4

We have the formule for a model with logistic growth:

$$ N_t = N_{t-1} + g\, N_{t-1}\left( 1 -\dfrac{N_{t-1}}{K}\right)$$

where $g$ defines the growth rate and $K$ is the carrying capacity.

Let's say we have the following data:

$N_0 = 10$, $N_1 = 18$, $N_2 = 29$, $N_3 = 47$, $N_4 = 71$, $N_5 = 119$, $N_6 = 175$, $N_7 = 257$, $N_8 = 351$, $N_9 = 441$, $N_{10} = 513$, $N_{11} = 560$, $N_{12} = 595$ ,$N_{13} = 630$ ,$N_{14} = 641$ ,$N_{15} = 651$ ,$N_{16} = 656$ ,$N_{17} = 660$ ,$N_{18} = 662$.

How can we get a good approximation for what $g$ should be using this data? I just tried to fill in 2 consecutive points but it obviously didn't work out because the function isn't linear and you get a different $g$ between every 2 consecutive points. So how should I approach this problem?

Davide Giraudo
  • 172,925
iAm90offFrom90
  • 55
  • Or do you have to fill all of the points in and find the average $g$? That would seem inefficient to me.. – iAm90offFrom90 Jun 19 '13 at 19:18
  • You may also try an overall regression for example your data against: $N_{t+1}/N_t=a-b N_t$; this is just one possiblity perhaps, then apply your data and try to regress, then see the deviation and statistical quality of $g$. This might resemble what they did. Honestly I am not a friend of this approaches since models should really be able to describe dynamic behaviour and if they dont, so they dont. – al-Hwarizmi Jun 19 '13 at 19:35

1 Answers1

0

If $g$ is presumed to be independent of $N$ then your data as such does not fit a logistic progression over $N$ for $0 \leq t \leq 18$ (results in contradiction). It would fulfil certain segments probably where the equation can be solved for constant $g$ and $K$.

For example:

$$18=10 a - 100 b$$ $$29=18 a - 18^2 b$$

gives certain solution for $a=1+g$ and $b=g/k$.

So what you did is correct but the $g$ seems not be constant over the whole bandwidth $N$ for $0 \leq t \leq 18$.

What you could do instead is to test stepwise and find $g$ for each progression and possibly apply a regression that gives certain approxm. relation between $N \rightarrow g$ in other words $g$ as function of $N$.

al-Hwarizmi
  • 4,290
  • 2
  • 19
  • 36
  • The answer model says that with $N_0 = 9.6$ (you had to read it off a graph) and $g = 0.6$ you have quite a good model. How did they get this number though? – iAm90offFrom90 Jun 19 '13 at 19:27
  • see comment above! – al-Hwarizmi Jun 19 '13 at 19:37
  • One more tiny note, commonly the above is called the "logistic map" and the corresponding "logistic equation" is the differential see for both terms on wiki with some examples. – al-Hwarizmi Jun 19 '13 at 19:47