mkl@nec-labs.com
PRE 061901


Dear Dr. Kloster,

We are grateful for the interest in our article and thank you for your feedback.

Your argument is based on the assumption of non-interacting species. In such a case the ratio $f(k)/d(k)$ would indeed determine the stead-state abundance of species. However, our species are interacting. In particular the growth rate is not given by $f(k)$ but it is modified by the environmental capacity term that includes densities of other species (i.e., interactions). But one cannot exclude that interactions are is some cases not so important and your argument is approximately correct.


>In the second case, d(k_i)=(1-\rho_0)^{-k_i}, the death rate for k_i 
>close to one is sharply increased. The optimal k_i is roughly the k_i 
>for which f(k_i)/d(k_i) is maximal, which in this case is 
>-1/\log(1-\rho_0). Judging from figure 5, I expect that \rho_0 in this 
>case was around 1-e^{-2}.


Our simulations give $\rho_0 = 0.88...$ which is very close to $1-e^{-2}=0.86...$.


>In the first case, d(k_i)=\sqrt{k_i}, the death rate is always higher 
>than f(k_i) except at the boundaries 0 and 1, and the ratio 
>f(k_i)/d(k_i) has a clear maximum at k_i=1. From this one would expect a 
>single maximum in the distribution. However, for small k_i, both f(k_i) 
>and d(k_i) are very small, thus the dynamics for these species are very 
>slow. In particular, the time required for a species to die diverges as 
>1/d(k_i) = k_i^{-1/2}. Thus, the peak for small k_i does not indicate 
>two types of species living happily together; rather, I believe your 
>model is littered with slowly dying species with small k_i.


We did a study -- that was not reported in our article -- of the lifetime as a function of $k$ and the curves for the lifetime as a function of $k$ and for the $k$-probability density look quite the same for different models. So numerical results confirm that for small $k$'s the lifetime increases.

The peak for small values of $k$ does indeed not show the existence of a second species in the sense that there is no maximum of $f(k_i)/d(k_i)$ (which would be a definition of a species in a non-interacting point of view). We ran a much longer simulation for Fig.5 with $f(k)=k$ and $d(k)=\sqrt(k)$ with a smaller $k$ resolution. The $k=0$ peak is stable and clearly persists (see attached figure). This indicates the relevance of interactions in our model. If your argument would be valid, only the $k=1$ peak would survive and interactions would be negligible in the steady-state. On the other hand we observe that the small-$k$ species are renewed by the dynamics, due to interaction with the environmental capacity. It thus shows the existence of another species ($k$ small) living on a long timescale. So this situation may be considered as two types of species living together.

In any case we missed in our paper a similar discussion. Considering the ratio $f(k_i)/d(k_i)$ as a rough criterion combined with the study of the lifetime as a function of $k$ already provides analytical criterions and an interesting interpretation of the peaks. We are grateful to you for having pointed this to us.

Sincerely yours,
Francois Coppex.
