We introduce a class of stochastic models called population-size-dependent birth-and-death processes and describe their general evolutionary behaviours. We then further explain the notion of carrying capacity of such processes, that is, the population size where the process fluctuates around. Finally, we highlight our interest in the distribution of the total progeny size at extinction of such processes as one of our main research aspects.
The evolution of species in nature tends to be influenced by many factors, including the availability of food, habitat, nesting sites, and so on. Consider a small population of a particular species living in a safe habitat with plenty of food. The suitability of the environment tends to push the population to grow larger, until there is too much competition for limited resources to enable individuals to survive. From this example, we can see that the current size of a population can influence its future evolution.
To model the evolution of a population with limited resources, we use stochastic models called population-size-dependent birth-and-death processes. The evolution of such processes is governed by their birth and death rates, which both depend on the current size of the population. To match most evolutionary trends in nature, the birth rate tends to be larger than the death rate when the population size is small (that is, when there is no competition for resources), which gives the population a chance to grow. However, as the population size increases, the birth rate tends to decrease or the death rate tends to increase, and eventually, the death rate becomes larger than the birth rate, which prevents the process to grow to infinity.
These processes tend to fluctuate around what is known as the carrying capacity, which is the population size at which the birth rate equals the death rate. A particular property of the process is that the population tends to fluctuate around the carrying capacity: once the population size becomes greater than the carrying capacity, the process tends to undergo more death events than birth events, until the population is pulled down to lower population levels; and, conversely, once the population size decreases under the carrying capacity, the process tends to experience more birth events than death events, until the population is pulled up to large population levels again. Due stochastic events, the population will eventually become extinct, that is, reach population size zero.
A major part of the research project was to determine the distribution of the total progeny size at the time of extinction of such processes, that is, the total cumulative number of children produced until extinction of the population. Instead of using the empirical approach of simulating a large number of population trajectories, we modified algorithmic techniques that are designed for a more general type of Markov processes called quasi-birth-and-death processes. The original version of the algorithms could only be applied to processes with a known maximum population size and a known maximum total progeny size, which is not the case in the processes we are interested in. Therefore, in this project, we adapted the algorithms to produce the desired distributions in a computationally efficient way.
The University of Melbourne