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Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
Invasion fronts on graphs: The Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs
1. | Franklin W. Olin College of Engineering, Needham, MA 02492, USA |
2. | Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
We study the dynamics of the Fisher-KPP equation on the infinite homogeneous tree and Erdős-Réyni random graphs. We assume initial data that is zero everywhere except at a single node. For the case of the homogeneous tree, the solution will either form a traveling front or converge pointwise to zero. This dichotomy is determined by the linear spreading speed and we compute critical values of the diffusion parameter for which the spreading speed is zero and maximal and prove that the system is linearly determined. We also study the growth of the total population in the network and identify the exponential growth rate as a function of the diffusion coefficient, α. Finally, we make predictions for the Fisher-KPP equation on Erdős-Rényi random graphs based upon the results on the homogeneous tree. When α is small we observe via numerical simulations that mean arrival times are linearly related to distance from the initial node and the speed of invasion is well approximated by the linear spreading speed on the tree. Furthermore, we observe that exponential growth rates of the total population on the random network can be bounded by growth rates on the homogeneous tree and provide an explanation for the sub-linear exponential growth rates that occur for small diffusion.
References:
[1] |
A.-L. Barabási and R. Albert,
Emergence of scaling in random networks, Science, 286 (1999), 509-512.
doi: 10.1126/science.286.5439.509. |
[2] |
V. Batagelj and U. Brandes, Efficient generation of large random networks,
Phys. Rev. E, 71 (2005), 036113.
doi: 10.1103/PhysRevE.71.036113. |
[3] |
A. Bers, Space-time evolution of plasma instabilities-absolute and convective, in Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics, Volume 1 eds. A. A. Galeev & R. N. Sudan, (1984), 451-517. |
[4] |
B. Bollobás,
Random Graphs, volume 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2001.
doi: 10.1017/CBO9780511814068. |
[5] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,
Mem. Amer. Math. Soc., 44 (1983), iv+190pp.
doi: 10.1090/memo/0285. |
[6] |
L. Brevdo and T. J. Bridges,
Absolute and convective instabilities of spatially periodic flows, Philos. Trans. Roy. Soc. London Ser. A, 354 (1996), 1027-1064.
doi: 10.1098/rsta.1996.0040. |
[7] |
R. J. Briggs,
Electron-Stream Interaction with Plasmas, MIT Press, Cambridge, 1964. |
[8] |
D. Brockmann and D. Helbing,
The hidden geometry of complex, network-driven contagion phenomena, Science, 342 (2013), 1337-1342.
doi: 10.1126/science.1245200. |
[9] |
R. Burioni, S. Chibbaro, D. Vergni and A. Vulpiani, Reaction spreading on graphs,
Phys. Rev. E, 86 (2012), 055101.
doi: 10.1103/PhysRevE.86.055101. |
[10] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[11] |
G. Chinta, J. Jorgenson and A. Karlsson,
Heat kernels on regular graphs and generalized Ihara zeta function formulas, Monatsh. Math., 178 (2015), 171-190.
doi: 10.1007/s00605-014-0685-4. |
[12] |
F. Chung and S. -T. Yau, Coverings, heat kernels and spanning trees,
Electron. J. Combin., 6 (1999), Research Paper 12, 21 pp. |
[13] |
V. Colizza, R. Pastor-Satorras and A. Vespignani,
Reaction—diffusion processes and metapopulation models in heterogeneous networks, Nat Phys, 3 (2007), 276-282.
doi: 10.1038/nphys560. |
[14] |
R. Durrett,
Random Graph Dynamics, volume 20 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2007. |
[15] |
P. Erdős and A. Rényi,
On random graphs. I, Publ. Math. Debrecen, 6 (1959), 290-297.
|
[16] |
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. |
[17] |
J. Hindes, S. Singh, C. R. Myers and D. J. Schneider, Epidemic fronts in complex networks with metapopulation structure,
Phys. Rev. E, 88 (2013), 012809. |
[18] |
M. Holzer,
A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations, Discrete Contin. Dyn. Syst., 36 (2016), 2069-2084.
doi: 10.3934/dcds.2016.36.2069. |
[19] |
M. Holzer and A. Scheel,
Criteria for pointwise growth and their role in invasion processes, J. Nonlinear Sci., 24 (2014), 661-709.
doi: 10.1007/s00332-014-9202-0. |
[20] |
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), 1-25.
|
[21] |
N. E. Kouvaris, H. Kori and A. S. Mikhailov, Traveling and pinned fronts in bistable reaction-diffusion systems on networks,
PLoS ONE, 7 (2012), e45029.
doi: 10.1371/journal.pone.0045029. |
[22] |
H. Matano, F. Punzo and A. Tesei,
Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc. (JEMS), 17 (2015), 1199-1227.
doi: 10.4171/JEMS/529. |
[23] |
B. Mohar and W. Woess,
A survey on spectra of infinite graphs, Bull. London Math. Soc., 21 (1989), 209-234.
doi: 10.1112/blms/21.3.209. |
[24] |
M. E. J. Newman,
The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[25] |
M. A. Porter and J. P. Gleeson,
Dynamical Systems on Networks, volume 4 of Frontiers in Applied Dynamical Systems: Reviews and Tutorials. Springer, Cham, 2016. A tutorial.
doi: 10.1007/978-3-319-26641-1. |
[26] |
B. Sandstede and A. Scheel,
Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277.
doi: 10.1016/S0167-2789(00)00114-7. |
[27] |
S. H. Strogatz,
Exploring complex networks, Nature, 410 (2001), 268-276.
doi: 10.1038/35065725. |
[28] |
W. van Saarloos,
Front propagation into unstable states, Physics Reports, 386 (2003), 29-222.
|
[29] |
A. Vespignani,
Modelling dynamical processes in complex socio-technical systems, Nature Physics, 8 (2012), 32-39.
doi: 10.1038/nphys2160. |
[30] |
D. J. Watts and S. H. Strogatz,
Collective dynamics of "small-world" networks, nature, 393 (1998), 440-442.
|
[31] |
H. F. Weinberger,
Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.
doi: 10.1137/0513028. |
[32] |
B. Zinner, G. Harris and W. Hudson,
Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.
doi: 10.1006/jdeq.1993.1082. |
show all references
References:
[1] |
A.-L. Barabási and R. Albert,
Emergence of scaling in random networks, Science, 286 (1999), 509-512.
doi: 10.1126/science.286.5439.509. |
[2] |
V. Batagelj and U. Brandes, Efficient generation of large random networks,
Phys. Rev. E, 71 (2005), 036113.
doi: 10.1103/PhysRevE.71.036113. |
[3] |
A. Bers, Space-time evolution of plasma instabilities-absolute and convective, in Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics, Volume 1 eds. A. A. Galeev & R. N. Sudan, (1984), 451-517. |
[4] |
B. Bollobás,
Random Graphs, volume 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2001.
doi: 10.1017/CBO9780511814068. |
[5] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,
Mem. Amer. Math. Soc., 44 (1983), iv+190pp.
doi: 10.1090/memo/0285. |
[6] |
L. Brevdo and T. J. Bridges,
Absolute and convective instabilities of spatially periodic flows, Philos. Trans. Roy. Soc. London Ser. A, 354 (1996), 1027-1064.
doi: 10.1098/rsta.1996.0040. |
[7] |
R. J. Briggs,
Electron-Stream Interaction with Plasmas, MIT Press, Cambridge, 1964. |
[8] |
D. Brockmann and D. Helbing,
The hidden geometry of complex, network-driven contagion phenomena, Science, 342 (2013), 1337-1342.
doi: 10.1126/science.1245200. |
[9] |
R. Burioni, S. Chibbaro, D. Vergni and A. Vulpiani, Reaction spreading on graphs,
Phys. Rev. E, 86 (2012), 055101.
doi: 10.1103/PhysRevE.86.055101. |
[10] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[11] |
G. Chinta, J. Jorgenson and A. Karlsson,
Heat kernels on regular graphs and generalized Ihara zeta function formulas, Monatsh. Math., 178 (2015), 171-190.
doi: 10.1007/s00605-014-0685-4. |
[12] |
F. Chung and S. -T. Yau, Coverings, heat kernels and spanning trees,
Electron. J. Combin., 6 (1999), Research Paper 12, 21 pp. |
[13] |
V. Colizza, R. Pastor-Satorras and A. Vespignani,
Reaction—diffusion processes and metapopulation models in heterogeneous networks, Nat Phys, 3 (2007), 276-282.
doi: 10.1038/nphys560. |
[14] |
R. Durrett,
Random Graph Dynamics, volume 20 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2007. |
[15] |
P. Erdős and A. Rényi,
On random graphs. I, Publ. Math. Debrecen, 6 (1959), 290-297.
|
[16] |
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. |
[17] |
J. Hindes, S. Singh, C. R. Myers and D. J. Schneider, Epidemic fronts in complex networks with metapopulation structure,
Phys. Rev. E, 88 (2013), 012809. |
[18] |
M. Holzer,
A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations, Discrete Contin. Dyn. Syst., 36 (2016), 2069-2084.
doi: 10.3934/dcds.2016.36.2069. |
[19] |
M. Holzer and A. Scheel,
Criteria for pointwise growth and their role in invasion processes, J. Nonlinear Sci., 24 (2014), 661-709.
doi: 10.1007/s00332-014-9202-0. |
[20] |
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), 1-25.
|
[21] |
N. E. Kouvaris, H. Kori and A. S. Mikhailov, Traveling and pinned fronts in bistable reaction-diffusion systems on networks,
PLoS ONE, 7 (2012), e45029.
doi: 10.1371/journal.pone.0045029. |
[22] |
H. Matano, F. Punzo and A. Tesei,
Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc. (JEMS), 17 (2015), 1199-1227.
doi: 10.4171/JEMS/529. |
[23] |
B. Mohar and W. Woess,
A survey on spectra of infinite graphs, Bull. London Math. Soc., 21 (1989), 209-234.
doi: 10.1112/blms/21.3.209. |
[24] |
M. E. J. Newman,
The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[25] |
M. A. Porter and J. P. Gleeson,
Dynamical Systems on Networks, volume 4 of Frontiers in Applied Dynamical Systems: Reviews and Tutorials. Springer, Cham, 2016. A tutorial.
doi: 10.1007/978-3-319-26641-1. |
[26] |
B. Sandstede and A. Scheel,
Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277.
doi: 10.1016/S0167-2789(00)00114-7. |
[27] |
S. H. Strogatz,
Exploring complex networks, Nature, 410 (2001), 268-276.
doi: 10.1038/35065725. |
[28] |
W. van Saarloos,
Front propagation into unstable states, Physics Reports, 386 (2003), 29-222.
|
[29] |
A. Vespignani,
Modelling dynamical processes in complex socio-technical systems, Nature Physics, 8 (2012), 32-39.
doi: 10.1038/nphys2160. |
[30] |
D. J. Watts and S. H. Strogatz,
Collective dynamics of "small-world" networks, nature, 393 (1998), 440-442.
|
[31] |
H. F. Weinberger,
Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.
doi: 10.1137/0513028. |
[32] |
B. Zinner, G. Harris and W. Hudson,
Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.
doi: 10.1006/jdeq.1993.1082. |








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