Local Population of Eritrichium caucasicum as an Object of Mathematical Modelling. II. How Short Does the Short-Lived Perennial Live?

Resumo

In the previous publication (Logofet et al., 2017), we reported on constructing a matrix model for a local population of Eritrichium caucasicum at high altitudes of north-western Caucasus. The model described the population structure according to the stages of ontogeny and field data for 6 years of observation. Calibrated from the data, the matrices, L(t), of stage-specific vital rates, which projected the population vector at time t (t = 2009, 2010, …, 2013) to the next year, were dependent on t and naturally different, reflecting indirectly the temporal differences in habitat conditions that occurred during the observations. Therefore, the model turned out to be non-autonomous. In addition to the range of variations in the adaptation measure λ₁(L), we also obtained certain “age traits from a stage-structured model,” such as the average stage duration and the life expectancy for each stage. Those traits were uniquely determined for each given matrix L by a known (from the English literature) VAMC (virtual absorbing Markov chain) technique, while their variations for different years t pointed out the need to solve a mathematical problem of finding the geometric mean (G) of five matrices L(t) with a fixed pattern. The problem has no exact solution, whereas the best approximate one (presented here) results in the estimate of life expectancy as 3.5 years and that of the mean age at first flowering as 12 years. Given the data of 6-year observations, the forecast of whether the local population increases/declines in the long term draws on the range of possible variations in the measure λ₁(G) under reproductive uncertainty, and this range localizes entirely to the right of 1, though very close to λ₁ = 1 meaning a stable population.