Matrix population models applied to viability analysis and conservation: Theory and practice using the ULM software

We outline a general method to carry out population viability analyses (PVA) by making we of matrix population models. We consider a structured population (by age, sex, reproductive status space, etc.) whose demographic parameters are known from field study. To assess the extinction risk and definite a management program, we advocate a three-step PVA: (1) Setting up a constant matrix model that includes the mean values of demographic rates. The sizes of each population class are linked from one year to the next by a transition matrix that contains all vital rates. When these parameters are taken to be constant (fixed to their mean), the matrix analysis yields the determine population growth rate, population structure, stage-specific reproductive values and the sensitivities of the growth rate to variations in demographic rate. (2) Assessing the extinction risk due to stochastic factors: demographic stochasticity, environmental stochasticity and catastrophes. We show how to compute the stochastic growth rate, extinction probabilities and the distribution of time to extinction, from computations based on the constant matrix model (step 1) together with Monte-Carlo simulations. (3) Determining action on demographic parameters and amelioration of monitoring programs. The extinction risk can be reduced by increasing the population growth rate, decreasing its temporal variability or boosting current population size. Which parameters should be fine-tuned in order to cause the largest increase in population growth can be found out by computing the growth rate elasticities to demographic rates. Furthermore, variance in population growth can be decomposed into the produced by mean parameter values, and that produced by fluctuations in parameters. Finally, reproductive values and their sensitivities indicate which classes should be reinforced to obtain a long-lasting raise of population size. The ULM software allows one to apply this agenda automatically to any particular case study. The software can be conveniently used to model populations with an kind of life cycle. The user will enter the model by making use of a friendly, simplified programming language that leaves him or her entirely free to decide of the matrix structure, parameter values and factors of parameter variations (stochastic factors, density-dependence...). All PVA-related parameters mentioned above (growth rate, sensitivities, elasticities, extinction probabilities, distribution of extinction time, etc.) are computed by the software. Here this is illustrated with an overview of two case studies, that of a natural, declining population of snakes (Vipera ursinii ursinii) and that of a reintroduced, growing population of raptors (Gyps fulvus fulvus).