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How to fit a logistic discrete function!

2 messages · Alexandre F. Souza, Frank Berninger

Original
Alexandre F. Souza
# 
Dear friends,

Sorry for the somewhat lengthy message, but I would really like to hear
your opinion on this.

I am organizing an undergraduation population ecology course and would like
to work the logistic equation as proposed by Bellows 1981 and presented by
Begon et al. 2006 in their Ecology textbook:

Nt+1 =(Nt*R)/((1+a*Nt)^b)

Where R is lambda, the discrete growth rate (R = Nt+1/Nt), and a and b are
constants.

The beauty of this equation, in contrast to other more traditional
formulatons, is because it bypasses the support capacity K, which is hard
to know in advance, and makes it a function of a and R as K = (R - 1)/a. So
K becomes a consequence of population properties and not an unrealistic
prerequisite. Furthermore, a is meaningful in its own right, measuring the
per capita susceptibility to crowding: the larger the value of a, the
greater the effect of density on the actual rate of increase in the
population. Finally, b is thought to portray undercompensation (b < 1),
perfect compensation (b = 1), scramble-like overcompensation (b > 1) or
even density independence (b = 0).

However, because the the equation is an iterative one, I am having a hard
time figuring out how to fit it to an empirical dataset made by a time
series of population counts. I believe this is key to make it more concrete
for the students, to show how to make it practical and not merely a
theoretical construct.

I could not find this equation used much often, and could not find an
example of how fitting it.

Does anyone have any suggestions?

Thank your very much in advance,

Alexandre
Frank Berninger
# 
Actually an interesting question. I think it depends on the data you have.

If you have single survey of your population (e.g a survey where you
have population sizes and ages, each population is measured one time)
the only method would ordinarly least square (e.g. the nls function).

If you have populations measured several times you might look at a range
of options usually going to non-linear mixed model (best is probably the
nlme package). Option one would be to fit population size to age using
nlme and then introduce the identity of the population as random effect
and putting possibly autoregressive function as correlation into the
function.

Another strategy would be to estimate growth rate (instead of population
size) by derivating your function. This might help you to get rid of
your autoregressive terms but might also cause other problems (less
reliable parameters etc).

Both ways are not trivial you would expect problems in convergence and
you might have to simplify the random effect structure to be able to
estimate the model.
On 01/10/2016 03:13 AM, Alexandre F. Souza wrote: