Many studies have shown that Physarum polycephalum slime mold is able to find the shortest path in a maze. In this paper we study this behavior in a network, using a hyperbolic model of chemotaxis. Suitable transmission and boundary conditions at each node are considered to mimic the behavior of such an organism in the feeding process. Several numerical tests are presented for special network geometries to show the qualitative agreement between our model and the observed behavior of the mold.
BonifaciV.MehlhornK. and VarmaG., Physarum can compute the shortest path, Journal of Theoretical Biology309 (2012), 121–133.
2.
BorscheS.KlarA. and PhamT.N.H., Kinetic and related macroscopic models for chemotaxis on networks, preprint ArXiv: 1512.07001v1 (2015),
3.
BorscheS.GöttlichS.KlarA. and SchillenP., The scalar Keller-Segel model on networks, Mathematical Models and Methods in Applied Sciences24(2) (2014), 221–247.
4.
BrettiG.NataliniR. and RibotM., A hyperbolic model of chemotaxis on a network: A numerical study, Mathematical Modelling and Numerical Analysis48(1) (2014), 231–258.
5.
CamilliF. and CorriasL., Parabolic models for chemotaxis on weighted networks, preprint ArXiv: 151107279 (2015).
6.
DolakY. and HillenT., Cattaneo models for chemosensitive movement, numerical solution and pattern formation, J Math Biol46 (2003), 153–170; corrected version after misprinted 160 in J Math Biol46 (2003), 461–478.
7.
FilbetF.LaurençotP. and PerthameB., Derivation of hyperbolic models for chemosensitive movement, J Math Biol50(2) (2005), 189–207.
8.
GambaA.AmbrosiD.ConiglioA.de CandiaA.Di TaliaS.GiraudoE.SeriniG.PreziosiL. and BussolinoF., Percolation, morphogenesis, and burgers dynamics in blood vessels formation, Phys Rev Letters90 (2003), 118101.1–118101.4.
9.
GaravelloM. and PiccoliB., Traffic flow on networks: Conservation laws models, AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). xvi+243 ISBN: 978-1-60133-000-0; 1-60133-000-6.
10.
GosseL., Asymptotic-preserving and well-balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes, J Math Anal Appl388(2) (2012), 964–983.
11.
GosseL., Well-balanced numerical approximations display asymptotic decay toward Maxwellian distributions for a model of chemotaxis in a bounded interval, SIAM J Sci Comput34(1) (2012), A520–A545.
12.
GreenbergJ.M. and AltW., Stability results for a diffusion equation with functional drift approximating a chemotaxis model, Trans Amer Math Soc300 (1987), 235–258.
13.
GuarguagliniF.R.MasciaC.NataliniR. and RibotM., Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis, Discrete Contin Dyn Syst Ser B12 (2009), 39–76.
14.
GuarguagliniF.R. and NataliniR., Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J Math Anal, in press.
15.
HorstmannD., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber Deutsch Math-Verein105 (2003), 103–165.
16.
KellerE.F. and SegelL.A., Initiation of slime mold aggregation viewed as an instability, J Theor Biol26 (1970), 399–415.
17.
KedemO. and KatchalskyA., Thermodynamic analysis of the permeability of biological membrane to non-electrolytes, Biochimica et Biophysica Acta27 (1958), 229–246.
18.
MiyajiT. and OhnishiI., Mathematical analysis to an adaptive network of the plasmodium system, Hokkaido Math J36 (2007), 445–465.
19.
MiyajiT. and OhnishiI., Physarum can solve the shortest path problem on Riemannian surface mathematically rigorously, Int J Pure Appl Math46 (2008), 353–369.
20.
MurrayJ.D., Mathematical biology I. An introduction. Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York 2002; II. Spatial models and biomedical applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York 2003.
21.
NakagakiT.YamadaH. and TóthA., Maze-solving by an amoeboid organism, Nature407 (2000), 470.
22.
NataliniR. and RibotM., An asymptotic high order mass-preserving scheme for a hyperbolic model of chemotaxis, SIAM Journal of Numerical Analysis50 (2012), 883–905.
23.
PerthameB., Transport equations in biology, Frontiers in Mathematics, Birkhäuser (2007).
24.
TeroA.KobayashiR. and NakagakiT., A coupled-oscilator model with a conservation law for the rhythmic amoeboid movement of plasmodial slime molds, Phys D205 (2005), 125–135.
25.
TeroA.KobayashiR. and NakagakiT., A mathematical model for adaptive transport network in path finding by true slime mold, J Theoret Biol244 (2007), 553–564.
26.
TeroA.TakagiS.SaigusaT.ItoK.BebberD.P.FrickerM.D.YumikiK.KobayashiR. and NakagakiT., Rules for biologically inspired adaptive network design, Science327 (2010), 439–442.
27.
SegelL.A., A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J Appl Math32 (1977), 653–665.
28.
WatanabeS.TeroA.TakamatsuA. and NakagakiT., Traffic optimization in railroad networks using an algorithm mimicking an amoeba-like organism, Physarum plasmodium, ByoSystems105 (2011), 225–232.
