Logistic Equation

dN/dt = rmN(K-N) = rmN(1-N/K)

K

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.

What are the assumptions of the logistic equation?

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

.

.

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

. . . . . . .

. . . . . . .

0 0 0 0 0 sk-1 0

See example in class

MNt = Nt+1

M

Stage Model