Transcript Exponential growth or decline

Growth and decline
Exponential growth
pop. size
at time
t+t
N(t+ t)
=
pop. size
at time t
+
growth
increment
=
N(t)
+
N
Hypothesis:  N = r N t
r - rate constant of growth
Differential equation for exponential growth
dN
 rN
dt
N (t )  N0 exp(rt )
Exponential growth r=0.1
10
9
8
7
N(t)
6
5
4
3
2
1
0
0
2
4
6
8
10
12
Time - t
14
16
18
20
Exponential growth in discrete time
Nt+1 = Nt + r Nt
Nt+1 = (1+r) Nt
Nt = (1+r)t N0
Exponential decline
dN
  rN
dt
N (t )  N0 exp(rt )
r - mortality rate
Exponential decline r=0.1
120
100
80
N(t)
60
40
20
0
0
2
4
6
8
10
12
Time – t
14
16
18
20
Limited growth
Factors that affect population dynamics
• reproduction (growth rate)
• mortality
• environmental capacity
Monomolecular model for limited
growth
First order chemical reaction: A  P
A – reactant, P – product, R(t) – reactant concentration
k – reaction rate
dR
  kR
dt
- Exponential decay
C(t) – product concentration
dC
 k ( A  C)
dt
A = R(0)
Monomolecular growth
50
45
40
Product C(t)=A(1-exp(-kt))
Concentrations
35
30
25
20
15
Reactant R(t)=A exp(-kt)
10
5
0
0
1
2
3
4
5
time
6
7
8
9
10
Logistic growth model
Relies on the hypothesis that population
growth is limited by environmental capacity
dN
 N
 rN 1  
dt
 K
K – environmental capacity
K
N (t ) 
 K

1  
 1 exp(rt )
 N0 
150
100
N(t)
50
time
0
0
2
4
6
8
10
12
14
16
18
20
Logistic growth with time delay
dN(t )
 N (t  TD ) 
 rN (t ) 1 

dt
K


Factor that limits growth acts after some time TD
No analytical solution
1800
1600
1400
1200
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
80
90
100
dN (t )
 N (t  TD ) 
 0.5 N (t ) 1 

dt
600


Discrete logistic model
 N (i ) 
N (i  1)  RN (i) 1 

K


Growth of individual organisms
Von Bertalanffy’s 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
Von Bertalanffy’s model
dW
 HS  CW
dt
S  a2 L , W  a3 L
2
dL
 k ( Lmax  L)
dt
S – surface area
W – weight
L - length
H,C - parameters
3
(monomolecular growth)


L0
L(t )  Lmax 1 
exp(kt) 
 Lmax

Richards’ family of models
1 m


dW
kW  Wmax 

  1

dt (1  m)  W 

W (t )  Wmax 1  A exp kt 
1
1m
Has all of previous models as special cases
Allometric growth
Allometry – study of relative sizes of different
parts of organisms X, Y
Hypothesis:
1 dY
1 dX
b
Y dt
X dt
Y  AX b
Computations
Matlab script files and functions
Simulink block diagrams
Computations
Matlab functions:
exp(x) - exponential
plot(x,y) - plot
ode45 – compute solution to ODE
X=A\B - least squares (help slash)
fmins - minimize function over arguments