Myopic models of population dynamics on infinite networks

Previous Article
On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains
NHM Home
This Issue
Next Article
Continuum surface energy from a lattice model

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

Abstract
Related Papers

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 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.

[1]	Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations and Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81
[2]	Keng Deng. On a nonlocal reaction-diffusion population model. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[3]	Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85
[4]	Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
[5]	Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
[6]	Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[7]	Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks and Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23
[8]	T. S. Evans, A. D. K. Plato. Network rewiring models. Networks and Heterogeneous Media, 2008, 3 (2) : 221-238. doi: 10.3934/nhm.2008.3.221
[9]	Tanka Nath Dhamala. A survey on models and algorithms for discrete evacuation planning network problems. Journal of Industrial and Management Optimization, 2015, 11 (1) : 265-289. doi: 10.3934/jimo.2015.11.265
[10]	Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170
[11]	Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[12]	José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[13]	Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
[14]	Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321
[15]	Philippe Michel, Suman Kumar Tumuluri. A note on a neuron network model with diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3659-3676. doi: 10.3934/dcdsb.2020085
[16]	Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41
[17]	Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19
[18]	Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133
[19]	M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[20]	Patrick Guidotti. A family of nonlinear diffusions connecting Perona-Malik to standard diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 581-590. doi: 10.3934/dcdss.2012.5.581

2021 Impact Factor: 1.41

Tools

Metrics

Other articles
by authors

on AIMS
Robert Carlson
on Google Scholar
Robert Carlson

[Back to Top]