doi: 10.3934/dcdsb.2020323

Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations

1. 

School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

2. 

Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

* Corresponding author: Tianshou Zhou

Received  June 2020 Revised  August 2020 Published  November 2020

Fund Project: This work was partially supported by the National Natural Science foundation of P.R. China through Grant grants 11931019, 11775314 and 11701115

Fisher-KPP equations are an important class of mathematical models with practical background. Previous studies analyzed the asymptotic behaviors of the front and back of the wavefront and proved the existence of stochastic traveling waves, by imposing decrease constraints on the growth function. For the Fisher-KPP equation with a stochastically fluctuated growth rate, we find that if the decrease restrictions are removed, the same results still hold. Moreover, we show that with increasing the noise intensity, the original equation with Fisher-KPP nonlinearity evolves into first the one with degenerated Fisher-KPP nonlinearity and then the one with Nagumo nonlinearity. For the Fisher-KPP equation subjected to the environmental noise, the established asymptotic behavior of the front of the wavefront still holds even if the decrease constraint on the growth function is ruled out. If this constraint is removed, however, the established asymptotic behavior of the back of the wavefront will no longer hold, implying that the decrease constraint on the growth function is a sufficient and necessary condition to ensure the asymptotic behavior of the back of the wavefront. In both cases of noise, the systems can allow stochastic traveling waves.

Citation: Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020323
References:
[1]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Memoirs of the American Mathematical Society, 44 (1983). doi: 10.1090/memo/0285.  Google Scholar

[2]

P. C. Bressloff and M. A. Webber, Front propagation in stochastic neural fields, SIAM Journal on Applied Dynamical Systems, 11 (2012), 708-740.  doi: 10.1137/110851031.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM Journal on Applied Mathematics, 57 (1997), 327-346.  doi: 10.1137/S0036139995284681.  Google Scholar

[5]

J. G. Conlon and C. R. Doering, On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation, Journal of Statistical Physics, 120 (2005), 421-477.  doi: 10.1007/s10955-005-5960-2.  Google Scholar

[6]

C. Cosner, Reaction-diffusion equations and ecological modeling, in Tutorials in Mathematical Biosciences â…£ (eds. A. Friedman), Springer, (2008), 77-115. doi: 10.1007/978-3-540-74331-6_3.  Google Scholar

[7]

S. P. DawsonS. Chen and G. D. Doolen, Lattice Boltzmann computations for reaction-diffusion equations, Journal of Chemical Physics, 98 (1993), 1514-1523.  doi: 10.1063/1.464316.  Google Scholar

[8]

H. DuZ. XuJ. D. Shrout and M. Alber, Multiscale modeling of pseudomonas aeruginosa swarming, Mathematical Models and Methods in Applied Sciences, 21 (2011), 939-954.  doi: 10.1142/S0218202511005428.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Human Genetics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

M. Freidlin, Coupled reaction-diffusion equations, The Annals of Probability, 19 (1991), 29-57.  doi: 10.1214/aop/1176990535.  Google Scholar

[11]

C. H. S. Hamster and H. J. Hupkes, Stability of traveling waves for reaction-diffusion equations with multiplicative noise, SIAM Journal on Applied Dynamical Systems, 18 (2019), 205-278.  doi: 10.1137/17M1159518.  Google Scholar

[12]

C. H. S. Hamster and H. J. Hupkes, Travelling waves for reaction-diffusion equations forced by translation invariant noise, Physica D: Nonlinear Phenomena, 401 (2020), 132233. doi: 10.1016/j.physd.2019.132233.  Google Scholar

[13]

Z. Huang and Z. Liu, Stochastic traveling wave solution to stochastic generalized KPP equation, NoDEA-nonlinear Differential Equations and Applications, 22 (2015), 143-173.  doi: 10.1007/s00030-014-0279-9.  Google Scholar

[14]

Z. HuangZ. Liu and Z. Wang, Stochastic traveling wave solution to a stochastic KPP equation, Journal of Dynamics and Differential Equations, 28 (2016), 389-417.  doi: 10.1007/s10884-015-9485-3.  Google Scholar

[15]

Z. Huang and Z. Liu, Random traveling wave and bifurcations of asymptotic behaviors in the stochastic KPP equation driven by dual noises, Journal of Differential Equations, 261 (2016), 1317-1356.  doi: 10.1016/j.jde.2016.04.003.  Google Scholar

[16]

J. Inglis and J. Maclaurin, A general framework for stochastic traveling waves and patterns, with application to neural field equations, SIAM Journal on Applied Dynamical Systems, 15 (2016), 195-234.  doi: 10.1137/15M102856X.  Google Scholar

[17]

M. IpsenL. Kramer and P. G. Sorensen, Amplitude equations for description of chemical reaction-diffusion systems, Physics Reports, 337 (2000), 193-235.  doi: 10.1016/S0370-1573(00)00062-4.  Google Scholar

[18]

B. L. Keyfitz, Shock waves and reaction-diffusion equations. By Joel Smoller, American Mathematical Monthly, 93 (1986), 315-318.  doi: 10.2307/2323701.  Google Scholar

[19]

J. Kruger and W. Stannat, Front propagation in stochastic neural fields: a rigorous mathematical framework, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1293-1310.  doi: 10.1137/13095094X.  Google Scholar

[20]

E. Lang, A multiscale analysis of traveling waves in stochastic neural fields, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1581-1614.  doi: 10.1137/15M1033927.  Google Scholar

[21]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM Journal on Applied Mathematics, 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.  Google Scholar

[22]

M. MahalakshmiG. Hariharan and K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry, Journal of Mathematical Chemistry, 51 (2013), 2361-2385.  doi: 10.1007/s10910-013-0216-x.  Google Scholar

[23]

C. Mueller and R. B. Sowers, Random traveling waves for the KPP equation with noise, Journal of Functional Analysis, 128 (1995), 439-498.  doi: 10.1006/jfan.1995.1038.  Google Scholar

[24]

C. MuellerL. Mytnik and J. Quastel, Small noise asymptotics of traveling waves, Markov Processes and Related Fields, 14 (2008), 333-342.   Google Scholar

[25]

C. MuellerL. Mytnik and J. Quastel, Effect of noise on front propagation in reaction-diffusion equations of KPP type, Inventiones Mathematicae, 184 (2011), 405-453.  doi: 10.1007/s00222-010-0292-5.  Google Scholar

[26]

C. Mueller, L. Mytnik and L. Ryzhik, The speed of a random front for stochastic reaction-diffusion equations with strong noise, arXiv: 1903.03645. Google Scholar

[27]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Annales De L Institut Henri Poincare-Analyse Non Lineaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[28]

J. Nolen, An invariance principle for random traveling waves in one dimension, SIAM Journal on Mathematical Analysis, 43 (2011), 153-188.  doi: 10.1137/090746513.  Google Scholar

[29]

B. ØksendalG. Våge and H. Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 130 (2000), 1363-1381.  doi: 10.1017/S030821050000072X.  Google Scholar

[30]

B. ØksendalG. Våge and H. Z. Zhao, Two properties of stochastic KPP equations: Ergodicity and pathwise property, Nonlinearity, 14 (2001), 639-662.  doi: 10.1088/0951-7715/14/3/311.  Google Scholar

[31]

W. Shen, Traveling waves in diffusive random media, Journal of Dynamics and Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[32]

W. Shen and Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Transactions of the American Mathematical Society, 369 (2017), 2573-2613.  doi: 10.1090/tran/6726.  Google Scholar

[33]

W. Shen and Z. Shen, Transition fronts in time heterogeneous and random media of ignition type, Journal of Differential Equations, 262 (2017), 454-485.  doi: 10.1016/j.jde.2016.09.030.  Google Scholar

[34]

W.-J. Sheng and J.-B. Wang, Entire solutions of time periodic bistable reaction-advection-diffusion equations in infinite cylinders, Journal of Mathematical Physics, 56 (2015), 081501. doi: 10.1063/1.4927712.  Google Scholar

[35]

T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canadian Journal of Mathematics, 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8.  Google Scholar

[36]

W. Stannat, Stability of travelling waves in stochastic Nagumo equations, arXiv: 1301.6378. Google Scholar

[37]

R. Tribe, A travelling wave solution to the kolmogorov equation with noise, Stochastics and Stochastics Reports, 56 (1996), 317-340.  doi: 10.1080/17442509608834047.  Google Scholar

[38]

R. Tribe and N. Woodward, Stochastic order methods applied to stochastic travelling waves, Electronic Journal of Probability, 16 (2011), 436-469.  doi: 10.1214/EJP.v16-868.  Google Scholar

[39]

W. WangY. CaiM. WuK. Wang and Z. Li, Complex dynamics of a reaction-diffusion epidemic model, Nonlinear Analysis-real World Applications, 13 (2011), 2240-2258.  doi: 10.1016/j.nonrwa.2012.01.018.  Google Scholar

[40]

Z. Wang, Z. Huang and Z. Liu, Stochastic traveling waves of a stochastic Fisher-KPP equation and bifurcations for asymptotic behaviors, Stochastics and Dynamics, 19 (2019), 1950028. doi: 10.1142/S021949371950028X.  Google Scholar

show all references

References:
[1]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Memoirs of the American Mathematical Society, 44 (1983). doi: 10.1090/memo/0285.  Google Scholar

[2]

P. C. Bressloff and M. A. Webber, Front propagation in stochastic neural fields, SIAM Journal on Applied Dynamical Systems, 11 (2012), 708-740.  doi: 10.1137/110851031.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd, Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM Journal on Applied Mathematics, 57 (1997), 327-346.  doi: 10.1137/S0036139995284681.  Google Scholar

[5]

J. G. Conlon and C. R. Doering, On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation, Journal of Statistical Physics, 120 (2005), 421-477.  doi: 10.1007/s10955-005-5960-2.  Google Scholar

[6]

C. Cosner, Reaction-diffusion equations and ecological modeling, in Tutorials in Mathematical Biosciences â…£ (eds. A. Friedman), Springer, (2008), 77-115. doi: 10.1007/978-3-540-74331-6_3.  Google Scholar

[7]

S. P. DawsonS. Chen and G. D. Doolen, Lattice Boltzmann computations for reaction-diffusion equations, Journal of Chemical Physics, 98 (1993), 1514-1523.  doi: 10.1063/1.464316.  Google Scholar

[8]

H. DuZ. XuJ. D. Shrout and M. Alber, Multiscale modeling of pseudomonas aeruginosa swarming, Mathematical Models and Methods in Applied Sciences, 21 (2011), 939-954.  doi: 10.1142/S0218202511005428.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Human Genetics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

M. Freidlin, Coupled reaction-diffusion equations, The Annals of Probability, 19 (1991), 29-57.  doi: 10.1214/aop/1176990535.  Google Scholar

[11]

C. H. S. Hamster and H. J. Hupkes, Stability of traveling waves for reaction-diffusion equations with multiplicative noise, SIAM Journal on Applied Dynamical Systems, 18 (2019), 205-278.  doi: 10.1137/17M1159518.  Google Scholar

[12]

C. H. S. Hamster and H. J. Hupkes, Travelling waves for reaction-diffusion equations forced by translation invariant noise, Physica D: Nonlinear Phenomena, 401 (2020), 132233. doi: 10.1016/j.physd.2019.132233.  Google Scholar

[13]

Z. Huang and Z. Liu, Stochastic traveling wave solution to stochastic generalized KPP equation, NoDEA-nonlinear Differential Equations and Applications, 22 (2015), 143-173.  doi: 10.1007/s00030-014-0279-9.  Google Scholar

[14]

Z. HuangZ. Liu and Z. Wang, Stochastic traveling wave solution to a stochastic KPP equation, Journal of Dynamics and Differential Equations, 28 (2016), 389-417.  doi: 10.1007/s10884-015-9485-3.  Google Scholar

[15]

Z. Huang and Z. Liu, Random traveling wave and bifurcations of asymptotic behaviors in the stochastic KPP equation driven by dual noises, Journal of Differential Equations, 261 (2016), 1317-1356.  doi: 10.1016/j.jde.2016.04.003.  Google Scholar

[16]

J. Inglis and J. Maclaurin, A general framework for stochastic traveling waves and patterns, with application to neural field equations, SIAM Journal on Applied Dynamical Systems, 15 (2016), 195-234.  doi: 10.1137/15M102856X.  Google Scholar

[17]

M. IpsenL. Kramer and P. G. Sorensen, Amplitude equations for description of chemical reaction-diffusion systems, Physics Reports, 337 (2000), 193-235.  doi: 10.1016/S0370-1573(00)00062-4.  Google Scholar

[18]

B. L. Keyfitz, Shock waves and reaction-diffusion equations. By Joel Smoller, American Mathematical Monthly, 93 (1986), 315-318.  doi: 10.2307/2323701.  Google Scholar

[19]

J. Kruger and W. Stannat, Front propagation in stochastic neural fields: a rigorous mathematical framework, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1293-1310.  doi: 10.1137/13095094X.  Google Scholar

[20]

E. Lang, A multiscale analysis of traveling waves in stochastic neural fields, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1581-1614.  doi: 10.1137/15M1033927.  Google Scholar

[21]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM Journal on Applied Mathematics, 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.  Google Scholar

[22]

M. MahalakshmiG. Hariharan and K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry, Journal of Mathematical Chemistry, 51 (2013), 2361-2385.  doi: 10.1007/s10910-013-0216-x.  Google Scholar

[23]

C. Mueller and R. B. Sowers, Random traveling waves for the KPP equation with noise, Journal of Functional Analysis, 128 (1995), 439-498.  doi: 10.1006/jfan.1995.1038.  Google Scholar

[24]

C. MuellerL. Mytnik and J. Quastel, Small noise asymptotics of traveling waves, Markov Processes and Related Fields, 14 (2008), 333-342.   Google Scholar

[25]

C. MuellerL. Mytnik and J. Quastel, Effect of noise on front propagation in reaction-diffusion equations of KPP type, Inventiones Mathematicae, 184 (2011), 405-453.  doi: 10.1007/s00222-010-0292-5.  Google Scholar

[26]

C. Mueller, L. Mytnik and L. Ryzhik, The speed of a random front for stochastic reaction-diffusion equations with strong noise, arXiv: 1903.03645. Google Scholar

[27]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Annales De L Institut Henri Poincare-Analyse Non Lineaire, 26 (2009), 1021-1047.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[28]

J. Nolen, An invariance principle for random traveling waves in one dimension, SIAM Journal on Mathematical Analysis, 43 (2011), 153-188.  doi: 10.1137/090746513.  Google Scholar

[29]

B. ØksendalG. Våge and H. Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 130 (2000), 1363-1381.  doi: 10.1017/S030821050000072X.  Google Scholar

[30]

B. ØksendalG. Våge and H. Z. Zhao, Two properties of stochastic KPP equations: Ergodicity and pathwise property, Nonlinearity, 14 (2001), 639-662.  doi: 10.1088/0951-7715/14/3/311.  Google Scholar

[31]

W. Shen, Traveling waves in diffusive random media, Journal of Dynamics and Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[32]

W. Shen and Z. Shen, Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Transactions of the American Mathematical Society, 369 (2017), 2573-2613.  doi: 10.1090/tran/6726.  Google Scholar

[33]

W. Shen and Z. Shen, Transition fronts in time heterogeneous and random media of ignition type, Journal of Differential Equations, 262 (2017), 454-485.  doi: 10.1016/j.jde.2016.09.030.  Google Scholar

[34]

W.-J. Sheng and J.-B. Wang, Entire solutions of time periodic bistable reaction-advection-diffusion equations in infinite cylinders, Journal of Mathematical Physics, 56 (2015), 081501. doi: 10.1063/1.4927712.  Google Scholar

[35]

T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canadian Journal of Mathematics, 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8.  Google Scholar

[36]

W. Stannat, Stability of travelling waves in stochastic Nagumo equations, arXiv: 1301.6378. Google Scholar

[37]

R. Tribe, A travelling wave solution to the kolmogorov equation with noise, Stochastics and Stochastics Reports, 56 (1996), 317-340.  doi: 10.1080/17442509608834047.  Google Scholar

[38]

R. Tribe and N. Woodward, Stochastic order methods applied to stochastic travelling waves, Electronic Journal of Probability, 16 (2011), 436-469.  doi: 10.1214/EJP.v16-868.  Google Scholar

[39]

W. WangY. CaiM. WuK. Wang and Z. Li, Complex dynamics of a reaction-diffusion epidemic model, Nonlinear Analysis-real World Applications, 13 (2011), 2240-2258.  doi: 10.1016/j.nonrwa.2012.01.018.  Google Scholar

[40]

Z. Wang, Z. Huang and Z. Liu, Stochastic traveling waves of a stochastic Fisher-KPP equation and bifurcations for asymptotic behaviors, Stochastics and Dynamics, 19 (2019), 1950028. doi: 10.1142/S021949371950028X.  Google Scholar

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