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), 33-76. 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), 255-262. 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), 2146-2189. 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), 451-480. 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), 173-213. 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), 1101-1146. 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), 469-507. 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), 115619, 23pp. 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), 1014-1065. 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), 5541-5566. 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), 3553-3563. 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), 361-400.  Google Scholar

[13]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. Google Scholar

[14]

M. Freidlin, On wave front propagation in periodic media, in Stochastic Analysis and Applications (ed. M. Pinsky), Advances in Probability and Related Topics, 7, 1984, 147-166. Google Scholar

[15]

M. Freidlin, Functional Integration and Partial Differential Equations, Ann. Math. Stud., 109, Princeton University Press, Princeton, NJ, 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), 521-525.  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), 1229-1247. 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), 509-519. 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), 309-358. 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), 1-26. 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), 5605-5633. 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), 335-375. 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), 2388-2406. 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), 169-185. 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), 167-194. 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), 1021-1047. 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), 815-839. 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), 421-442. 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), 493-532. 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, Esztergom, Hungary, 1979, published in Seria Colloquia Mathematica Societatis Janos Bolyai, 27, North Holland, 1981, 835-873.  Google Scholar

[31]

L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows, Commun. Math. Sci., 5 (2007), 575-593. 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), 143-160. 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, Oxford University Press, Oxford, 1997. Google Scholar

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis, 20 (1999), 1-11.  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), 441-453. 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), 33-76. 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), 255-262. 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), 2146-2189. 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), 451-480. 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), 173-213. 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), 1101-1146. 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), 469-507. 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), 115619, 23pp. 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), 1014-1065. 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), 5541-5566. 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), 3553-3563. 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), 361-400.  Google Scholar

[13]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. Google Scholar

[14]

M. Freidlin, On wave front propagation in periodic media, in Stochastic Analysis and Applications (ed. M. Pinsky), Advances in Probability and Related Topics, 7, 1984, 147-166. Google Scholar

[15]

M. Freidlin, Functional Integration and Partial Differential Equations, Ann. Math. Stud., 109, Princeton University Press, Princeton, NJ, 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), 521-525.  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), 1229-1247. 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), 509-519. 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), 309-358. 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), 1-26. 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), 5605-5633. 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), 335-375. 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), 2388-2406. 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), 169-185. 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), 167-194. 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), 1021-1047. 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), 815-839. 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), 421-442. 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), 493-532. 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, Esztergom, Hungary, 1979, published in Seria Colloquia Mathematica Societatis Janos Bolyai, 27, North Holland, 1981, 835-873.  Google Scholar

[31]

L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows, Commun. Math. Sci., 5 (2007), 575-593. 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), 143-160. 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, Oxford University Press, Oxford, 1997. Google Scholar

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis, 20 (1999), 1-11.  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), 441-453. doi: 10.1007/s00205-009-0282-1.  Google Scholar

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