American Institute of Mathematical Sciences

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

Myopic models of population dynamics on infinite networks

Robert Carlson 1,

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.
Keywords: Network reaction-diffusion problems, network population models, nonlinear diffusions..
Mathematics Subject Classification: Primary: 34B4.
Citation: Robert Carlson. Myopic models of population dynamics on infinite networks. Networks & 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.  Google Scholar

[2]

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

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

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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.  Google Scholar

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

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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.  Google Scholar

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R. Diestel, Graph Theory, Springer, 2005.  Google Scholar

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

[9]

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

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

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

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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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

R. Lyons and Y. Peres, Probability on Trees and Networks,, preprint., ().   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

R. Diestel, Graph Theory, Springer, 2005.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

R. Lyons and Y. Peres, Probability on Trees and Networks,, preprint., ().   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

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