Practical work 2 : STOCHASTICITY
I. Stochastic modelling applied to the Alpine marmot
I. 1. The biological model
The Alpine marmot Marmota marmota is a highly social grounddwelling rodent that lives in family groups in alpine meadows in the Alps. A family group is established on a territory and is typically composed of a dominant pair, of sexually mature subordinates, immature yearlings and juveniles (pups of the year). This species is socially monogamous and no apparent dimorphism is observable. The Alpine marmot hibernates from early October to the end of March. Then, the mating season is brief and lasts only 2 weeks. Females give birth after 30 days of gestation to an average of 3.8 pups. Pups emerge above ground around weaning after 40 days of lactation. All individuals have to accumulate fat reserves before the next hibernation.
I.2. Demographic parameters
The growing season is short, from April to late September, so females reproduce only once a year. Sexual maturity is obtained at two years old but individuals usually do not reproduce before being three years old. Only 2% of two years old females reproduce (r2 = 0.02). Among older females, the proportion of reproductive females is remarkably stable (r3 = 0.68) and does not depend on female’s age after two years old.
As noted above, females give birth to 3.8 pups in average and this litter size seems not to depend on female’s age. We will consider that the sexratio (proportion of males) at birth is equal to 0.5 (it is not really the case). So, a female that reproduces gives birth in average to
F2 = F3 = 3.8/2 females
The survival rates have been estimated by CaptureMarkRecapture (CMR) models. The mean survival rates over a period of 20 years of study are:
s0 = 0.53 (survival of pups to age 1)
s1 = 0.63 (survival of yearlings to age 2)
s2 = 0.58
s3 = 0.78 (survival of adults)
These survival rates do not differ according to sex in a given age class (as expected for a nondimorphic monogamous species).
I. 3. Life cycle
The life cycle is presented in the next figure:
The age class 0, 1, 2 and 3 correspond to pups, yearlings, two years old and older respectively
I. 4. Deterministic approach
The number of individuals (N_{i}) in each age class i is determined just before reproduction.

Wrote the corresponding Leslie matrix L and express the projection model

Use the software ‘ULM’ to identify the stable annual multiplication rate , the stable age structure and the stable reproductive values per age.

Give an interpretation of your results
I.5. Environmental stochasticity
The deterministic model supposes that the vital rates are constant over time. This simplifies the analysis but is not completely realistic. This supposition may in fact be relaxed by introducing for example environmental and/or demographic stochasticity. Let us start with environmental stochasticity. Different approaches are possible.

To be more realistic, we have in fact recognized 3 types of years: ‘good’, ‘average’ and ‘bad’ years. Each type of year occurs randomly and affects vital rates. Build a ‘.ulm’ file to analyse this type of environmental stochasticity on the annual multiplication rate . Interpret your results.

Modify the definition of the types of year and look at the effect on the demography of the population.

In a final step, we can consider that the vital rates are completely stochastic. In this case, the model can be rewritten as following:
N_{t+1 }= M_{t} x N_{t} where M_{t} is a projection matrix that differs at each time.
The stochastic annual multiplication rate can be found by simulation. Choose a vital rate and change its constant value in the file ‘marmot_det.ulm’ to a distribution. Compare with the asymptotic annual multiplication rate . Redo the simulation by changing another vital rate (the first one being kept then constant).

Build a ‘.ulm’ file containing stochastic distributions for all vital rates. Analyse the results.
I.6. Demographic stochasticity
The Alpine marmot has been reintroduced in the ‘Massif Central’, a mountain in the central part of France. The number of reintroduced individuals is quite low: 15 pairs of adults. This small population is then subject to demographic stochasticity. Build a ‘.ulm’ file to analyse the effects of demographic stochasticity on the annual multiplication rate . 