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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?

Gregoire Nadin 1,

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.
Keywords: Eigenvalue optimization, reaction-diffusion equations, random stationary ergodic coefficients, spreading properties, biological invasions..
Mathematics Subject Classification: 34F05, 35B40, 35K57, 35P15, 92D2.
Citation: Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

M. Freidlin, Functional Integration and Partial Differential Equations, Ann. Math. Stud., 109, Princeton University Press, Princeton, NJ, 1985.

[16]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 249 (1979), 521-525.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997.

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis, 20 (1999), 1-11.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

M. Freidlin, Functional Integration and Partial Differential Equations, Ann. Math. Stud., 109, Princeton University Press, Princeton, NJ, 1985.

[16]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 249 (1979), 521-525.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997.

[34]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Analysis, 20 (1999), 1-11.

[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.

[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.

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