Hi Marcos,

This is strongly dpending on your system (plant or animal and type e.g. tree or herbaceous species, etc...), I think and there is likely no general answer.

Thanks, Patrick:

the question is "open" since the model I am dealing with is quite general. It combines individual dispersal with demography (described by a density dependent Leslie matrix). I know very little about density-dependent fertility but, which other density-dependent function could make sense?

Dear Marcos

I don’t know how far the suggestion would be useful to you, but in human demographic literature, where fertility is obliviously age-dependent (and restricted within a limited age-range (usually between the ages of 15-49 years) researchers use various functions including the Gaussian distribution to mimic the shape of female fertility. Other common functions that capture the shape of human fertility include: Hadwiger distribution, the Pearson Type 1 curve and Type III curves, or Beta and Gamma distributions. If this is what you are looking for, you may find this paper useful to you (T. Chandola, D. A. Coleman and R. W. Hiorns, 1999. “Recent European Fertility Patterns: Fitting Curves to 'Distorted' Distributions”, Population Studies, Vol. 53, No. 3, pp. 317-329.

Good luck

Dear Yohannes;

many thanks for your comment. We deal with an age structured population but, at this moment, we consider just reproductive and non-reproductive age classes, so that we can not incorporate the age-dependent fertility structure. But I think it is interesting for a future work.

Thanks!

Dear all;

regarding the original question, I want to share with you a couple of references

\bibitem{Ugarcovici04} I.~Ugarcovici, H.~Weiss.

\emph{Chaotic dynamics of a nonlinear density dependent

population model}. Nonlinearity 17 (2004) pp. 1689--1711.

\bibitem{Ugarcovici07} I.~Ugarcovici, H.~Weiss.

\emph{Chaotic attractors and physical measures for some density dependent Leslie population models}. Nonlinearity 20 (2007) pp. 2897--2906.

Best

Dear Marcos,

I think that you deal with Leslie matrix as defined here: https://en.wikipedia.org/wiki/Leslie_matrix . What do you understand as density? I got used to understand density as population density, like people per sq.km in spatial models. But then, if space is finite, density increases with the total population stock, and this can influence fertility, etc.

It was correctly mentioned that for human population fertility is age-specific, i.e., it varies for different age cohorts, that enter the formula in wiki as vector n_i. The dependence of fertility on age is likely to be one-peaked, probably at thirty (plus minus) age, depending on culture.

Dear Yuri;

thanks for your answer. I wanted to mean "the total population size" but,  as the environment size is implicitly considerd constant, I would say that a variation in the total population size is equivalent (proportional) to a variation of the density.

Dear Marcos,

then do you mean how fertility will depend on population density? I think that both cases - decreasing function and with one peak - are possible.

I have a couple of papers that study economic impact of population density. One of them proves that there exist an optimal density for certain conditions, that gives the highest economic growth. I have only abstract of it in my RG page, 

"Socio-economic influences of population density" in Chinese Business Review (2009), but can send you this article if you are interested and give me your email.