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December  2019, 14(4): 633-657. doi: 10.3934/nhm.2019025

On a model of target detection in molecular communication networks

Hirotada Honda ,

1-7-11, Akabanedai, Kita-Ku, Tokyo 115-0053, Japan

* Corresponding author: Hirotada Honda

Received  June 2018 Revised  June 2019 Published  October 2019

This paper is concerned with a target-detection model using bio-nanomachines in the human body that is actively being discussed in the field of molecular communication networks. Although the model was originally proposed as spatially one-dimensional, here we extend it to two dimensions and analyze it. After the mathematical formulation, we first verify the solvability of the stationary problem, and then the existence of a strong global-in-time solution of the non-stationary problem in Sobolev–Slobodetskiĭ space. We also show the non-negativeness of the non-stationary solution.

Keywords: Molecular communication, target detection, global-in-time solvability.
Mathematics Subject Classification: Primary: 35K61, 35Q92; Secondary: 92C17.
Citation: Hirotada Honda. On a model of target detection in molecular communication networks. Networks and Heterogeneous Media, 2019, 14 (4) : 633-657. doi: 10.3934/nhm.2019025
References:
[1]

K. Ahn and K. Kang, On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical, Discrete Contin. Dyn. Syst., 34 (2014), 5165-5179.  doi: 10.3934/dcds.2014.34.5165.

[2]

J. T. Beale, Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-352.  doi: 10.1007/BF00250586.

[3]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.

[4]

A. Einolghozati, M. Sardari, A. Beirami and F. Fekri, Capacity of discrete molecular diffusion channels, Proc. IEEE International Symposium on Information Theory, (2011). doi: 10.1109/ISIT.2011.6034228.

[5]

A. Einolghozati, M. Sardari and F. Fekri, Capacity of diffusion-based molecular communication with ligand receptors, Proc. IEEE Information Theory Workshop, (2011). doi: 10.1109/ITW.2011.6089591.

[6]

B. D. Ewald and R. Temam, Maximum principles for the primitive equations of the atmosphere, Discrete Contin. Dynam. Systems, 7 (2001), 343-362.  doi: 10.3934/dcds.2001.7.343.

[7]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[8]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-162.  doi: 10.1016/S0022-247X(02)00147-6.

[9]

F. R. Guarguaglini and R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Comm. Pure and Appl. Anal., 6 (2007), 287-309.  doi: 10.3934/cpaa.2007.6.287.

[10]

F. R. Guarguaglini and R. Natalini, Nonlinear transmission problems for quasilinear diffusion systems, Networks and Heterogeneous Media, 2 (2007), 359-381.  doi: 10.3934/nhm.2007.2.359.

[11]

H. Honda and A. Tani, Some boundedness of solutions for the primitive equations of the atmosphere and the ocean, ZAMM Journal of Applied Mathematics and Mechanics, 95 (2015), 38-48.  doi: 10.1002/zamm.201200216.

[12]

H. Honda, Local-in-time solvability of target detection model in molecular communication network, International Journal of Applied Mathematics, 31 (2018), 427-455. 

[13]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Dtsch. Math.-Verein., 105 (2003), 103-165. 

[14]

S. Iwasaki, Convergence of solutions to simplified self-organizing target-detection model, Sci. Math. Japnonicae, 81 (2016), 115-129. 

[15]

S. Iwasaki, J. Yang and T. Nakano, A mathematical model of mon-diffusion-based mobile molecular communication networks, IEEE Comm. Lettr., 21 (2017), 1967-1972.  doi: 10.1109/LCOMM.2017.2681061.

[16]

K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the 1D Keller-Segel Model, IMA J. Appl. Math., 72 (2007), 140-162.  doi: 10.1093/imamat/hxl028.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. 

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Society, Providence, R.I., 1968.

[19] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der Mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[21] T. Nakano, A. Eckford and T. Haraguchi, Molecular Communication, Cambridge University Press, Cambridge, 2013. 
[22]

T. Nakano and et al., Performance evaluation of leader-follower-based mobile molecular communication networks for target detection applications, IEEE Trans. Comm., 65 (2017), 663–676.

[23]

L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675.  doi: 10.1002/cpa.3160080414.

[24]

Y. Okaie and et al., Modeling and performance evaluation of mobile bionanocensor networks for target tracking, Proc. IEEE ICC, (2014), 3969–3974.

[25]

Y. Okaie and et al., Cooperative target tracking by a mobile bionanosensor network, IEEE Trans. Nanobioscience, 13 (2014), 267–277.

[26]

K. Osaki and A. Yagi, Atsushi Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. 

[27]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[28]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224. 

[29]

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 109-121. 

[30]

Y. Sugiyama, Y. Tsutsui and J. J. L. Velázquez, Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl., 410 (2014), 908-917.  doi: 10.1016/j.jmaa.2013.08.065.

[31]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[32]

N. Tanaka and A. Tani, Surface waves for a compressible viscous fluid, J. Math. Fluid Mech., 5 (2003), 303-363.  doi: 10.1007/s00021-003-0078-2.

[33]

G. Wang and J. Wei, Steady state solutions of a rReaction-diffusion systems modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.  doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D.

[34]

J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982,500 pp.

show all references

References:
[1]

K. Ahn and K. Kang, On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical, Discrete Contin. Dyn. Syst., 34 (2014), 5165-5179.  doi: 10.3934/dcds.2014.34.5165.

[2]

J. T. Beale, Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-352.  doi: 10.1007/BF00250586.

[3]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.

[4]

A. Einolghozati, M. Sardari, A. Beirami and F. Fekri, Capacity of discrete molecular diffusion channels, Proc. IEEE International Symposium on Information Theory, (2011). doi: 10.1109/ISIT.2011.6034228.

[5]

A. Einolghozati, M. Sardari and F. Fekri, Capacity of diffusion-based molecular communication with ligand receptors, Proc. IEEE Information Theory Workshop, (2011). doi: 10.1109/ITW.2011.6089591.

[6]

B. D. Ewald and R. Temam, Maximum principles for the primitive equations of the atmosphere, Discrete Contin. Dynam. Systems, 7 (2001), 343-362.  doi: 10.3934/dcds.2001.7.343.

[7]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[8]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-162.  doi: 10.1016/S0022-247X(02)00147-6.

[9]

F. R. Guarguaglini and R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Comm. Pure and Appl. Anal., 6 (2007), 287-309.  doi: 10.3934/cpaa.2007.6.287.

[10]

F. R. Guarguaglini and R. Natalini, Nonlinear transmission problems for quasilinear diffusion systems, Networks and Heterogeneous Media, 2 (2007), 359-381.  doi: 10.3934/nhm.2007.2.359.

[11]

H. Honda and A. Tani, Some boundedness of solutions for the primitive equations of the atmosphere and the ocean, ZAMM Journal of Applied Mathematics and Mechanics, 95 (2015), 38-48.  doi: 10.1002/zamm.201200216.

[12]

H. Honda, Local-in-time solvability of target detection model in molecular communication network, International Journal of Applied Mathematics, 31 (2018), 427-455. 

[13]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Dtsch. Math.-Verein., 105 (2003), 103-165. 

[14]

S. Iwasaki, Convergence of solutions to simplified self-organizing target-detection model, Sci. Math. Japnonicae, 81 (2016), 115-129. 

[15]

S. Iwasaki, J. Yang and T. Nakano, A mathematical model of mon-diffusion-based mobile molecular communication networks, IEEE Comm. Lettr., 21 (2017), 1967-1972.  doi: 10.1109/LCOMM.2017.2681061.

[16]

K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the 1D Keller-Segel Model, IMA J. Appl. Math., 72 (2007), 140-162.  doi: 10.1093/imamat/hxl028.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. 

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Society, Providence, R.I., 1968.

[19] O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der Mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[21] T. Nakano, A. Eckford and T. Haraguchi, Molecular Communication, Cambridge University Press, Cambridge, 2013. 
[22]

T. Nakano and et al., Performance evaluation of leader-follower-based mobile molecular communication networks for target detection applications, IEEE Trans. Comm., 65 (2017), 663–676.

[23]

L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675.  doi: 10.1002/cpa.3160080414.

[24]

Y. Okaie and et al., Modeling and performance evaluation of mobile bionanocensor networks for target tracking, Proc. IEEE ICC, (2014), 3969–3974.

[25]

Y. Okaie and et al., Cooperative target tracking by a mobile bionanosensor network, IEEE Trans. Nanobioscience, 13 (2014), 267–277.

[26]

K. Osaki and A. Yagi, Atsushi Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. 

[27]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[28]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224. 

[29]

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 109-121. 

[30]

Y. Sugiyama, Y. Tsutsui and J. J. L. Velázquez, Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl., 410 (2014), 908-917.  doi: 10.1016/j.jmaa.2013.08.065.

[31]

A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[32]

N. Tanaka and A. Tani, Surface waves for a compressible viscous fluid, J. Math. Fluid Mech., 5 (2003), 303-363.  doi: 10.1007/s00021-003-0078-2.

[33]

G. Wang and J. Wei, Steady state solutions of a rReaction-diffusion systems modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.  doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D.

[34]

J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982,500 pp.

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