Abstract
In this chapter we discuss the origin and role of nonlinearities in some classes of biological models. We describe underlying biological mechanisms that generate nonlinearities and how they have been modeled in subfields, such as ecology and epidemiology. We present examples of recent models to highlight the importance of indirect effects and the emergence of alternative stable states, and trade offs. At the same time, we emphasize recent developments and unresolved challenges in biological modeling, such as data-theory coupling, parameter estimation and the generalization of results from low- to high-dimensional systems. We finish with recent examples of mathematical models of the glucose-insulin regulatory system, cancer treatment, limb development and pattern formation, and DNA.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D.J.T. Sumpter, M. Beekman, Anim. Behav. 66, 273 (2003)
M.J. Ryan, R.A. Page, K.L. Hunter, R.C. Taylor, Anim. Behav. 147, 189 (2019)
M.J. Hatcher, J.T.A. Dick, A.M. Dunn, Ecol. Lett. 9, 1253 (2006)
E.E. Werner, S.D. Peacor, Ecology 84, 1083 (2003)
Z. Rapti, T.E. Stewart Merrill, B. Mueller-Brennan, J.H. Kavouras, C.E. Cáceres, Theor. Popul. Biol. 130, 132 (2019)
C.R. Bertram, M. Pinkowski, S.R. Hall, M.A. Duffy, C.E. Cáceres, Oecologia 173, 1023 (2013)
A. Lotka, Elements of Physical Biology (Williams and Wilkins, Baltimore, 1925)
V. Volterra, Mem. R. Accad. Naz. dei Lincei 2, 31 (1926)
C.S. Holling, Can. Entomol. 91, 385 (1959)
M.L. Rosenzweig, R.H. MacArthur, Amer. Nat. 97, 209 (1963)
J.H.P. Dawes, M.O. Souza, J. Theor. Biol. 327, 11 (2013)
P.A. Abrams, Amer. Nat. 121, 887 (1983)
J.S. Weitz, J. Dushoff, Theor. Ecol. 1, 13 (2008)
J. Grilli, G. Barabás. M.J. Michalska-Smith, S. Allesina, Nature 548, 210 (2017)
W.O. Kermack, A.G. McKendrick, Proc. R. Soc. A 115, 700 (1927)
H.W. Hethocote, SIAM Rev. 42, 599 (2000)
J. Monod, Annu. Rev. Microbiol. 3, 371 (1949)
K.A. Johnson, R.S. Goody, Biochemistry 50, 8464 (2011)
S.M. Clifton, T. Kim, J.H. Chandrashekhar, G.A. O’Toole, Z. Rapti, R.J. Whitaker, mSystems 4, e00221-19 (2019)
L.A. Real, Amer. Nat. 111, 289 (1977)
A.V. Hill, J. Physiol. 40(Suppl.), i–vii (1910)
J.-D. Bézanet, M. Bischofberger, E. Tiecke, A. Gonçalves, J.F. Martin, A. Zuniga, F. Naef, R. Zeller, Science 323, 1050 (2009)
K.E. Sears, J.A. Maier, M. Rivas-Astroza, R. Poe, S. Zhong, K. Kosog, J.D. Marcot, R.R. Behringer, C.J. Cretekos, J.J. Rasweiler IV, Z. Rapti, PLoS Genet. 11, e1005398 (2015)
K.D. Lafferty. G. DeLeo, C.J. Briggs, A.P. Dobson, T. Gross, A.M. Kuris, Science 349, 854 (2015)
S. Estrela, C.H. Trisos, S.P. Brown, Amer. Nat. 180, 566–576 (2012)
C.E. Cáceres, G. Davis, S. Duple, S.R. Hall, A. Koss, P. Lee, Z. Rapti, Math. Biosci. 258, 148 (2014)
O.S. Venturelli, H. El-Samad, R.M. Murray, Proc. Natl. Acad. Sci. USA 109, E3324 (2012)
T.S. Gardner, C.R. Cantor, J.J. Collins, Nature 403, 339 (2000)
M. Beekman, D.J.T. Sumpter, F.L.W. Ratnieks, Proc. Natl. Acad. Sci. USA 98, 9703 (2001)
J.C. de Roode, A.J. Yates, S. Altizer, Proc. Natl. Acad. Sci. USA 105, 7489 (2008)
R.E. Lenski, R.M. May, J. Theor. Biol. 169, 253 (1994)
Z. Rapti, C.E. Cáceres, B. Math, Biol. 78, 235 (2016)
C.-J. Lin, K.A. Deger, J.H. Tien, Math. Biosci. 277, 15 (2016)
S. Alizon, Amer. Nat. 172, E113 (2008)
M.A. Gil, Ecology 94, 1871 (2013)
R.H. Condon, C.M. Duarte, K.A. Pitt, K.L. Robinson, C.H. Lucas, K.R. Sutherland, H.W. Mianzan, M. Bogeberg, J.E. Purcell, M.B. Decker, S-i. Uye, L.P. Madin, R.D. Brodeur, S.H.D. Haddock, A. Malej, G.D. Parry, E. Eriksen, J. Quiñones, M. Acha, M. Harvey, J.M. Arthur, W.M. Graham, Proc. Natl. Acad. Sci. USA 110, 1000 (2013)
R.M. May, Proc. Roy. Soc. Lond. B 228, 241 (1985)
N.N. Mangan, S.L. Brunton, J.L. Proctor, J.N. Kutz, I.E.E.E. Trans, Mol. Biol. Multi-Scale Commun. 2, 52 (2016)
Y.H. Kao, M. Eisenberg, Epidemics 25, 89 (2018)
W.J. Sutherland, R.P. Freckleton, H.C.J. Godfray, S.R. Beissinger, T. Benton, D.D. Cameron, Y. Carmel, D.A. Coomes, T. Coulson, M.C. Emmerson, R.S. Hails, G.C. Hays, D.J. Hodgson, M.J. Hutchings, D. Johnson, J.P.G. Jones, M.J. Keeling, H. Kokko, W.E. Kunin, X. Lambin, O.T. Lewis, Y. Mahli, N. Mieszkowska, E.J. Milner-Gullard, K. Norris, A.B. Phillimore, D.W. Purves, J.M. Reid, D.C. Reuman, K. Thompson, J.M.J. Travis, L.A. Turnbull, D.A. Wardle, T. Wiegand, J. Ecol. 101, 58 (2013)
K. Faust, L. Lahti, D. Gonze, W.M. de Vos, J. Raes, Curr. Opin. Microbiol. 25, 56 (2015)
C. Bandt, B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)
Y. Cao, W-w. Tung, J.B. Gao, V.A. Protopopescu, L.M. Hively, Phys. Rev. E 70, 046217 (2004)
M. Zanin, L. Zunino, O.A. Rosso, D. Papo, Entropy 14, 1553 (2012)
P.J. Mumbly, A. Hastings, H.J. Edwards, Nature 450, 98 (2007)
M. Scheffer, S.H. Hosper, M.-L. Meijer, B. Moss, E. Jeppesen, Trends Ecol. Evol. 8, 275 (1993)
M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkin, S.R. Carpenter, V. Dakos, H. Held, E.H. van Nes, M. Rietkerk, G. Sugihara, Nature 461, 53 (2009)
K. Meyer, A. Hoyer-Leitzel, S. Iams, I. Klusky, V. Lee, S. Ligtenberg, E. Bussmann, M.L. Zeeman, Nat. Sustain. 1, 671 (2018)
E.C. Lee, M.R. Kelly Jr., B.M. Ochocki, S.M. Akinwumi, K.E.S. Hamre, J.H. Tien, M.C. Eisenberg, J. Theor. Biol. 420, 68 (2017)
National Diabetes Statistics Report, 2017 (Centers for Disease Control and Prevention, U.S. Department of Health and Human Services, Atlanta, USA, 2017)
P. Palumbo, S. Ditlevsen, A. Bertuzzi, A. De Gaetano, Math. Biosci. 244, 69 (2013)
J. Li, Y. Kuang, SIAM J. Appl. Math. 67, 757 (2007)
S.M. Kissler, C. Cichowitz, S. Sankaranarayanan, D.M. Bortz, J. Theor. Biol. 359, 101 (2014)
P.S. Shabestari, S. Panahi, B. Hatef, S. Jafari, J.C. Sprott, Chaos. Soliton Fract. 112, 44 (2018)
F. Bray, J. Ferlay, I. Soerjomataram, R.L. Siegel, L.A. Torre, A. Jemal, CA-Cancer J. Clin. 68, 394 (2018)
H.M. Byrne, Nat. Rev. Cancer 10, 221 (2010)
R.A. Gatenby, E.T. Gawlinksi, Cancer Res. 56, 5745 (1996)
G.J. Kimmel, F.L. Locke, P.M. Altrock, Evolutionary dynamics of CAR T cell therapy (2019). https://doi.org/10.1101/717074
L.G. de Pillis, W. Gu, A.E. Radunskaya, J. Theor. Biol. 238, 841 (2006)
T. Alarcón, H.M. Byrne, P.K. Maini, J. Theor. Biol. 229, 395 (2004)
A.K. Barreiro, J.C. Bronski, Z. Rapti, SIAM J. Math. Anal. 51, 256 (2019)
K.J. Painter, T. Hillen, J. Theor. Biol. 323, 25 (2013)
A.M. Turing, Phil. Trans. Royal Soc. B 237, 37 (1952)
J.D. Murray, Sci. Am. 258, 80 (1988)
J. Zhu, Y.T. Yang, M.S. Alber, S.A. Newman, PLoS ONE 5, e10892 (2010)
K. Onimaru, L. Marcon, M. Musy, M. Tanaka, J. Sharpe, Nat. Commun. 7, 11582 (2016)
A. Badugu, C. Kraemer, P. Germann, D. Menshykau, D. Iber, Sci. Rep. 2, 991 (2012)
Z. Rapti, M.A. Duennes, S.A. Cameron, Biol. J. Linn. Soc. 113, 384 (2014)
M. Peyrard, A.R. Bishop, Phys. Rev. Lett. 62, 2755 (1989)
M. Peyrard, Nonlinearity 17, R1 (2004)
J. Cuevas-Maraver, P.G. Kevrekidis, in A Dynamical Perspective on the\(\phi ^4\)Model: Past, Present and Future, ed. by P.G. Kevrekidis, J. Cuevas-Maraver (Springer, Cham, 2019), p. 137
N.K. Voulgarakis, A. Redondo, A.R. Bihop, K.Ø. Rasmussen, Phys. Rev. Lett. 96, 248101 (2006)
L.V. Yakushevich, Nonlinear Physics of the DNA, 2nd edn. (Wiley, Hoboken, 2004)
T. Dauxois, M. Peyrard, A.R. Bishop, Phys. Rev. E 47, R44 (1993)
Z. Rapti, A. Smerzi, K.Ø. Rasmussen, A.R. Bishop, C.H. Choi, A. Usheva, Europhys. Lett. 74, 540 (2006)
Z. Rapti, A. Smerzi, K.Ø. Rasmussen, A.R. Bishop, C.H. Choi, A. Usheva, Phys. Rev. E 73, 051902 (2006)
Z. Rapti, K.Ø. Rasmussen, A.R. Bishop, J. Nonlinear Math. Phys. 18, 381 (2011)
C.H. Choi, Z. Rapti, V. Gelev, M.R. Hacker, B. Alexandrov, E.J. Park, J.S. Park, N. Horikoshi, A. Smerzi, K.Ø. Rasmussen, A.R. Bishop, A. Usheva, Biophys. J. 95, 597 (2008)
A. Krueger, E. Protozanova, M.D. Frank-Kamenetskii, Biophys. J. 90, 3091 (2006)
M.R. Kantorovitz, Z. Rapti, V. Gelev, A. Usheva, BMC Bioinform. 11, 604 (2010)
J. Cuevas-Maraver, P.G. Kevrekidis, D.E. Pelinovsky, Stud. Appl. Math. 137, 214 (2016)
Z. Rapti, Phys. Lett. A 377, 1543 (2013)
V. Koukouloyannis, P.G. Kevrekidis, J. Cuevas, V. Rothos, Phys. D 241, 16 (2013)
Z. Rapti, Eur. Phys. J. E 32, 209 (2010)
B.S. Alexandrov, V. Gelev, S.W. Yoo, L.B. Alexandrov, Y. Fukuyo, A.R. Bishop, K.Ø. Rasmussen, A. Usheva, Nucleic Acids Res. 38, 1790 (2010)
P. Maniadis, B.S. Alexandrov, A.R. Bishop, K.Ø. Rasmussen, Phys. Rev. E 83, 011904 (2011)
B. Blasius, L. Rudolf, G. Weithoff, U. Gaedke, G.F. Fussmann, Nature 577, 226 (2020)
Acknowledgements
We gratefully acknowledge support through grants NSF-DMS-1815764 and UIUC RB17060.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Rapti, Z. (2020). Nonlinearity and Biology. In: Kevrekidis, P., Cuevas-Maraver, J., Saxena, A. (eds) Emerging Frontiers in Nonlinear Science. Nonlinear Systems and Complexity, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44992-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-44992-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-44991-9
Online ISBN: 978-3-030-44992-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)