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Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. Population dynamics deals with the way populations are affected by birth and death rates, and by immigration and emigration, and studies topics such as ageing populations or population decline.
One common mathematical model for population dynamics is the exponential growth model. With the exponential model, the rate of change of any given population is proportional to the already existing population.
Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. The first principle of population dynamics is widely regarded as the exponential law of Malthus, as modeled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.
A more general model formulation was proposed by F.J. Richards in 1959, 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. The Lotka–Volterra predator-prey equations are another famous example, as well as the alternative Arditi-Ginzburg equations. The computer game SimCity and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.
In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. 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 analyzed, and provide important results that may be applied to health policy decisions.
Fisheries and wildlife management
- Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers to the age a fish can be caught and counted in nets
- Population growth rate, which measures the growth of individuals in size and length. More important in fisheries, where population is often measured in biomass.
- Mortality, which includes harvest mortality and natural mortality. Natural mortality includes non-human predation, disease and old age.
If N1 is the number of individuals at time 1 then
- N1 = N0 + B - D + I - E
where N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.
If we measure these rates over many time intervals, we can determine how a population's density changes over time. Immigration and emigration are present, but are usually not measured.
All of these are measured to determine the harvestable surplus, which is the number of individuals that can be harvested from a population without affecting long term stability, or average population size. The harvest within the harvestable surplus is considered compensatory mortality, where the harvest deaths are substituting for the deaths that would occur naturally. It started in Europe. Harvest beyond that is additive mortality, harvest in addition to all the animals that would have died naturally. These terms are not the universal good and evil of population management, for example, in deer, the DNR are trying to reduce deer population size overall to an extent, since hunters have reduced buck competition and increased deer population unnaturally.
Intrinsic rate of increase
The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase.
Where (dN/dt) is the rate of increase of the population and N is the population size, r is the intrinsic rate of increase. This is therefore the theoretical maximum rate of increase of a population per individual . The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.
- Brotz, Lucas; Cheung, William W. L.; Kleisner Kristin; Pakhomov, Evgeny; Pauly, Daniel (2012). "Increasing jellyfish populations: trends in Large Marine Ecosystems". Hydrobiologia 688.
- Khamsi, Mohamed Amine. "Population Dynamics". Sosmath.com. Retrieved 26 January 2013.
- Jahn, GC, LP Almazan, and J Pacia. 2005. Effect of nitrogen fertilizer on the intrinsic rate of increase of the rusty plum aphid, Hysteroneura setariae (Thomas) (Homoptera: Aphididae) on rice (Oryza sativa L.). Environmental Entomology 34 (4): 938-943. 
- Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth by Andrey Korotayev, Artemy Malkov, and Daria Khaltourina. ISBN 5-484-00414-4
- Turchin, P. 2003. Complex Population Dynamics: a Theoretical/Empirical Synthesis. Princeton, NJ: Princeton University Press.
- Weiss, V. 2007. The population cycle drives human history - from a eugenic phase into a dysgenic phase and eventual collapse. The Journal of Social, Political and Economic Studies 32: 327-358 
- GreenBoxes code sharing network. Greenboxes (Beta) is a repository for open-source population modelling and PVA code. Greenboxes allows users an easy way to share their code and to search for others shared code.
- The Virtual Handbook on Population Dynamics. An online compilation of state-ot-the-art basic tools for the analysis of population dynamics with emphasis on benthic invertebrates.
- Creatures! High School interactive simulation program that implements an agent based simulation of grass, rabbits and foxes.