August  2015, 20(6): 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

1. 

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France

Received  October 2013 Revised  March 2014 Published  June 2015

We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. A classical result of Freidlin and Gartner [16] yields that the solutions of the initial value problems associated with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in [8] in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null average or rescaling the coefficients by the change of variables $x\to x/L$, with $L>1$, speeds up the propagation. From a modelling point of view, these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.
Citation: Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide,, C. R. Acad. Sci. Paris, 328 (2000), 255.  doi: 10.1016/S1287-4620(00)00115-0.  Google Scholar

[3]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena,, Comm. Math. Phys., 253 (2005), 451.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[5]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for kpp type problems. I - periodic framework,, J. Europ. Math. Soc., 7 (2005), 173.  doi: 10.4171/JEMS/26.  Google Scholar

[6]

H. Berestycki, F. Hamel and L.Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[7]

H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains,, Ann. Mat. Pura Appl., 186 (2007), 469.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[8]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4764932.  Google Scholar

[9]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, Comm. Pure Appl. Math., 68 (2014), 1014.  doi: 10.1002/cpa.21536.  Google Scholar

[10]

A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations,, Trans. Amer. Math. Soc., 366 (2014), 5541.  doi: 10.1090/S0002-9947-2014-06105-9.  Google Scholar

[11]

M. ElSmaily, The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow,, Proceedings of the American Math. Society, 141 (2013), 3553.  doi: 10.1090/S0002-9939-2013-11728-4.  Google Scholar

[12]

M. ElSmaily and S. Kirsch, The speed of propagation for KPP reaction-diffusion equations within large drift,, Advances in Diff. Equations, 16 (2011), 361.   Google Scholar

[13]

R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.   Google Scholar

[14]

M. Freidlin, On wave front propagation in periodic media,, in Stochastic Analysis and Applications (ed. M. Pinsky), (1984), 147.   Google Scholar

[15]

M. Freidlin, Functional Integration and Partial Differential Equations,, Ann. Math. Stud., (1985).   Google Scholar

[16]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 249 (1979), 521.   Google Scholar

[17]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., 60 (2011), 1229.  doi: 10.1512/iumj.2011.60.4370.  Google Scholar

[18]

C. J. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition,, Comm. Pure Appl. Math., 31 (1978), 509.  doi: 10.1002/cpa.3160310406.  Google Scholar

[19]

A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 309.  doi: 10.1016/S0294-1449(01)00068-3.  Google Scholar

[20]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), (1937), 1.   Google Scholar

[21]

X. Liang, X. Lin and H. Matano, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media,, Trans. Amer. Math. Soc., 362 (2010), 5605.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[22]

P.-L. Lions and P. E. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335.  doi: 10.1081/PDE-200050077.  Google Scholar

[23]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 41 (2009), 2388.  doi: 10.1137/080743597.  Google Scholar

[24]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation,, Eur. J. Appl. Math., 22 (2011), 169.  doi: 10.1017/S0956792511000027.  Google Scholar

[25]

J. Nolen, A central limit theorem for pulled fronts in a random medium,, Networks and Heterogeneous Media, 6 (2011), 167.  doi: 10.3934/nhm.2011.6.167.  Google Scholar

[26]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional random medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[27]

J. Nolen and J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows,, Ann. de l'Inst. Henri Poincare - Analyse Non Lineaire, 26 (2008), 815.  doi: 10.1016/j.anihpc.2008.02.005.  Google Scholar

[28]

J. Nolen and J. Xin, KPP fronts in 1D random drift,, Discrete and Continuous Dynamical Systems B, 11 (2009), 421.  doi: 10.3934/dcdsb.2009.11.421.  Google Scholar

[29]

J. Nolen and J. Xin, Variational principle of KPP front speeds in temporally random shear flows with applications,, Communications in Mathematical Physics, 269 (2007), 493.  doi: 10.1007/s00220-006-0144-8.  Google Scholar

[30]

G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, in Proceedings of Conference on Random Fields, (1979), 835.   Google Scholar

[31]

L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.  doi: 10.4310/CMS.2007.v5.n3.a4.  Google Scholar

[32]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Population Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[33]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997).   Google Scholar

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications,, Asymptotic Analysis, 20 (1999), 1.   Google Scholar

[35]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).  doi: 10.1007/978-3-642-84659-5.  Google Scholar

[36]

A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

B. Audoly, H. Berestycki and Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide,, C. R. Acad. Sci. Paris, 328 (2000), 255.  doi: 10.1016/S1287-4620(00)00115-0.  Google Scholar

[3]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Func. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena,, Comm. Math. Phys., 253 (2005), 451.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[5]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for kpp type problems. I - periodic framework,, J. Europ. Math. Soc., 7 (2005), 173.  doi: 10.4171/JEMS/26.  Google Scholar

[6]

H. Berestycki, F. Hamel and L.Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts,, J. Math. Pures Appl., 84 (2005), 1101.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[7]

H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains,, Ann. Mat. Pura Appl., 186 (2007), 469.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[8]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4764932.  Google Scholar

[9]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains,, Comm. Pure Appl. Math., 68 (2014), 1014.  doi: 10.1002/cpa.21536.  Google Scholar

[10]

A. Ducrot, T. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations,, Trans. Amer. Math. Soc., 366 (2014), 5541.  doi: 10.1090/S0002-9947-2014-06105-9.  Google Scholar

[11]

M. ElSmaily, The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow,, Proceedings of the American Math. Society, 141 (2013), 3553.  doi: 10.1090/S0002-9939-2013-11728-4.  Google Scholar

[12]

M. ElSmaily and S. Kirsch, The speed of propagation for KPP reaction-diffusion equations within large drift,, Advances in Diff. Equations, 16 (2011), 361.   Google Scholar

[13]

R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.   Google Scholar

[14]

M. Freidlin, On wave front propagation in periodic media,, in Stochastic Analysis and Applications (ed. M. Pinsky), (1984), 147.   Google Scholar

[15]

M. Freidlin, Functional Integration and Partial Differential Equations,, Ann. Math. Stud., (1985).   Google Scholar

[16]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media,, Sov. Math. Dokl., 249 (1979), 521.   Google Scholar

[17]

F. Hamel, G. Nadin and L. Roques, A viscosity solution method for the spreading speed formula in slowly varying media,, Indiana Univ. Math. J., 60 (2011), 1229.  doi: 10.1512/iumj.2011.60.4370.  Google Scholar

[18]

C. J. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition,, Comm. Pure Appl. Math., 31 (1978), 509.  doi: 10.1002/cpa.3160310406.  Google Scholar

[19]

A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 309.  doi: 10.1016/S0294-1449(01)00068-3.  Google Scholar

[20]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), (1937), 1.   Google Scholar

[21]

X. Liang, X. Lin and H. Matano, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media,, Trans. Amer. Math. Soc., 362 (2010), 5605.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[22]

P.-L. Lions and P. E. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335.  doi: 10.1081/PDE-200050077.  Google Scholar

[23]

G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator,, SIAM J. Math. Anal., 41 (2009), 2388.  doi: 10.1137/080743597.  Google Scholar

[24]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation,, Eur. J. Appl. Math., 22 (2011), 169.  doi: 10.1017/S0956792511000027.  Google Scholar

[25]

J. Nolen, A central limit theorem for pulled fronts in a random medium,, Networks and Heterogeneous Media, 6 (2011), 167.  doi: 10.3934/nhm.2011.6.167.  Google Scholar

[26]

J. Nolen and L. Ryzhik, Traveling waves in a one-dimensional random medium,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1021.  doi: 10.1016/j.anihpc.2009.02.003.  Google Scholar

[27]

J. Nolen and J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows,, Ann. de l'Inst. Henri Poincare - Analyse Non Lineaire, 26 (2008), 815.  doi: 10.1016/j.anihpc.2008.02.005.  Google Scholar

[28]

J. Nolen and J. Xin, KPP fronts in 1D random drift,, Discrete and Continuous Dynamical Systems B, 11 (2009), 421.  doi: 10.3934/dcdsb.2009.11.421.  Google Scholar

[29]

J. Nolen and J. Xin, Variational principle of KPP front speeds in temporally random shear flows with applications,, Communications in Mathematical Physics, 269 (2007), 493.  doi: 10.1007/s00220-006-0144-8.  Google Scholar

[30]

G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients,, in Proceedings of Conference on Random Fields, (1979), 835.   Google Scholar

[31]

L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows,, Commun. Math. Sci., 5 (2007), 575.  doi: 10.4310/CMS.2007.v5.n3.a4.  Google Scholar

[32]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments,, Theor. Population Biol., 30 (1986), 143.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[33]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997).   Google Scholar

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications,, Asymptotic Analysis, 20 (1999), 1.   Google Scholar

[35]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).  doi: 10.1007/978-3-642-84659-5.  Google Scholar

[36]

A. Zlatos, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows,, Arch. Ration. Mech. Anal., 195 (2009), 441.  doi: 10.1007/s00205-009-0282-1.  Google Scholar

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