Population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History. Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years, although over the last century the scope of mathematical biology has greatly expanded. 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. This principle provided  the basis for the subsequent predictive theories, such as the 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. Logistic function. 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 ('null hypothesis') of no external influence. Models can be more mathematically complex where "...several competing hypotheses are simultaneously confronted with the data." 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: formula_1 where is the total number of individuals in the specific experimental population being studied, is the number of births and  D is the number of deaths per individual in a particular experiment or model. The algebraic symbols , and stand for the rates of birth, death, and the rate of change per individual in the general population, the intrinsic rate of increase. This formula can be read as the rate of change in the population () is equal to births minus deaths (). Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the : formula_2 where is the biomass density, is the maximum per-capita rate of change, and is the carrying capacity of the population . The formula can be read as follows: the rate of change in the population () is equal to growth () that is limited by carrying capacity . 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. 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". It is formula_3 where the derivative formula_4 is the rate of increase of the population, is the population size, and is the intrinsic rate of increase. Thus "r" is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is commonly used in insect population ecology or management to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth . Epidemiology. 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. Geometric populations. The mathematical formula below can used to model geometric populations. Geometric populations grow in discrete reproductive periods between intervals of abstinence, as opposed to populations which grow without designated periods for reproduction. Say that "N" denotes the number of individuals in each generation of a population that will reproduce. formula_5 where is the population  size in generation , and is the population size in the generation directly after ; is the sum of births in the population between generations and (i.e. the birth rate); is the sum of immigrants added to the population between generations; is the sum of deaths between generations (death rate); and is the sum of emigrants moving out of the population between generations. When there is no migration to or from the population, formula_6 Assuming in this case that the birth and death rates are constants, then the birth rate minus the death rate equals "R", the geometric rate of increase. formula_7  where is the finite rate of increase. Therefore: formula_8 where is the Finite rate of increase raised to the power of the number of generations (e.g. for [two generations] → , for [one generation] → , and for [before any generations - at time zero] → Doubling time. The doubling time () of a population is the time required for the population to grow to twice its size. We can calculate the doubling time of a geometric population using the equation: by exploiting our knowledge of the fact that the population () is twice its size () after the doubling time. formula_9 The doubling time can  be found by taking logarithms. For instance: formula_10 Or: formula_11 Therefore: formula_12 Half-life of geometric populations. The half-life of a population is the time taken for the population to decline to half its size. We can calculate the half-life of a geometric population using the equation: by exploiting our knowledge of the fact that the population () is half its size () after a half-life. formula_13 where is the half-life. The half-life can be calculated by taking logarithms (see above). formula_14 Geometric (R) growth constant. formula_15 formula_16 where is the change in population size between two  generations (between generation and ). formula_17 Finite ("λ") growth constant. formula_18 formula_19 Mathematical relationship between geometric and logistic populations. In geometric populations, and represent growth constants (see 2 and 2.3). In logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase () is the relevant growth constant. Since generations of reproduction in a geometric population do not overlap (e.g. reproduce once a year) but do in an exponential population, geometric and exponential populations are usually considered to be mutually exclusive. However, both sets of constants share the mathematical relationship below. The growth equation for exponential populations is formula_20  where is Euler's number, a universal constant often applicable in logistic equations, and is the intrinsic growth rate. To find the relationship between a geometric population and a logistic population, we assume the is the same for both models, and we expand to the following equality: formula_21 Giving us formula_22 and formula_23 Evolutionary game theory. Evolutionary game theory was first developed by Ronald Fisher in his 1930 article "The Genetic Theory of Natural Selection". In 1973 John Maynard Smith formalised a central concept, the evolutionarily stable strategy. Population dynamics have been used in several control theory applications. Evolutionary game  theory can be used in different industrial or other contexts. Industrially, it is mostly used in multiple-input-multiple-output (MIMO) systems, although it can be adapted for use in single-input-single-output (SISO) systems. Some other examples of applications are military campaigns, water distribution, dispatch of distributed generators, lab experiments, transport problems, communication problems, among others. Trivia. The computer game "SimCity", "Sim Earth" and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.