Chapter 2 Brief history of dynamical systems theory

2.1 Early development

Generalization of the tangent.

  • P. de Fermat (1607-1665)
  • I. Barrow (1630-1677)
  • I. Newton (1642-1727)
  • G. W. Leibnitz (1646-1716)
  • B. Taylor (1685-1731)
  • L. Euler (1707-1783) First to solve differential equation first order (1739)
  • J-L. Lagrange (1736-1813)
  • Riccati equation https://en.wikipedia.org/wiki/Riccati_equation

2.1.1 Limit concept

  • J. L. Cauchy (1789-1857),
  • K. T. W. Weierstrass (1815-1897) et
  • R. Lipschitz (1832-1903)

2.1.2 Using geometry

This was a big step forwards to find solutions even for complex system (using differential geometry). But things got a bit more tricky when chaos was discover.

2.1.3 Chaos

  • A. Turing (1912-1954)
  • E. Lorenz (1907-2008)
  • Li
  • Lyapunov

2.1.3.1 Structural stability and catastrophe theory

  • R. Thom (1923-2002)
  • E. C. Zeeman FRS (1925 – 2016)
  • A. Andronov (1901-1952)
  • L. Pontryagin (1908-1988).}

2.1.4 Foliation

https://en.wikipedia.org/wiki/Foliation

  • C. Ehresmann (1905-1979)
  • G. H. Reeb (1920-1993)

2.1.5 Ref - Math

  • Lyapunov AM (1992) The general problem of the stability of motion. International Journal of Control 55 :531–534.

  • Tao T, Vu V, Krishnapur M (2010) Random matrices : Universality of esds and the circular law. The Annals of Probability 38 :2023–2065.

  • Stoll A (1998) Tangente à une courbe : résoudre des problèmes par le nouvement. Repères - IREM 30 :95–109.

  • Lorenz EN (1963) Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20 :130–141.

  • Li Ty, Yorke JA (1975) Period three implies chaos. The American Mathematical Monthly 82 :985. 4

  • Logofet DO (2005) Stronger-than-lyapunov notions of matrix stability, or how “flowers” help solve problems in mathematical ecology. Linear Algebra and its Applications 398 :75–100.

  • Tao T, Vu VAN (1991) Random matrices : the circular law. pp 1–46.

2.2 Ref - Ecology

  • Volterra V (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118 :558–560.

  • Elton C, Nicholson M (1942) The ten-year cycle in numbers of the lynx in canada. The Journal of Animal Ecology 11 :215.

  • Gause G (1932) Experimental studies on the struggle for existence i. mixed population of two species of yeast. Journal of Experimental Biology 9 :389–402.

  • Gardner MR, Ashby WR (1970) Connectance of large dynamic (cybernetic) systems : Critical values for stability. Nature 228 :784–784.

  • Holling CS (1959) The components of predation as revealed by a study of small-mammal predation of the european pine sawfly.

  • May RM (1972) Will a large complex system be stable ? Nature 238 :413–414.

  • Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483 :205–8.

  • Rohr RP, Saavedra S, Bascompte J (2014) On the structural stability of mutualistic systems. Science 345 :1253497–1253497.

  • GF Fussmann et al. (2000), “Crossing the Hopf Bifurcation in a Live Predator-Prey System”

  • Costantino R, Desharnais R, Cushing J, Dennis B (1997) Chaotic dynamics in an insect population. Science (New York, N.Y.) 275 :389–91. 4

  • McCann KS (2000) The diversity-stability debate. Nature 405 :228–33. 9, 10, 13 31

  • Montoya JM, Woodward G, Emmerson MC, Solé RV (2009) Press perturbations and indirect effects in real food webs. Ecology 90 :2426–33. 10

  • Ives AR, Carpenter SR (2007) Stability and diversity of ecosystems. Science (New York, N.Y.) 317 :58–62. 10, 26

  • Bacaër N (2008) Histoire de mathématiques et de populations p 211.

  • Caswell H, Neubert MG (2005) Reactivity and transient dynamics of discrete-time ecological systems. Journal of Difference Equations and Applications 11 :295–310.

  • Pimm SL, Lawton JH (1978) On feeding on more than one trophic level. Nature 275 :542–544. 12

  • Yodzis P (1981) The stability of real ecosystems. Nature 289 :674–676

  • McCann K, Hastings A, Huxel GR (1998) Weak trophic interactions and the balance of nature. Nature 395 :794–798. 13 32

  • Berlow EL (1999) Strong effects of weak interactions in ecological communities. 13

  • Wootton JT (1994) The nature and consequences of indirect effects in ecological communi- ties. Annual Review of Ecology and Systematics 25 :443–466. 13

  • Rooney N, McCann K, Gellner G, Moore JC (2006) Structural asymmetry and the stability of diverse food webs. Nature 442 :265–9. 13

  • Schneider FD, Scheu S, Brose U (2012) Body mass constraints on feeding rates determine the consequences of predator loss. Ecology letters 15 :436–43. 13

  • Jackson JB, et al. (2001) Historical overfishing and the recent collapse of coastal ecosystems. Science (New York, N.Y.) 293 :629–37. 13

  • Estes Ja, et al. (2011) Trophic downgrading of planet earth. Science (New York, N.Y.) 333 :301–6.

  • Ripple WJ, Beschta RL (2003) Wolf reintroduction, predation risk, and cottonwood recovery in yellowstone national park. Forest Ecology and Management 184 :299–313.

  • Kéfi S, et al. (2007) Spatial vegetation patterns and imminent desertification in mediterranean arid ecosystems. Nature 449 :213–7.

  • Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483 :205–208.

  • Tang S, Pawar S, Allesina S (2014) Correlation between interaction strengths drives stability in large ecological networks supporting information organization of the supporting informa- tion. 14

  • Brown JH, Gillooly JF, Allen AP, Savage VM, West GB (2004) Toward a metabolic theory of ecology. Ecology 85 :1771–1789.

  • Thébault E, Fontaine C (2010) Stability of ecological communities and the architecture of mutualistic and trophic networks. Science (New York, N.Y.) 329 :853–6.

  • Saavedra S, Stouffer DB, Uzzi B, Bascompte J (2011) Strong contributors to network persistence are the most vulnerable to extinction. Nature 478 :233–5.

  • James A, Pitchford JW, Plank MJ (2012) Disentangling nestedness from models of ecological complexity. Nature 487 :227–30. 15

  • Saavedra S, Stouffer DB (2013) Disentangling nestedness" disentangled. Nature 500 :E1–2.

  • Kéfi S, et al. (2012) More than a meal. . . integrating non-feeding interactions into food webs. Ecology letters 15 :291–300.

  • Tang S, Allesina S (2014) Reactivity and stability of large ecosystems. Frontiers in Ecology and Evolution 2 :1–8.

  • Neubert MG, Caswell H (1997) Alternatives to resilience for measuring the responses of eco- logical systems to perturbations. Ecology 78 :653–665.

  • Bastolla U, et al. (2009) The architecture of mutualistic networks minimizes competition and increases biodiversity. Nature 458 :1018–20.

  • Säterberg T, Sellman S, Ebenman B (2013) High frequency of functional extinctions in ecolo- gical networks. Nature 499 :468–70.

  • Knight TM, McCoy MW, Chase JM, McCoy Ka, Holt RD (2005) Trophic cascades across ecosystems. Nature 437 :880–3.

  • Yoshida T, Jones LE, Ellner SP, Fussmann GF, Hairston NG (2003) Rapid evolution drives eco- logical dynamics in a predator-prey system. Nature 424 :303–6.