-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy path02-History.Rmd
228 lines (137 loc) · 7.88 KB
/
02-History.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
# Brief history of dynamical systems theory {#overview}
- http://www.scholarpedia.org/article/History_of_dynamical_systems
## 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
### Limit concept
- J. L. Cauchy (1789-1857),
- K. T. W. Weierstrass (1815-1897) et
- R. Lipschitz (1832-1903)
### Using geometry
- L. I. Fuchs (1833-1902)
- H. Poincaré (1854-1912),
- Stephen Smale https://en.wikipedia.org/wiki/Stephen_Smale (1930 -)
- Birkhoff
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.
### Chaos
- A. Turing (1912-1954)
- E. Lorenz (1907-2008)
- Li
- Lyapunov
#### Structural stability and catastrophe theory
- R. Thom (1923-2002)
- E. C. Zeeman FRS (1925 – 2016)
- A. Andronov (1901-1952)
- L. Pontryagin (1908-1988).}
### Foliation
https://en.wikipedia.org/wiki/Foliation
- C. Ehresmann (1905-1979)
- G. H. Reeb (1920-1993)
### 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.
## 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.
## Choose your computational tools
Among others:
- Julia: http://docs.juliadiffeq.org/latest/
- Octave: https://octave.org/doc/v4.0.3/Ordinary-Differential-Equations.html
- Python: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.ode.html
- Rust: https://docs.rs/GSL/0.4.26/rgsl/types/ordinary_differential_equations/index.html
- FORTRAN C/C++
### R
- R: https://cran.r-project.org/web/packages/deSolve/index.html https://www.r-pkg.org/badges/version/odeintr
### ODE with `desolve`
[deSolve](https://cran.r-project.org/web/packages/deSolve/index.html) package
is commonly used to solve ODE. I recommend you start by looking at the vignettes
- https://cran.r-project.org/web/packages/deSolve/vignettes/compiledCode.pdf
- https://cran.r-project.org/web/packages/deSolve/vignettes/deSolve.pdf