Transcript Logistic Equation dN/dt = rmN(K-N) K = rmN(1-N/K) What are the assumptions of the logistic equation? 1) population started with a stable age distribution.

Logistic Equation
dN/dt = rmN(K-N)
K
= rmN(1-N/K)
What are the assumptions of the logistic equation?
1) population started with a stable age distribution - already
discussed
2) density is measured equally - i.e. how is density measured? This
can be a problem when the biomass of different life stages
are different, also size/individual is different under different
conditions.
3) depressive effects of density on the rate of increase operates
instantaneously without time lags.
Time Lags
dN/dt = rmN (K - Nt-tau)
K
Time Lags
tau
Time Lags
Key point - oscillations can be intrinsic in the
system and not driven by external
environmental changes.
Nicholson’s Blowflies
Nicholson’s Blowflies
Nicholson’s Blowflies
Nicholson’s Blowflies
The recruitment of new individuals into the population was
determined at the egg-laying stage by the influence of food
supply on per capita fecundity. As a result, fluctuations were
significantly reduced.
The time lag in density-dependent response was reduced.
Daphnia
Daphnia
Population Projection
N0,t
N1,t
Population Vector
Nt =
N2,t
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Nn,t
Population Projection
The number of new organisms (age 0) at time t+1 will be
the contribution of fertility from all aged mothers.
N0,t+1 = foN0 + f1N1 + f2N2 + f3N3 + … + fkNk
The number of organisms age 1 at time t+1 = s0N0
The number of organisms age 2 at time t+1 = s1N1
The number of organisms age 3 at time t+1 = s2N2
f0
f1
f2
f3
f4
…… fk-1
fk
s0
0
0
0
0
…… 0
0
0
s1
0
0
0
…… 0
0
0
0
s3
0
0
…… 0
0
0
0
0
s4
0
…… 0
0
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0
0
0
0
0
sk-1
0
See example in class
MNt = Nt+1
M
Stage Model