# American Institute of Mathematical Sciences

July 2018, 23(5): 1873-1893. doi: 10.3934/dcdsb.2018185

## Generalized network transport and Euler-Hille formula

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa 2 Institute of Mathematics, Łódź University of Technology, Łódź, Poland 3 Advanced System Analysis Program, International Institute for Applied System Analysis, Laxenburg, Austria

* Corresponding author: Jacek Banasiak

The research was partially conducted during the scholarship of A. P. at the International Institute for Applied System Analysis and supported by a grant for young scientists of the Institute of Mathematics of Lódź University of Technology. J. B. was partially supported by the Incentive Funding of the National Research Foundation of South Africa.

Received  April 2017 Revised  August 2017 Published  May 2018

In this article we consider asymptotic properties of network flow models with fast transport along the edges and explore their connection with an operator version of the Euler formula for the exponential function. This connection, combined with the theory of the regular convergence of semigroups, allows for proving that for fast transport along the edges and slow rate of redistribution of the flow at the nodes, the network flow semigroup (or its suitable projection) can be approximated by a finite dimensional dynamical system related to the boundary conditions at the nodes of the network. The novelty of our results lies in considering more general boundary operators than that allowed for in previous papers.

Citation: Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185
##### References:
 [1] H. Amann and J. Escher, Analysis II, Birkhäuser, Basel, 2008. [2] F. M. Atay and L. Roncoroni, Lumpability of linear evolution equations in Banach spaces, Evolution Equation and Control Theory, 6 (2017), 15-34. doi: 10.3934/eect.2017002. [3] P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, Journal of Evolution Equations, 11 (2011), 121-154. doi: 10.1007/978-3-540-78273-5_5. [4] J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: An asymptotic analysis approach, Journal of Evolution Equations, 11 (2011), 121-154. doi: 10.1007/s00028-010-0086-7. [5] J. Banasiak, A. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks, Semigroup Forum, 93 (2016), 427-443. doi: 10.1007/s00233-015-9730-4. [6] J. Banasiak and A. Falkiewicz, Some Transport and diffusion processes on networks and their graph realizability, Applied Mathematical Letters, 45 (2015), 25-30. doi: 10.1016/j.aml.2015.01.006. [7] J. Banasiak and A. Falkiewicz, A singular limit for an age structured mutation problem, Mathematical Biosciences and Engineering, 14 (2017), 17-30. doi: 10.3934/mbe.2017002. [8] J. Banasiak, A. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with application to population problems, Mathematical Models and Methods in Applied Sciences, 26 (2016), 215-247. doi: 10.1142/S0218202516400017. [9] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhaüser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6. [10] J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, Networks and Heterogeneous Media, 9 (2014), 197-216. doi: 10.3934/nhm.2014.9.197. [11] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1. [12] A. Bobrowski, On Hille-type approximation of degenerate semigroups of operators, Linear Algebra and its Applications, 511 (2016), 31-53. doi: 10.1016/j.laa.2016.08.036. [13] A. Bobrowski, Convergence of One-parameter Operator Semigroups in Models of Mathematical Biology and Elsewhere, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663. [14] A. Bobrowski, From diffusions on graphs to Markov chains via asymptotic state lumping, Annales Henri Poincare, 13 (2012), 1501-1510. doi: 10.1007/s00023-012-0158-z. [15] B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2. [16] K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York Berlin Heidelberg, 2000. [17] Y. Iwasa, V. Andreasen and S. Levin, Aggregation in model ecosystems. Ⅰ. Perfect aggregation, Ecological Modelling, 37 (1987), 287-302. [18] Y. Iwasa, V. Andreasen and S. Levin, Aggregation in model ecosystems. Ⅱ. Approximate aggregation, IMA Journal of Mathematics Applied in Medicine and Biology, 6 (1989), 1-23. doi: 10.1093/imammb/6.1.1-a. [19] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [20] M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357. [21] M. Kimmel, A. Świerniak and A. Polański, Infinite-dimensional model of evolution of drug resistance of cancer cells, J. Math. Systems Estimation Control, 8 (1998), 1-16. [22] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Mathematische Zeitschrift, 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3. [23] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Theor. Biol., 1 (1974), 17-36. doi: 10.1007/BF02339486. [24] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512. [25] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [26] M. Rotenberg, Transport theory for growing cell population, Journal of Theoretical Biology, 103 (1983), 181-199. doi: 10.1016/0022-5193(83)90024-3.

show all references

##### References:
 [1] H. Amann and J. Escher, Analysis II, Birkhäuser, Basel, 2008. [2] F. M. Atay and L. Roncoroni, Lumpability of linear evolution equations in Banach spaces, Evolution Equation and Control Theory, 6 (2017), 15-34. doi: 10.3934/eect.2017002. [3] P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, Journal of Evolution Equations, 11 (2011), 121-154. doi: 10.1007/978-3-540-78273-5_5. [4] J. Banasiak, A. Goswami and S. Shindin, Aggregation in age and space structured population models: An asymptotic analysis approach, Journal of Evolution Equations, 11 (2011), 121-154. doi: 10.1007/s00028-010-0086-7. [5] J. Banasiak, A. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks, Semigroup Forum, 93 (2016), 427-443. doi: 10.1007/s00233-015-9730-4. [6] J. Banasiak and A. Falkiewicz, Some Transport and diffusion processes on networks and their graph realizability, Applied Mathematical Letters, 45 (2015), 25-30. doi: 10.1016/j.aml.2015.01.006. [7] J. Banasiak and A. Falkiewicz, A singular limit for an age structured mutation problem, Mathematical Biosciences and Engineering, 14 (2017), 17-30. doi: 10.3934/mbe.2017002. [8] J. Banasiak, A. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with application to population problems, Mathematical Models and Methods in Applied Sciences, 26 (2016), 215-247. doi: 10.1142/S0218202516400017. [9] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhaüser/Springer, Cham, 2014. doi: 10.1007/978-3-319-05140-6. [10] J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, Networks and Heterogeneous Media, 9 (2014), 197-216. doi: 10.3934/nhm.2014.9.197. [11] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1. [12] A. Bobrowski, On Hille-type approximation of degenerate semigroups of operators, Linear Algebra and its Applications, 511 (2016), 31-53. doi: 10.1016/j.laa.2016.08.036. [13] A. Bobrowski, Convergence of One-parameter Operator Semigroups in Models of Mathematical Biology and Elsewhere, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663. [14] A. Bobrowski, From diffusions on graphs to Markov chains via asymptotic state lumping, Annales Henri Poincare, 13 (2012), 1501-1510. doi: 10.1007/s00023-012-0158-z. [15] B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2. [16] K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York Berlin Heidelberg, 2000. [17] Y. Iwasa, V. Andreasen and S. Levin, Aggregation in model ecosystems. Ⅰ. Perfect aggregation, Ecological Modelling, 37 (1987), 287-302. [18] Y. Iwasa, V. Andreasen and S. Levin, Aggregation in model ecosystems. Ⅱ. Approximate aggregation, IMA Journal of Mathematics Applied in Medicine and Biology, 6 (1989), 1-23. doi: 10.1093/imammb/6.1.1-a. [19] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [20] M. Kimmel and D. N. Stivers, Time-continuous branching walk models of unstable gene amplification, Bull. Math. Biol., 50 (1994), 337-357. [21] M. Kimmel, A. Świerniak and A. Polański, Infinite-dimensional model of evolution of drug resistance of cancer cells, J. Math. Systems Estimation Control, 8 (1998), 1-16. [22] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Mathematische Zeitschrift, 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3. [23] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Theor. Biol., 1 (1974), 17-36. doi: 10.1007/BF02339486. [24] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512. [25] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1. [26] M. Rotenberg, Transport theory for growing cell population, Journal of Theoretical Biology, 103 (1983), 181-199. doi: 10.1016/0022-5193(83)90024-3.
Commutativity of the aggregation diagram
The graph G representing the canal network in Example 1
The line graph of the graph shown on Fig. 2
Graphical representation of the Kimmel–Stivers model
Kimmel–Stievers model with vital dynamics
Discrete Lebowitz–Rubinow–Rotenberg model
 [1] Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735 [2] Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465 [3] Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419 [4] Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351 [5] Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 [6] Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098 [7] Lavinia Roncoroni. Exact lumping of feller semigroups: A $C^{\star}$-algebras approach. Conference Publications, 2015, 2015 (special) : 965-973. doi: 10.3934/proc.2015.0965 [8] Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159 [9] Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617 [10] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [11] Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 [12] Yang Wang. The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem. Communications on Pure & Applied Analysis, 2011, 10 (2) : 731-744. doi: 10.3934/cpaa.2011.10.731 [13] Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341 [14] Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 [15] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-23. doi: 10.3934/dcdsb.2017199 [16] Mounir Balti, Ramzi May. Asymptotic for the perturbed heavy ball system with vanishing damping term. Evolution Equations & Control Theory, 2017, 6 (2) : 177-186. doi: 10.3934/eect.2017010 [17] Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 [18] Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1 [19] Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499 [20] Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041

2016 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables