September  2014, 9(3): 477-499. doi: 10.3934/nhm.2014.9.477

Myopic models of population dynamics on infinite networks

1. 

Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80918, United States

Received  November 2013 Revised  June 2014 Published  October 2014

Reaction-diffusion equations are treated on infinite networks using semigroup methods. To blend high fidelity local analysis with coarse remote modeling, initial data and solutions come from a uniformly closed algebra generated by functions which are flat at infinity. The algebra is associated with a compactification of the network which facilitates the description of spatial asymptotics. Diffusive effects disappear at infinity, greatly simplifying the remote dynamics. Accelerated diffusion models with conventional eigenfunction expansions are constructed to provide opportunities for finite dimensional approximation.
Citation: Robert Carlson. Myopic models of population dynamics on infinite networks. Networks and Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477
References:
[1]

R. Carlson, Boundary value problems for infinite metric graphs, Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368. doi: 10.1090/pspum/077/2459880.

[2]

E. A. Coddington and R. Carlson, Linear Ordinary Differential Equations, SIAM, 1997. doi: 10.1137/1.9781611971439.

[3]

F. Chung, Spectral Graph Theory, American Mathematical Society, Providence, 1997.

[4]

V. Colizza, R. Pastor-Satorras and A. Vespignani, Reaction-diffusion processes and metapopulation models in heterogeneous networks, Nature Physics, 3 (2007), 276-282. doi: 10.1038/nphys560.

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1990.

[6]

E. B. Davies, Large deviations for heat kernels on graphs, J. London Math. Soc. (2), 47 (1993), 65-72. doi: 10.1112/jlms/s2-47.1.65.

[7]

R. Diestel, Graph Theory, Springer, 2005.

[8]

J. Dodziuk, Elliptic operators on infinite graphs, in Analysis, Geometry and Topology of Elliptic Operators, World Sci. publ., (2006), 353-368.

[9]

P. Doyle and J. Snell, Random Walks and Electrical Networks, Mathematical Association of America, Washington, D.C., 1984.

[10]

P. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, 1979.

[11]

A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012.

[12]

S. Haeseler, M. Keller, D. Lenz and R. Wojciechowski, Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions, J. Spectral Theory, 2 (2012), 397-432.

[13]

P. Hartman, Ordinary Differential Equations, Wiley, 1973.

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, 1981.

[15]

J. Hocking and G. Young, Topology, Addison-Wesley, 1961.

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1995.

[17]

M. Keeling and K. Eames, Networks and epidemic models, Journal of the Royal Society Interface, 2 (2005). doi: 10.1098/rsif.2005.0051.

[18]

M. Keller and D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom., 5 (2010), 198-224. doi: 10.1051/mmnp/20105409.

[19]

P. Lax, Functional Analysis, John Wiley & Sons, 2002.

[20]

T. Liggett, Continuous Time Markov Processes, American Mathematical Society, Providence, 2010.

[21]

R. Lyons and Y. Peres, Probability on Trees and Networks, preprint.

[22]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014.

[23]

J. Murray, Mathematical Biology I: An Introduction, Springer, New York, 2002.

[24]

J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, New York, 2003.

[25]

M. Newman, A. Barabasi and D. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006.

[26]

M. Newman, Spread of epidemic disease on networks, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.016128.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

J. Ramirez, Population persistence under advection-diffusion in river networks, Journal of Mathematical Biology, 65 (2012), 919-942. doi: 10.1007/s00285-011-0485-6.

[29]

H. Royden, Real Analysis, Macmillan, New York, 1988.

[30]

J. Sarhad, R Carlson and K. Anderson, Population persistence in river networks, Journal of Mathematical Biology, online, (2013). doi: 10.1007/s00285-013-0710-6.

[31]

M. Yamasaki, Parabolic and hyberbolic infinite networks, Hiroshima Math. J., 7 (1977), 135-146.

show all references

References:
[1]

R. Carlson, Boundary value problems for infinite metric graphs, Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368. doi: 10.1090/pspum/077/2459880.

[2]

E. A. Coddington and R. Carlson, Linear Ordinary Differential Equations, SIAM, 1997. doi: 10.1137/1.9781611971439.

[3]

F. Chung, Spectral Graph Theory, American Mathematical Society, Providence, 1997.

[4]

V. Colizza, R. Pastor-Satorras and A. Vespignani, Reaction-diffusion processes and metapopulation models in heterogeneous networks, Nature Physics, 3 (2007), 276-282. doi: 10.1038/nphys560.

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1990.

[6]

E. B. Davies, Large deviations for heat kernels on graphs, J. London Math. Soc. (2), 47 (1993), 65-72. doi: 10.1112/jlms/s2-47.1.65.

[7]

R. Diestel, Graph Theory, Springer, 2005.

[8]

J. Dodziuk, Elliptic operators on infinite graphs, in Analysis, Geometry and Topology of Elliptic Operators, World Sci. publ., (2006), 353-368.

[9]

P. Doyle and J. Snell, Random Walks and Electrical Networks, Mathematical Association of America, Washington, D.C., 1984.

[10]

P. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, 1979.

[11]

A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012.

[12]

S. Haeseler, M. Keller, D. Lenz and R. Wojciechowski, Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions, J. Spectral Theory, 2 (2012), 397-432.

[13]

P. Hartman, Ordinary Differential Equations, Wiley, 1973.

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, 1981.

[15]

J. Hocking and G. Young, Topology, Addison-Wesley, 1961.

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1995.

[17]

M. Keeling and K. Eames, Networks and epidemic models, Journal of the Royal Society Interface, 2 (2005). doi: 10.1098/rsif.2005.0051.

[18]

M. Keller and D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom., 5 (2010), 198-224. doi: 10.1051/mmnp/20105409.

[19]

P. Lax, Functional Analysis, John Wiley & Sons, 2002.

[20]

T. Liggett, Continuous Time Markov Processes, American Mathematical Society, Providence, 2010.

[21]

R. Lyons and Y. Peres, Probability on Trees and Networks, preprint.

[22]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014.

[23]

J. Murray, Mathematical Biology I: An Introduction, Springer, New York, 2002.

[24]

J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, New York, 2003.

[25]

M. Newman, A. Barabasi and D. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006.

[26]

M. Newman, Spread of epidemic disease on networks, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.016128.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

J. Ramirez, Population persistence under advection-diffusion in river networks, Journal of Mathematical Biology, 65 (2012), 919-942. doi: 10.1007/s00285-011-0485-6.

[29]

H. Royden, Real Analysis, Macmillan, New York, 1988.

[30]

J. Sarhad, R Carlson and K. Anderson, Population persistence in river networks, Journal of Mathematical Biology, online, (2013). doi: 10.1007/s00285-013-0710-6.

[31]

M. Yamasaki, Parabolic and hyberbolic infinite networks, Hiroshima Math. J., 7 (1977), 135-146.

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