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Myopic models of population dynamics on infinite networks
1. | Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80918, United States |
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] | |
[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] | |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, 1981. |
[15] | |
[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] | |
[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] | |
[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] | |
[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] | |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, 1981. |
[15] | |
[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] | |
[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] | |
[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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