Population dynamics

Population dynamics is also closely related to other mathematical biology fields such as epidemiology, and also uses techniques from evolutionary game theory in its modelling.

Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years,[1] although over the last century the scope of mathematical biology has greatly expanded.

[citation needed] The beginning of population dynamics is widely regarded as the work of Malthus, formulated as the Malthusian growth model.

According to Malthus, assuming that the conditions (the environment) remain constant (ceteris paribus), a population will grow (or decline) exponentially.

[2]: 18  This principle provided the basis for the subsequent predictive theories, such as the demographic studies such as the work of Benjamin Gompertz[3] and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.

[4] A more general model formulation was proposed by F. J. Richards in 1959,[5] further expanded by Simon Hopkins, in which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general formulation.

[14][15] Simplified population models usually start with four key variables (four demographic processes) including death, birth, immigration, and emigration.

Mathematical models used to calculate changes in population demographics and evolution hold the assumption of no external influence.

Models can be more mathematically complex where "...several competing hypotheses are simultaneously confronted with the data.

"[16] For example, in a closed system where immigration and emigration does not take place, the rate of change in the number of individuals in a population can be described as:

This formula can be read as the rate of change in the population (dN/dt) is equal to births minus deaths (B − D).

[2][13][17] Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation:

The formula can be read as follows: the rate of change in the population (dN/dt) is equal to growth (rN) that is limited by carrying capacity (1 − N/K).

From these basic mathematical principles the discipline of population ecology expands into a field of investigation that queries the demographics of real populations and tests these results against the statistical models.

The field of population ecology often uses data on life history and matrix algebra to develop projection matrices on fecundity and survivorship.

This information is used for managing wildlife stocks and setting harvest quotas.

[18] Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations.

Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.

[citation needed] The mathematical formula below is used to model geometric populations.

where For the sake of simplicity, we suppose there is no migration to or from the population, but the following method can be applied without this assumption.

Then, we assume the birth and death rates do not depend on the time t (which is equivalent to assume that the number of births and deaths are effectively proportional to the population size).

This equation means that the sequence (Nt) is geometric with first term N0 and common ratio 1 + R, which we define to be λ. λ is also called the finite rate of increase.

where λt is the finite rate of increase raised to the power of the number of generations.

[24] We can calculate the doubling time of a geometric population using the equation: Nt = λt N0 by exploiting our knowledge of the fact that the population (N) is twice its size (2N) after the doubling time.

where e is Euler's number, a universal constant often applicable in logistic equations, and r is the intrinsic growth rate.

[26] In 1973 John Maynard Smith formalised a central concept, the evolutionarily stable strategy.

Some other examples of applications are military campaigns, water distribution, dispatch of distributed generators, lab experiments, transport problems, communication problems, among others.

[28] Plant dynamics experience a higher degree of this seasonality than do mammals, birds, or bivoltine insects.

[28] When combined with perturbations due to disease, this often results in chaotic oscillations.

[28] The computer game SimCity, Sim Earth and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.