Population Lifespan Is Governed by Mathematical Laws
Researchers at HSE University and MSU have established a universal law governing the time to extinction of a population in a random environment. Their analysis of the evolution of branching processes—complex probabilistic systems—shows that, regardless of the initial population size, extinction follows strict mathematical laws. The results have been published in the Journal of Applied Probability.
Bisexual branching processes are mathematical models in probability theory that describe the evolution of populations in which reproduction requires the formation of male–female pairs. In these processes, each family randomly produces a number of male and female offspring, who then form new pairs and reproduce in subsequent generations. Such models are used to study, for example, species population dynamics and the spread of genetic mutations.
Researchers Anton Zhiyanov from HSE University and Alexander Shklyaev from Moscow State University used mathematical methods to estimate the lifespan of a population under the assumption that each family produces, on average, one new family.
Anton Zhiyanov
'We considered a scenario in which the number of offspring produced by each couple is random, but the average number of offspring pairs generated by each family is one. In other words, each couple produces children who go on to form, on average, exactly one new couple, meaning that one family is effectively replaced by one new family. At first glance, such a population might be expected to survive indefinitely and remain stable. However, our results show that this is not the case,' says study co-author Anton Zhiyanov, Research Fellow at the Laboratory of Molecular Physiology of the HSE Faculty of Biology and Biotechnology.
The researchers found that such populations are inevitably doomed to extinction, and that the expected time to extinction grows in proportion to the square of the logarithm of the initial population size.
'With a thousandfold increase in population size—for example, from one thousand to one million individuals—the lifespan increases only about fourfold. This is known as logarithmic dependence. Thus, although the number of individuals and their rapid early growth may seem significant, they eventually have little effect on the overall lifespan of the population,' Zhiyanov explains.
The authors tested the model under different mechanisms of pair formation, including both monogamous and polygamous systems. The resulting law proved to be universal and holds in all cases. According to the researchers, this suggests that the overall dynamics of the system are governed not by specific model details, but by underlying probabilistic principles.
Understanding the evolution of branching processes helps predict population stability, inform demographic policy, estimate the duration of epidemics, and model the behaviour of complex random systems across a wide range of fields, from biology to economics.
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