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Article Contents

# Invasion fronts on graphs: The Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs

• 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.

Mathematics Subject Classification: Primary: 37L60; Secondary: 35R02, 35C07, 05C80.

 Citation:

• Figure 1.  The linear spreading speed for (5), calculated numerically as a function of $\alpha$ for $k = 3$ (red), $k = 4$ (black) and $k = 5$ (blue). Note the critical values $\alpha_2(k)$ for which the spreading speed is zero and $\alpha_1(k)$ where the speed is maximal. Also note that as $\alpha\to 0$, these spreading speeds appear to approach a common curve

Figure 2.  Critical rates of diffusion for period trees with period $m = 2$. On the left, we plot $\alpha_1$ as a function of $k_1$ with $k_2$ fixed to preserve the mean degree. On the right, we plot $\alpha_2$ as a function of $k_1$. Note that in both case the periodic heterogeneity increases the critical diffusion rates

Figure 3.  Numerical simulations of (2) with $k = 3$ and for $\alpha = 0.2$ (left), $\alpha = 0.8$ (middle) and $\alpha = 2.2$ (right). The blue curves are $u_n(t)$ while the red curves depict the normalized population at each level, i.e. $w_n(t)/\max_n(w_n(t))$. Note that $0.2 < \alpha_1(3) < 0.8 < \alpha_2(3) < 2.2$. For $\alpha = 0.2$, we observe that the maximal population is concentrated at the front interface. For $\alpha = 0.8$, the maximal population is concentrated ahead of the front interface. Finally, for $\alpha = 2.2$ the local population at any fixed node converges to zero, but the total population grows and eventually is concentrated at the final level of the tree

Figure 4.  On the left, we compare predictions for the exponential growth rate of the maximum of $w_n(t)$ as a function of $\alpha$ (blue line) against the exponential growth rates of $M(t)$ observed in direct numerical simulations (asterisks) for $k = 5$. On the right, we compare numerically observed spreading speeds for $w_n(t)$ (asterisks) versus linear spreading speeds determined numerically from the pinched double root criterion applied to $\tilde{d}_s(\gamma,\lambda)$ (blue line). Here we have taken $k = 5$

Figure 5.  Arrival times for an Erdős-Réyni graph with $N = 60,000$ and expected degree $k_{ER} = 2$. Various values of $\alpha$ are considered. In green is the best fit linear approximation for the mean arrival times for nodes with distance between $3$ and $12$ from the initial location

Figure 6.  Speed associated to the mean arrival times in numerical simulations on an Erdős-Réyni graph with $N = 60,000$ and expected degree $k_{ER} = 2$ are shown in asterisks. The blue curve is the spreading speed predicted by the analysis in Section 2 for the homogeneous tree with $k = 2.54$, found by numerically computing roots of (6). This value is chosen since it is one less than the mean degree of the network over those nodes with distance between $3$ and $12$ from the original location

Figure 7.  Growth rate of the total population for Erdős-Rényi graph. On the left, $N = 500,000$ and $\alpha = 0.1,0.35,0.6,0.85$. Larger values of $\alpha$ correspond to faster growth rates. On the right is the case of $N = 60,000$ with the same values of $\alpha$

Figure 8.  Numerically calculated exponential growth rate for the Erdős-Rényi graph. On the left, $N = 500,000$ and observed growth rates are plotted as circles. The asterisks are the corresponding growth rates in the homogeneous tree with depth $13$. The lower curve is degree $k = 3$ while the larger curve is degree $k = 4$. On the right are the same computations, but for the Erdős-Rényi graph with $N = 60,000$ and for homogeneous trees with $k = 2$ and $k = 3$

•  A.-L. Barabási  and  R. Albert , Emergence of scaling in random networks, Science, 286 (1999) , 509-512.  doi: 10.1126/science.286.5439.509. V. Batagelj and U. Brandes, Efficient generation of large random networks, Phys. Rev. E, 71 (2005), 036113. doi: 10.1103/PhysRevE.71.036113. 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. B. Bollobás, Random Graphs, volume 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2001. doi: 10.1017/CBO9780511814068. 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. 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. R. J. Briggs, Electron-Stream Interaction with Plasmas, MIT Press, Cambridge, 1964. D. Brockmann  and  D. Helbing , The hidden geometry of complex, network-driven contagion phenomena, Science, 342 (2013) , 1337-1342.  doi: 10.1126/science.1245200. 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. X. Chen , Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997) , 125-160. 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. F. Chung and S. -T. Yau, Coverings, heat kernels and spanning trees, Electron. J. Combin., 6 (1999), Research Paper 12, 21 pp. 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. R. Durrett, Random Graph Dynamics, volume 20 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2007. P. Erdős  and  A. Rényi , On random graphs. I, Publ. Math. Debrecen, 6 (1959) , 290-297. 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. 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. 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. 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. 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. 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. 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. 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. M. E. J. Newman , The structure and function of complex networks, SIAM Review, 45 (2003) , 167-256.  doi: 10.1137/S003614450342480. 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. 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. S. H. Strogatz , Exploring complex networks, Nature, 410 (2001) , 268-276.  doi: 10.1038/35065725. W. van Saarloos , Front propagation into unstable states, Physics Reports, 386 (2003) , 29-222. A. Vespignani , Modelling dynamical processes in complex socio-technical systems, Nature Physics, 8 (2012) , 32-39.  doi: 10.1038/nphys2160. D. J. Watts  and  S. H. Strogatz , Collective dynamics of "small-world" networks, nature, 393 (1998) , 440-442. 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. 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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