An approximation of populations on a habitat with large carrying capacity.

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton-Watson process from the very beginning.

On the establishment of a mutant.

How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147-162, 2011. https://doi.org/10.1080/17513758.2010.506041) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident populations, with sizes proportional to a large carrying capacity K. Actually, we assume carrying capacities to be [Formula: see text] and [Formula: see text] for the resident and the mutant populations, respectively, and study the dynamics for [Formula: see text]. We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to [Formula: see text]. We introduce the time of establishment through the asymptotic behaviour of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed [Formula: see text], where [Formula: see text] is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times [Formula: see text], we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at a random point. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times [Formula: see text] with [Formula: see text].

What can be observed in real time PCR and when does it show?

Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299-304, 2003. doi: 10.1016/S0022-5193(03)00166-8 ), and based on Michaelis-Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433-440, 1997). A crucial role is played by the Michaelis-Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when [Formula: see text], and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a "veil of uncertainty". This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.