April  2016, 36(4): 2069-2084. doi: 10.3934/dcds.2016.36.2069

A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations

1. 

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States

Received  September 2014 Revised  July 2015 Published  September 2015

This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the linearization about the unstable state. We prove that initial data consisting of compactly supported perturbations of Heaviside step functions spreads asymptotically with the anomalous speed. The proof makes use of a comparison principle and the explicit construction of sub and super solutions.
Citation: Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069
References:
[1]

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

[2]

S. Bell, A. White, J. Sherratt and M. Boots, Invading with biological weapons: The role of shared disease in ecological invasion,, Theoretical Ecology, 2 (2009), 53. doi: 10.1007/s12080-008-0029-x. Google Scholar

[3]

M. R. Booty, R. Haberman and A. A. Minzoni, The accommodation of traveling waves of Fisher's type to the dynamics of the leading tail,, SIAM J. Appl. Math., 53 (1993), 1009. doi: 10.1137/0153050. Google Scholar

[4]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,, Mem. Amer. Math. Soc., 44 (1983). doi: 10.1090/memo/0285. Google Scholar

[5]

E. C. Elliott and S. J. Cornell, Dispersal polymorphism and the speed of biological invasions,, PLoS ONE, 7 (2012). doi: 10.1371/journal.pone.0040496. Google Scholar

[6]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. Google Scholar

[7]

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

[8]

M. Freidlin, Coupled reaction-diffusion equations,, Ann. Probab., 19 (1991), 29. doi: 10.1214/aop/1176990535. Google Scholar

[9]

A. Ghazaryan, P. Gordon and A. Virodov, Stability of fronts and transient behaviour in KPP systems,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1769. doi: 10.1098/rspa.2009.0400. Google Scholar

[10]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-{KPP} equations,, Netw. Heterog. Media, 8 (2013), 275. doi: 10.3934/nhm.2013.8.275. Google Scholar

[11]

M. Holzer, Anomalous spreading in a system of coupled Fisher-KPP equations,, Phys. D, 270 (2014), 1. doi: 10.1016/j.physd.2013.12.003. Google Scholar

[12]

M. Holzer and A. Scheel, A slow pushed front in a Lotka Volterra competition model,, Nonlinearity, 25 (2012), 2151. doi: 10.1088/0951-7715/25/7/2151. Google Scholar

[13]

M. Holzer and A. Scheel, Criteria for pointwise growth and their role in invasion processes,, J. Nonlinear Sci., 24 (2014), 661. doi: 10.1007/s00332-014-9202-0. Google Scholar

[14]

M. Iida, R. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients,, SIAM Journal on Mathematical Analysis, 43 (2011), 1369. doi: 10.1137/100792846. Google Scholar

[15]

A. Kolmogorov, I. Petrovskii and N. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantite' de matiere et son application a un probleme biologique,, Moscow Univ. Math. Bull., 1 (1937), 1. Google Scholar

[16]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008. Google Scholar

[17]

G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system. II. The critical case,, J. Differential Equations, 146 (1998), 399. doi: 10.1006/jdeq.1997.3391. Google Scholar

[18]

W. van Saarloos, Front propagation into unstable states,, Physics Reports, 386 (2003), 29. Google Scholar

[19]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

[20]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207. doi: 10.1007/s00285-007-0078-6. Google Scholar

show all references

References:
[1]

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

[2]

S. Bell, A. White, J. Sherratt and M. Boots, Invading with biological weapons: The role of shared disease in ecological invasion,, Theoretical Ecology, 2 (2009), 53. doi: 10.1007/s12080-008-0029-x. Google Scholar

[3]

M. R. Booty, R. Haberman and A. A. Minzoni, The accommodation of traveling waves of Fisher's type to the dynamics of the leading tail,, SIAM J. Appl. Math., 53 (1993), 1009. doi: 10.1137/0153050. Google Scholar

[4]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,, Mem. Amer. Math. Soc., 44 (1983). doi: 10.1090/memo/0285. Google Scholar

[5]

E. C. Elliott and S. J. Cornell, Dispersal polymorphism and the speed of biological invasions,, PLoS ONE, 7 (2012). doi: 10.1371/journal.pone.0040496. Google Scholar

[6]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. Google Scholar

[7]

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

[8]

M. Freidlin, Coupled reaction-diffusion equations,, Ann. Probab., 19 (1991), 29. doi: 10.1214/aop/1176990535. Google Scholar

[9]

A. Ghazaryan, P. Gordon and A. Virodov, Stability of fronts and transient behaviour in KPP systems,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1769. doi: 10.1098/rspa.2009.0400. Google Scholar

[10]

F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-{KPP} equations,, Netw. Heterog. Media, 8 (2013), 275. doi: 10.3934/nhm.2013.8.275. Google Scholar

[11]

M. Holzer, Anomalous spreading in a system of coupled Fisher-KPP equations,, Phys. D, 270 (2014), 1. doi: 10.1016/j.physd.2013.12.003. Google Scholar

[12]

M. Holzer and A. Scheel, A slow pushed front in a Lotka Volterra competition model,, Nonlinearity, 25 (2012), 2151. doi: 10.1088/0951-7715/25/7/2151. Google Scholar

[13]

M. Holzer and A. Scheel, Criteria for pointwise growth and their role in invasion processes,, J. Nonlinear Sci., 24 (2014), 661. doi: 10.1007/s00332-014-9202-0. Google Scholar

[14]

M. Iida, R. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients,, SIAM Journal on Mathematical Analysis, 43 (2011), 1369. doi: 10.1137/100792846. Google Scholar

[15]

A. Kolmogorov, I. Petrovskii and N. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantite' de matiere et son application a un probleme biologique,, Moscow Univ. Math. Bull., 1 (1937), 1. Google Scholar

[16]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008. Google Scholar

[17]

G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system. II. The critical case,, J. Differential Equations, 146 (1998), 399. doi: 10.1006/jdeq.1997.3391. Google Scholar

[18]

W. van Saarloos, Front propagation into unstable states,, Physics Reports, 386 (2003), 29. Google Scholar

[19]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

[20]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207. doi: 10.1007/s00285-007-0078-6. Google Scholar

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