# 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  July 2018 Early access  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 and 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.

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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.

##### 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
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