Growth and Decline
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Transcript of Growth and Decline
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Exponential growth
pop. size
at time
t+t
=pop. size
at time t+
growth
increment
N(t+ t) = N(t) + N
Hypothesis: N = r N t
r - rate constant of growth
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Differential equation for exponential growth
rNdt
dN
)exp()( 0 rtNtN
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Exponential growth in discrete time
Nt+1= Nt + r Nt
Nt+1= (1+r) Nt
Nt= (1+r)t N0
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Exponential decline
rNdt
dN
)exp()( 0 rtNtN
r - mortality rate
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0 2 4 6 8 10 12 14 16 18 20
0
20
40
60
80
100
120
Timet
N(t)
Exponential decline r=0.1
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Limited growth
Factors that affect population dynamics
reproduction (growth rate)
mortality
environmental capacity
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Monomolecular model for limited
growthFirst order chemical reaction: A P
Areactant, Pproduct, R(t)reactant concentration
kreaction rate
kRdt
dR - Exponential decay
C(t)product concentration
)( CAkdt
dC A = R(0)
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0 1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
50
Conc
entrations
time
Product C(t)=A(1-exp(-kt))
Reactant R(t)=A exp(-kt)
Monomolecular growth
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Logistic growth model
Relies on the hypothesis that population
growth is limited by environmental capacity
K
NrN
dt
dN1
Kenvironmental capacity
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)exp(11
)(
0
rtNK
KtN
0 2 4 6 8 10 12 14 16 18 200
50
100
150
time
N(t)
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Logistic growth with time delay
Factor that limits growth acts after some time TD
No analytical solution
K
TtNtrN
dt
tdND)(
1)()(
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600
)(1)(5.0
)(DTtN
tNdt
tdN
0 10 20 30 40 50 60 70 80 90 1000
200
400
600
800
1000
1200
1400
1600
1800
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Discrete logistic model
K
iNiRNiN )(1)()1(
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Growth of individual organisms
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Von Bertalanffys model
Postulates:
Gain in weight is proportional to the surface
area of the organism
Loss in weight is proportional to the weight
of the organism
Organism maintain the same shape while
growing
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Von Bertalanffys model
CWHSdt
dW
Ssurface area
Wweight
L - length
H,C - parameters3
32
2 , LaWLaS
)( max LLk
dt
dL
)exp(1)(
max
0max kt
L
LLtL
(monomolecular growth)
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Richards family of models
1)1(
1
max
m
W
W
m
kW
dt
dW
mktAWtW 11
exp1)( max
Has all of previous models as special cases
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Allometric growth
Allometrystudy of relative sizes of different
parts of organisms X, Y
Hypothesis:dt
dX
Xb
dt
dY
Y
11
bAXY
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Computations
Matlab script files and functions
Simulink block diagrams
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Computations
Matlab functions:
exp(x) - exponential
plot(x,y) - plot
ode45compute solution to ODE
X=A\B - least squares (help slash)
fmins - minimize function over arguments