Another Way to Calculate Fitness from Life History Variables: Solution of the Age-Structured Logistic Equation

We consider a discrete time competition model. Populations compete for common limited resources but they have different fertilities and mortalities rates. We compare dynamical properties of this model with its continuous counterpart. We give sufficient conditions for competitive exclusion and the existence of periodic solutions related to the classical logistic, Beverton-Holt and Ricker models.

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (large-deviations) method, which requires only th...

The logistic equation has been extensively used to model biological phenomena across a variety of disciplines and has provided valuable insight into how our universe operates. Incorporating time-dependent parameters into the logistic equation allows the modeling of more complex behavior than its autonomous analog, such as a tumor's varying growth rate under treatment, or the expansion of bacterial colonies under varying resource conditions. Some of the most commonly used nume...

We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise in many complex situations. For sexual populations, even in the simplest setting, the equations are necessarily nonlinear due to the mixing of the parental genetic material. The solutions of such nonlinear equations display interesting fea...

We address a novel approach for stochastic individual-based modelling of a single species population. Individuals are distinguished by their remaining lifetimes, which are regulated by the interplay between the inexorable running of time and the individual's nourishment history. A food-limited environment induces intraspecific competition and henceforth the carrying capacity of the medium may be finite, often emulating the qualitative features of logistic growth. Inherently n...

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assump...

Modern developments in population dynamics emphasize the role of the turnover of individuals. In the new approaches stable population size is a dynamic equilibrium between different mortality and fecundity factors instead of an arbitrary fixed carrying capacity. The latest replicator dynamics models assume that regulation of the population size acts through feedback driven by density dependent juvenile mortality. Here, we consider a simplified model to extract the properties ...

Exact law of mortality dynamics in changing populations and environment is derived. The law is universal for all species, from single cell yeast to humans. It includes no characteristics of animal- environment interactions (metabolism etc) which are a must for life. Such law is unique for live systems with their homeostatic self-adjustment to environment. Its universal dynamics for all animals, with their drastically different biology, evolutionary history, and complexity, is...

This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encomp...

We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP may be solved by first changing the variable to R=(M-P)/P. This reduces the logistic differential equation to the simple linear differential equation dR/dt=-kR, which can be solved without using the customary but slightly more elaborate methods applied to the original logistic DE. The resulting solution in terms of R can be converted by simple algebra to the familiar sigmoid expression involving P. ...