populations models: part i
TRANSCRIPT
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Populations Models: Part I
Phil. Pollett
UQ School of Maths and Physics
Mathematics Enrichment Seminars
26 August 2015
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 1 / 48
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Growth of yeast
0 2 4 6 8 10 12 14 16 180
100
200
300
400
500
600
700
Am
ount
of y
east
Time (hours)
Carlson, T. (1913) Uber Geschwindigkeit und Grosse der Hefevermehrung in Wurze. Bio-
chemische Zeitschrift 57, 313–334.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 2 / 48
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Growth of yeast
0 2 4 6 8 10 12 14 16 180
100
200
300
400
500
600
700
Am
ount
of y
east
Time (hours)
xt =665
1 + e4.16−0.531 t
Carlson, T. (1913) Uber Geschwindigkeit und Grosse der Hefevermehrung in Wurze. Bio-
chemische Zeitschrift 57, 313–334.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 3 / 48
![Page 4: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/4.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
Davidson, J. (1938) On the growth of the sheep population in Tasmania. Trans. Roy. Soc.
Sth. Austral. 62, 342–346.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 4 / 48
![Page 5: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/5.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
nt = 1670/(1 + e240.81−0.13125 t )
Davidson, J. (1938) On the growth of the sheep population in Tasmania. Trans. Roy. Soc.
Sth. Austral. 62, 342–346.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 5 / 48
![Page 6: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/6.jpg)
Growth of yeast
0 2 4 6 8 10 12 14 16 180
100
200
300
400
500
600
700
Am
ount
of y
east
Time (hours)
xt =665
1 + e4.16−0.531 t
Carlson, T. (1913) Uber Geschwindigkeit und Grosse der Hefevermehrung in Wurze. Bio-
chemische Zeitschrift 57, 313–334.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 6 / 48
![Page 7: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/7.jpg)
A deterministic model
dn
dt= nf (n).
The net growth rate per individual is a function of the population size n.
We want f (n) to be positive for small n and negative for large n.
Simply setf (n) = r − sn to give
dn
dt= n(r − sn).
This is the Verhulst model (or logistic model):
Verhulst, P.F. (1838) Notice sur la loi que la population suit dans son accroisement. Corr. Math.
et Phys. X, 113–121.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 7 / 48
![Page 8: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/8.jpg)
A deterministic model
dn
dt= nf (n).
The net growth rate per individual is a function of the population size n.
We want f (n) to be positive for small n and negative for large n. Simply setf (n) = r − sn to give
dn
dt= n(r − sn).
This is the Verhulst model (or logistic model):
Verhulst, P.F. (1838) Notice sur la loi que la population suit dans son accroisement. Corr. Math.
et Phys. X, 113–121.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 7 / 48
![Page 9: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/9.jpg)
A deterministic model
dn
dt= nf (n).
The net growth rate per individual is a function of the population size n.
We want f (n) to be positive for small n and negative for large n. Simply setf (n) = r − sn to give
dn
dt= n(r − sn).
This is the Verhulst model (or logistic model):
Verhulst, P.F. (1838) Notice sur la loi que la population suit dans son accroisement. Corr. Math.
et Phys. X, 113–121.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 7 / 48
![Page 10: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/10.jpg)
The Verhulst model
Pierre Francois Verhulst (1804–1849, Brussels, Belgium)
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 8 / 48
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The Verhulst model
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 9 / 48
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The Verhulst model
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 10 / 48
![Page 13: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/13.jpg)
The Verhulst model
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“We will give the name logistic to the curve” - Verhulst 1845
Verhulst, P.F. (1845) Recherches mathematiques sur la loi d’accroissement de la population.
Nouveaux memoires de l’Academie Royale des Sciences et Belles-Lettres de Bruxelles
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 12 / 48
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The Verhulst model
An alternative formulation has r being the growth rate with unlimited resources and Kbeing the “natural” population size (the carrying capacity). We put f (n) = r(1− n/K)giving
dn
dt= rn(1− n/K),
which is the original model with s = r/K .
Integration gives
nt =K
1 +(
K−n0n0
)e−rt
(t ≥ 0).
This formulation is due to Raymond Pearl:
Pearl, R. and Reed, L. (1920) On the rate of growth of population of the United States since
1790 and its mathematical representation. Proc. Nat. Academy Sci. 6, 275–288.
Pearl, R. (1925) The biology of population growth, Alfred A. Knopf, New York.
Pearl, R. (1927) The growth of populations. Quart. Rev. Biol. 2, 532–548.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 13 / 48
![Page 16: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/16.jpg)
The Verhulst model
An alternative formulation has r being the growth rate with unlimited resources and Kbeing the “natural” population size (the carrying capacity). We put f (n) = r(1− n/K)giving
dn
dt= rn(1− n/K),
which is the original model with s = r/K .
Integration gives
nt =K
1 +(
K−n0n0
)e−rt
(t ≥ 0).
This formulation is due to Raymond Pearl:
Pearl, R. and Reed, L. (1920) On the rate of growth of population of the United States since
1790 and its mathematical representation. Proc. Nat. Academy Sci. 6, 275–288.
Pearl, R. (1925) The biology of population growth, Alfred A. Knopf, New York.
Pearl, R. (1927) The growth of populations. Quart. Rev. Biol. 2, 532–548.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 13 / 48
![Page 17: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/17.jpg)
The Verhulst model
An alternative formulation has r being the growth rate with unlimited resources and Kbeing the “natural” population size (the carrying capacity). We put f (n) = r(1− n/K)giving
dn
dt= rn(1− n/K),
which is the original model with s = r/K .
Integration gives
nt =K
1 +(
K−n0n0
)e−rt
(t ≥ 0).
This formulation is due to Raymond Pearl:
Pearl, R. and Reed, L. (1920) On the rate of growth of population of the United States since
1790 and its mathematical representation. Proc. Nat. Academy Sci. 6, 275–288.
Pearl, R. (1925) The biology of population growth, Alfred A. Knopf, New York.
Pearl, R. (1927) The growth of populations. Quart. Rev. Biol. 2, 532–548.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 13 / 48
![Page 18: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/18.jpg)
Population growth in USA
Pearl, R. and Reed, L. (1920) On the rate of growth of population of the United States since
1790 and its mathematical representation. Proc. Nat. Academy Sci. 6, 275–288.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 14 / 48
![Page 19: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/19.jpg)
Verhulst-Pearl model
Raymond Pearl (1879–1940, Farmington, N.H., USA)
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Pearl was a “social drinker”
Pearl was widely known for his lust for life and his love of food, drink, music and parties.He was a key member of the Saturday Night Club. Prohibition made no dent in Pearl’sdrinking habits (which were legendary).
In 1926, his book, Alcohol and Longevity, demonstrated that drinking alcohol inmoderation is associated with greater longevity than either abstaining or drinking heavily.
Pearl, R. (1926) Alcohol and Longevity , Alfred A. Knopf, New York.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 16 / 48
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Pearl was a “social drinker”
Pearl was widely known for his lust for life and his love of food, drink, music and parties.He was a key member of the Saturday Night Club. Prohibition made no dent in Pearl’sdrinking habits (which were legendary).
In 1926, his book, Alcohol and Longevity, demonstrated that drinking alcohol inmoderation is associated with greater longevity than either abstaining or drinking heavily.
Pearl, R. (1926) Alcohol and Longevity , Alfred A. Knopf, New York.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 16 / 48
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Verhulst-Pearl model
0 200 400 600 800 1000 1200 1400 1600 1800 20000
500
1000
1500
2000
2500
3000Trajectories of the logistic model: K = 1670, r = 0.007
t
nt
nt =K
1 +(
K−n0
n0
)e−rt
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Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
nt = 1670/(1 + e240.81−0.13125 t )
Davidson, J. (1938) On the growth of the sheep population in Tasmania. Trans. Roy. Soc.
Sth. Austral. 62, 342–346.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 18 / 48
![Page 24: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/24.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 1940−400
−300
−200
−100
0
100
200
300
400
500
600
700Growth of Tasmanian sheep population from 1818 to 1936
Year
Sta
ndar
dize
d nu
mbe
r of
she
ep (
thou
sand
s)
(With the deterministic trajectory subtracted)
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 19 / 48
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A stochastic model
We really need to account for the variation observed.
A common approach to stochastic modelling in Applied Mathematics can be summarisedas follows:
“I suspect that the world is not deterministic - I should add some noise”
Zen Maxim (for survival in a modern university): Before you criticize someone, you should
walk a mile in their shoes. That way, when you criticize them, you’ll be a mile away and
you’ll have their shoes.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 20 / 48
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A stochastic model
We really need to account for the variation observed.
A common approach to stochastic modelling in Applied Mathematics can be summarisedas follows:
“I suspect that the world is not deterministic - I should add some noise”
Zen Maxim (for survival in a modern university): Before you criticize someone, you should
walk a mile in their shoes. That way, when you criticize them, you’ll be a mile away and
you’ll have their shoes.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 20 / 48
![Page 27: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/27.jpg)
Coin tossing (fair coin)
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1Coin tossing: N=100
Number of tosses
Pro
port
ion
of h
eads
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.2
0.4
0.6
0.8
1Coin tossing: N=1000
Number of tosses
Pro
port
ion
of h
eads
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Coin tossing (fair coin)
Let pt be the proportion of “Heads” after t tosses. Then,
pt =1
2+ something random.
In fact, the Central Limit Theorem, as applied to coin tossing (de Moivre ('1733)),shows that, as t →∞,
2√t
(pt −
1
2
)D→ Z ∼ N(0, 1).
(STAT1201: the normal approximation to the binomial distribution.)
So, it would not be completely unreasonable for us to write
pt =1
2+
1
2√tZ .
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 22 / 48
![Page 29: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/29.jpg)
Coin tossing (fair coin)
Let pt be the proportion of “Heads” after t tosses. Then,
pt =1
2+ something random.
In fact, the Central Limit Theorem, as applied to coin tossing (de Moivre ('1733)),shows that, as t →∞,
2√t
(pt −
1
2
)D→ Z ∼ N(0, 1).
(STAT1201: the normal approximation to the binomial distribution.)
So, it would not be completely unreasonable for us to write
pt =1
2+
1
2√tZ .
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 22 / 48
![Page 30: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/30.jpg)
Coin tossing (fair coin)
Let pt be the proportion of “Heads” after t tosses. Then,
pt =1
2+ something random.
In fact, the Central Limit Theorem, as applied to coin tossing (de Moivre ('1733)),shows that, as t →∞,
2√t
(pt −
1
2
)D→ Z ∼ N(0, 1).
(STAT1201: the normal approximation to the binomial distribution.)
So, it would not be completely unreasonable for us to write
pt =1
2+
1
2√tZ .
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 22 / 48
![Page 31: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/31.jpg)
A stochastic model
We really need to account for the variation observed.
A common approach to stochastic modelling in Applied Mathematics can be summarisedas follows:
“I suspect that the world is not deterministic - I should add some noise”
Zen Maxim (for survival in a modern university): Before you criticize someone, you should
walk a mile in their shoes. That way, when you criticize them, you’ll be a mile away and
you’ll have their shoes.
In our case,
nt =K
1 +(
K−n0n0
)e−rt
+ something random
or (much better)dn
dt= rn
(1− n
K
)+ σ × noise.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 23 / 48
![Page 32: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/32.jpg)
A stochastic model
We really need to account for the variation observed.
A common approach to stochastic modelling in Applied Mathematics can be summarisedas follows:
“I suspect that the world is not deterministic - I should add some noise”
Zen Maxim (for survival in a modern university): Before you criticize someone, you should
walk a mile in their shoes. That way, when you criticize them, you’ll be a mile away and
you’ll have their shoes.
In our case,
nt =K
1 +(
K−n0n0
)e−rt
+ something random
or (much better)dn
dt= rn
(1− n
K
)+ σ × noise.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 23 / 48
![Page 33: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/33.jpg)
Noise?
The usual model for “noise” is white noise (or pure Gaussian noise).
Imagine a random process (ξt , t ≥ 0) with ξt ∼ N(0, 1) for all t and ξt1 , . . . , ξtnindependent for all finite sequences of times t1, . . . , tn.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 24 / 48
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White noise
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3
−2
−1
0
1
2
3
t
ξ t
White noise on [0,1] sampled 1000 times
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Brownian motion
The white noise process (ξt , t ≥ 0) is loosely defined as the derivative of standardBrownian motion (Bt , t ≥ 0).
Brownian motion (or Wiener process) can be constructed by way of a random walk. Aparticle starts at 0 and takes small steps of size +∆ or −∆ with equal probabilityp = 1/2 after successive time steps of size h.
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Symmetric random walk: ∆ = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4
−2
0
2
4
6
8
10
12Random walk simulation: h = 0.01, ∆ = 1
t
Xt
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Symmetric random walk: ∆ = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40
−30
−20
−10
0
10
20
30
40
50Random walk simulation: h = 0.0001, ∆ = 1
t
Xt
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 28 / 48
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Brownian motion
The white noise process (ξt , t ≥ 0) is loosely defined as the derivative of standardBrownian motion (Bt , t ≥ 0).
Brownian motion (or Wiener process) can be constructed by way of a random walk. Aparticle starts at 0 and takes small steps of size +∆ or −∆ with equal probabilityp = 1/2 after successive time steps of size h.
If ∆ ∼√h, as h→ 0, then the limit process is standard Brownian motion.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 29 / 48
![Page 39: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/39.jpg)
Brownian motion
The white noise process (ξt , t ≥ 0) is loosely defined as the derivative of standardBrownian motion (Bt , t ≥ 0).
Brownian motion (or Wiener process) can be constructed by way of a random walk. Aparticle starts at 0 and takes small steps of size +∆ or −∆ with equal probabilityp = 1/2 after successive time steps of size h.
If ∆ ∼√h, as h→ 0, then the limit process is standard Brownian motion.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 29 / 48
![Page 40: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/40.jpg)
Symmetric random walk: ∆ =√h
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Random walk simulation: h = 2.5e-005, ∆ = 0.005
t
Xt
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 30 / 48
![Page 41: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/41.jpg)
Brownian motion
The white noise process (ξt , t ≥ 0) is loosely defined as the derivative of standardBrownian motion (Bt , t ≥ 0).
Brownian motion (or Wiener process) can be constructed by way of a random walk. Aparticle starts at 0 and takes small steps of size +∆ or −∆ with equal probabilityp = 1/2 after successive time steps of size h.
If ∆ ∼√h, as h→ 0, then the limit process is standard Brownian motion.
This construction permits us to write dBt = ξt√dt, with the interpretation that a change
in Bt in time dt is a Gaussian random variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt , dBs) = 0 (s 6= t).
[Recall that ξt ∼ N(0, 1) for all t and ξt1 , . . . , ξtn independent for all finite sequences oftimes t1, . . . , tn.]
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 31 / 48
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Brownian motion
The white noise process (ξt , t ≥ 0) is loosely defined as the derivative of standardBrownian motion (Bt , t ≥ 0).
Brownian motion (or Wiener process) can be constructed by way of a random walk. Aparticle starts at 0 and takes small steps of size +∆ or −∆ with equal probabilityp = 1/2 after successive time steps of size h.
If ∆ ∼√h, as h→ 0, then the limit process is standard Brownian motion.
This construction permits us to write dBt = ξt√dt, with the interpretation that a change
in Bt in time dt is a Gaussian random variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt , dBs) = 0 (s 6= t).
[Recall that ξt ∼ N(0, 1) for all t and ξt1 , . . . , ξtn independent for all finite sequences oftimes t1, . . . , tn.]
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 31 / 48
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Brownian motion
This construction permits us to write dBt = ξt√dt, with the interpretation that a change
in Bt in time dt is a Gaussian random variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt , dBs) = 0 (s 6= t).
The correct (modern) interpretation is by way of the Ito integral:
Bt =∫ t
0dBs =
∫ t
0ξs ds.
General Brownian motion (Wt , t ≥ 0), with drift µ and variance σ2, can be constructed
in the same way but with ∆ ∼ σ√h and p = 1
2
(1 + (µ/σ)
√h)
, and we may write
dWt = µ dt + σ dBt ,
with the interpretation that a change in Wt in time dt is a Gaussian random variable withE(dWt) = µdt, Var(dWt) = σ2dt and Cov(dWt , dWs) = 0.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 32 / 48
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Brownian motion
This construction permits us to write dBt = ξt√dt, with the interpretation that a change
in Bt in time dt is a Gaussian random variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt , dBs) = 0 (s 6= t).
The correct (modern) interpretation is by way of the Ito integral:
Bt =∫ t
0dBs =
∫ t
0ξs ds.
General Brownian motion (Wt , t ≥ 0), with drift µ and variance σ2, can be constructed
in the same way but with ∆ ∼ σ√h and p = 1
2
(1 + (µ/σ)
√h)
, and we may write
dWt = µ dt + σ dBt ,
with the interpretation that a change in Wt in time dt is a Gaussian random variable withE(dWt) = µdt, Var(dWt) = σ2dt and Cov(dWt , dWs) = 0.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 32 / 48
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Brownian motion
This construction permits us to write dBt = ξt√dt, with the interpretation that a change
in Bt in time dt is a Gaussian random variable with E(dBt) = 0, Var(dBt) = dt andCov(dBt , dBs) = 0 (s 6= t).
The correct (modern) interpretation is by way of the Ito integral:
Bt =∫ t
0dBs =
∫ t
0ξs ds.
General Brownian motion (Wt , t ≥ 0), with drift µ and variance σ2, can be constructed
in the same way but with ∆ ∼ σ√h and p = 1
2
(1 + (µ/σ)
√h)
, and we may write
dWt = µ dt + σ dBt ,
with the interpretation that a change in Wt in time dt is a Gaussian random variable withE(dWt) = µdt, Var(dWt) = σ2dt and Cov(dWt , dWs) = 0.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 32 / 48
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Brownian motion
dWt = µ dt + σ dBt ,
This stochastic differential equation (SDE) can be integrated to give Wt = µt + σBt .
It does not require an enormous leap of faith for us now to write down, and properlyinterpret, the SDE
dnt = rnt (1− nt/K) dt + σdBt
as a model for growth.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 33 / 48
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Brownian motion
dWt = µ dt + σ dBt ,
This stochastic differential equation (SDE) can be integrated to give Wt = µt + σBt .
It does not require an enormous leap of faith for us now to write down, and properlyinterpret, the SDE
dnt = rnt (1− nt/K) dt + σdBt
as a model for growth.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 33 / 48
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Adding noise
The idea (indeed the very idea of an SDE) can be traced back to Paul Langevin’s 1908paper “On the theory of Brownian Motion”:
Langevin, P. (1908) Sur la theorie du mouvement brownien. Comptes Rendus 146, 530–533.
He derived a “dynamic theory” of Brownian Motion three years after Einstein’s groundbreaking paper on Brownian Motion:
Einstein, A. (1905) On the movement of small particles suspended in stationary liquids required
by the molecular-kinetic theory of heat. Ann. Phys. 17, 549–560. [English translation by Anna
Beck in The Collected Papers of Albert Einstein, Princeton University Press, Princeton, USA,
1989, Vol. 2, pp. 123–134.]
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Langevin
Langevin introduced a “stochastic force” (his phrase “complementaryforce”–complimenting the viscous drag µ) pushing the Brownian particle around invelocity space (Einstein worked in configuration space).
In modern terminology, Langevin described the Brownian particle’s velocity as anOrnstein-Uhlenbeck (OU) process and its position as the time integral of its velocity,while Einstein described its position as a Wiener process.
The Langevin equation (for a particle of unit mass) is
dvt = −µvt dt + σdBt .
This is Newton’s law (−µv = Force = mv) plus noise. The solution to this SDE is theOU process.
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Langevin
Langevin introduced a “stochastic force” (his phrase “complementaryforce”–complimenting the viscous drag µ) pushing the Brownian particle around invelocity space (Einstein worked in configuration space).
In modern terminology, Langevin described the Brownian particle’s velocity as anOrnstein-Uhlenbeck (OU) process and its position as the time integral of its velocity,while Einstein described its position as a Wiener process.
The Langevin equation (for a particle of unit mass) is
dvt = −µvt dt + σdBt .
This is Newton’s law (−µv = Force = mv) plus noise. The solution to this SDE is theOU process.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 35 / 48
![Page 51: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/51.jpg)
Langevin
Langevin introduced a “stochastic force” (his phrase “complementaryforce”–complimenting the viscous drag µ) pushing the Brownian particle around invelocity space (Einstein worked in configuration space).
In modern terminology, Langevin described the Brownian particle’s velocity as anOrnstein-Uhlenbeck (OU) process and its position as the time integral of its velocity,while Einstein described its position as a Wiener process.
The Langevin equation (for a particle of unit mass) is
dvt = −µvt dt + σdBt .
This is Newton’s law (−µv = Force = mv) plus noise. The solution to this SDE is theOU process.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 35 / 48
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Langevin
Paul Langevin (1872 – 1946, Paris, France)
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Langevin
Einstein said of Langevin
“... It seems to me certain that he would have developed the special theory of relativity ifthat had not been done elsewhere, for he had clearly recognized the essential points.”
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Langevin was a dark horse
In 1910 he had an affair with Marie Curie.
The person on the right is Langevin’s PhD supervisor Pierre Curie.
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Langevin was a dark horse
In 1910 he had an affair with Marie Curie.
The person on the right is Langevin’s PhD supervisor Pierre Curie.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 38 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt .
Differentiation (Itocalculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt . Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt . Differentiation (Ito
calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt . Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt . Differentiation (Ito
calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt .
Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt . Differentiation (Ito
calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt . Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt . Differentiation (Ito
calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt . Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Solution to Langevin’s equation
To solve dvt = −µvt dt + σdBt , consider the process yt = vteµt . Differentiation (Ito
calculus!) gives dyt = eµtdvt + µeµtvtdt.
But, from Langevin’s equation we have that
eµtdvt = −µeµtvt dt + σeµtdBt ,
and hence that dyt = σeµtdBt . Integration gives
yt = y0 +∫ t
0σeµsdBs ,
and so (the Ornstein-Uhlenbeck process)
vt = v0e−µt +
∫ t
0σe−µ(t−s)dBs .
We can deduce much from this. For example, vt is a Gaussian process with
E(vt) = v0e−µt and Var(vt) = σ2
2µ(1− e−2µt), and
Cov(vt , vt+s) = Var(vt)e−µ|s|.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 39 / 48
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Where were we?
We had just added noise to our logistic model:
dnt = rnt(
1− ntK
)dt + σ dBt .
A simple numerical method for solving SDEs like this is the Euler-Maruyama method .
In Matlab . . .
n = n + r*n*(1-n/K)*h + sigma*sqrt(h)*randn;
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 40 / 48
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Where were we?
We had just added noise to our logistic model:
dnt = rnt(
1− ntK
)dt + σ dBt .
A simple numerical method for solving SDEs like this is the Euler-Maruyama method .
In Matlab . . .
n = n + r*n*(1-n/K)*h + sigma*sqrt(h)*randn;
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 40 / 48
![Page 64: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/64.jpg)
Where were we?
We had just added noise to our logistic model:
dnt = rnt(
1− ntK
)dt + σ dBt .
A simple numerical method for solving SDEs like this is the Euler-Maruyama method .
In Matlab . . .
n = n + r*n*(1-n/K)*h + sigma*sqrt(h)*randn;
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 40 / 48
![Page 65: Populations Models: Part I](https://reader031.vdocuments.mx/reader031/viewer/2022020916/61b1d39f2e813e32bc2ff75b/html5/thumbnails/65.jpg)
Sheep in Tasmania
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
Growth of Tasmanian sheep population from 1818 to 1936
Year
Num
ber
of s
heep
(th
ousa
nds)
nt = 1670/(1 + e240.81−0.13125 t )
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 41 / 48
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Solution to SDE (Run 1)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
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Solution to SDE (Run 2)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
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Solution to SDE (Run 3)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 44 / 48
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Solution to SDE (Run 4)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
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Solution to SDE (Run 5)
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Solution to SDE (one sample path)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 46 / 48
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Solution to SDE
1820 1840 1860 1880 1900 1920 19400
500
1000
1500
2000
t
nt
Mean path of SDE solution with ± 2 standard deviations (1000 runs)
dnt = rnt
(1 − nt
K
)dt + σdBt
K = 1670, r = 0.13125, σ = 90
n0 = 73, t0 = 1818
(Solution to the deterministic model is in green)
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 47 / 48
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Logistic model with noise
A significant problem with this approach (deterministic dynamics plus noise) is thatvariation is not, but should be, an integral component of the dynamics.
Arguably a better approach is to use a continuous-time Markov chain to model nt .
This will be dealt with in Part II or, if you prefer, STAT3004 “Probability Models &Stochastic Processes”.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 48 / 48
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Logistic model with noise
A significant problem with this approach (deterministic dynamics plus noise) is thatvariation is not, but should be, an integral component of the dynamics.
Arguably a better approach is to use a continuous-time Markov chain to model nt .
This will be dealt with in Part II or, if you prefer, STAT3004 “Probability Models &Stochastic Processes”.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 48 / 48
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Logistic model with noise
A significant problem with this approach (deterministic dynamics plus noise) is thatvariation is not, but should be, an integral component of the dynamics.
Arguably a better approach is to use a continuous-time Markov chain to model nt .
This will be dealt with in Part II or, if you prefer, STAT3004 “Probability Models &Stochastic Processes”.
Phil. Pollett (UQ School of Maths and Physics) Populations Models: Part I 48 / 48