Local population of Eritrichium caucasicum as an object of mathematical modelling. II. How short does the short-lived perennial live?

Volume 78, N 1. 2017 pp. 56–66

D. O. Logofet1, E. S. Kazantseva2,I. N. Belova1, V. G. Onipchenko2

1Institute of Atmospheric Physics, RAS, Laboratory of Mathematical Ecology
119017 Moscow, Pyzhevsky Lane, 3
e-mail: danilal@postman.ru, iya@ifaran.ru
2M.V. Lomonosov Moscow State University, Biological Faculty,
Department of Geobotany
119234 Moscow, Leninskie Gory, 1, Bldg. 12
e-mail: biolenok@mail.ru, vonipchenko@mail.ru

In the previous publication (Logofet et al., 2016), we reported on constructing a matrix model of the Eritrichium caucasicum coenopopulation at high altitudes of north-western Caucasus. The model reflected 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. The model therefore turned out to be non-autonomous. In addition to the range of variations in the adaptation measure λ 1 (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 (in 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 average life expectancy estimate at 3.5 years for plants of the given species and the mean age of first flowering at 12 years. Given the data of 6-year observations, the forecast of whether the coenopopulation increases/declines in the long term draws on the range of potential variations in the measure λ 1 (G ) under reproductive uncertainty, and this range has localized entirely to the ricrht of 1 though very close to λ 1 = 1 meaning a stable population.

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