, and Matthew J. Simpson. "Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation." Applied Mathematics Letters 114 (April 2021): 106918. http://dx.doi.org/10.1016/j.aml.2020.106918.

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46

Bonizzoni, Francesca, Marcel Braukhoff, Ansgar Jüngel, and Ilaria Perugia. "A structure-preserving discontinuous Galerkin scheme for the Fisher–KPP equation." Numerische Mathematik 146, no. 1 (July 25, 2020): 119–57. http://dx.doi.org/10.1007/s00211-020-01136-w.

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47

Girardin, Léo. "Non-cooperative Fisher–KPP systems: Asymptotic behavior of traveling waves." Mathematical Models and Methods in Applied Sciences 28, no. 06 (May 21, 2018): 1067–104. http://dx.doi.org/10.1142/s0218202518500288.

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This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. In a previous paper, we established that these systems admit traveling wave solutions whose profiles connect the null state to a compact subset of the positive cone. The main object of this paper is the investigation of a more precise description of these profiles. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.

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48

Kong, Liang. "Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal." Journal of Mathematics Research 9, no. 1 (January 2, 2017): 1. http://dx.doi.org/10.5539/jmr.v9n1p1.

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The current paper investigate the persistence of positive solutions of KPP type evolution equations with random/nonlocal dispersal in locally spatially inhomogeneous habitat. By the constructions of super/sub solutions and comparison principle, we prove that such an equation has a unique globally stable positive stationary solution.

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49

Delgado-Vences, Francisco, and Franco Flandoli. "A spectral-based numerical method for Kolmogorov equations in Hilbert spaces." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 03 (August 31, 2016): 1650020. http://dx.doi.org/10.1142/s021902571650020x.

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We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher–KPP stochastic equation and a stochastic Burgers equation in dimension 1.

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50

Salako, Rachidi B., and Wenxian Shen. "Long-time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts." Stochastics and Dynamics 19, no. 06 (November 18, 2019): 1950046. http://dx.doi.org/10.1142/s0219493719500461.

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Abstract:

In the current series of two papers, we study the long-time behavior of the following random Fisher-KPP equation: [Formula: see text] where [Formula: see text], [Formula: see text] is a given probability space, [Formula: see text] is an ergodic metric dynamical system on [Formula: see text], and [Formula: see text] for every [Formula: see text]. We also study the long-time behavior of the following nonautonomous Fisher-KPP equation: [Formula: see text] where [Formula: see text] is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1.1) and (1.2). In this second part of the series, we investigate the existence and stability of transition fronts of (1.1) and (1.2). We first study the transition fronts of (1.1). Under some proper assumption on [Formula: see text], we show the existence of random transition fronts of (1.1) with least mean speed greater than or equal to some constant [Formula: see text] and the nonexistence of random transition fronts of (1.1) with least mean speed less than [Formula: see text]. We prove the stability of random transition fronts of (1.1) with least mean speed greater than [Formula: see text]. Note that it is proved in the first part that [Formula: see text] is the infimum of the spreading speeds of (1.1). We next study the existence and stability of transition fronts of (1.2). It is not assumed that [Formula: see text] and [Formula: see text] are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

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