*Ecological Archives* A025-009-A3

Alexandre Robert, Stéphane Chantepie, Samuel Pavard, François Sarrazin, and Céline Teplitsky. 2015. Actuarial senescence can increase the risk of extinction of mammal populations. *Ecological Applications* 25:116–124. http://dx.doi.org/10.1890/14-0221.1

Appendix C. Relationship between the equilibrium number of adult individuals (*K*) and the maximal number of mating events (ME_{max}).

Carrying capacities (*K*) were expressed in terms of maximal number of mature individuals throughout the article. However, in stochastic models, regulation was mechanistically implemented using a maximal number of mating events (*MEmax*). Such limitation may result from either a limited number of reproduction sites or a critical density of individuals. We determined the relationship between ME_{max} and *K* through linear regressions. Preliminary analysis indicated that (1) the relationship between ME_{max} and *K* varies with the life cycle considered (the relationship was thus different between the 58 species considered and between the *Senecence Model* and *Null Models* for each species, we thus performed distinct regressions for each of these scenarios); (2) in all populations with λ>1, when including a maximal number of mating events (ME_{max}≤500), pseudo-equilibria were reached rapidly (less than 100 time steps in all cases). We thus considered a similar time horizon of 100 time steps to determine the relationship between ME_{max} and *K *for all populations.

In each case, 500 population trajectories were run for 20 values of ME_{max} randomly drawn from the interval [10, 500]. The average number of mature individuals (=*K*) obtained after 100 time steps was recorded. ME_{max} was linearly regressed over *K* for these 20 values and the slope and intercept of regression were used to quantify the relationship between ME_{max} and *K*. The operation was repeated for the 232 scenarios (58 species, with one *Senescence* and three alternative *Null models* for each species). The linearity of regressions was checked by adding a quadratic component and comparing it to the linear model. In all cases, the quadratic component did not significantly improve model fit and *R*² was >0.99 in the simple linear regression.

For life cycles (either *Senescence* or *Null Models*) for which λ<1, the relationship between ME_{max} and *K* was impossible to assess due to the absence of non-zero pseudo equilibrium. In these cases, we assumed a number of nests equal to *x***K* (where *x* was the median ratio ME_{max}/*K* obtained with life cycles for which λ>1).