{"history":[],"article":{"id":"1e989eb9-eb70-47ff-91dc-42aea1cd7552","title":"Population dynamics","slug":"population-dynamics","content":":::summary\n:::\n\n# \n**Population dynamics** is the type of mathematics used to model and study the size and age composition of [populations](https://bliptext.com/articles/population) as [dynamical systems](https://bliptext.com/articles/dynamical-systems).\n\n# History\nPopulation dynamics has traditionally been the dominant branch of [mathematical biology](https://bliptext.com/articles/mathematical-biology), which has a history of more than 220 years, although over the last century the scope of [mathematical biology](https://bliptext.com/articles/mathematical-biology) has greatly expanded.\nThe beginning of population dynamics is widely regarded as the work of [Malthus](https://bliptext.com/articles/malthus), formulated as the [Malthus](https://bliptext.com/articles/malthus)ian growth model.\nAccording to Malthus, assuming that the conditions (the environment) remain constant ([ceteris paribus](https://bliptext.com/articles/ceteris-paribus)), a population will grow (or decline) [exponentially](https://bliptext.com/articles/exponential-growth).\nThis principle provided the basis for the subsequent predictive theories, such as the [demographic](https://bliptext.com/articles/demography) studies such as the work of [Benjamin Gompertz](https://bliptext.com/articles/benjamin-gompertz) and [Pierre François Verhulst](https://bliptext.com/articles/pierre-fran-ois-verhulst) in the early 19th century, who refined and adjusted the Malthusian [demographic](https://bliptext.com/articles/demography) model.\nA more general model formulation was proposed by [F. J. Richards](https://bliptext.com/articles/f-j-richards) in 1959, further expanded by [Simon Hopkins](https://bliptext.com/articles/simon-hopkins), in which the models of Gompertz, Verhulst and also [Ludwig von Bertalanffy](https://bliptext.com/articles/ludwig-von-bertalanffy) are covered as special cases of the general formulation.\nThe [Lotka–Volterra predator-prey equations](https://bliptext.com/articles/lotka-volterra-equation) are another famous example, as well as the alternative [Arditi–Ginzburg equations](https://bliptext.com/articles/arditi-ginzburg-equations).\n\n# Logistic function\nSimplified population models usually start with four key variables (four **demographic processes**) including death, birth, immigration, and emigration.\nMathematical models used to calculate changes in population demographics and evolution hold the assumption of no external influence.\nModels can be more mathematically complex where \"...several competing hypotheses are simultaneously confronted with the data.\"\nFor 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:\nwhere N is the total number of individuals in the specific experimental population being studied, B is the number of births and D is the number of deaths per individual in a particular experiment or model.\nThe [algebraic symbols](https://bliptext.com/articles/algebra) b, d and r stand for the rates of birth, death, and the rate of change per individual in the general population, the intrinsic rate of increase.\nThis formula can be read as the rate of change in the population (\ndN/dt\n) is equal to births minus deaths (\nB − D\n).\nUsing these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the [logistic equation](https://bliptext.com/articles/logistic-function):\nwhere N is the [population size](https://bliptext.com/articles/population-size), r is the intrinsic [rate of natural increase](https://bliptext.com/articles/rate-of-natural-increase), and K is the [carrying capacity](https://bliptext.com/articles/carrying-capacity) of the population.\nThe formula can be read as follows: the rate of change in the population (\ndN/dt\n) is equal to growth (\nrN\n) that is limited by carrying capacity\n(1 − N/K)\n. From these basic mathematical principles the discipline of population ecology expands into a field of investigation that queries the [demographics](https://bliptext.com/articles/demographics) of real populations and tests these results against the statistical models.\nThe field of population ecology often uses data on life history and matrix algebra to develop projection matrices on fecundity and survivorship.\nThis information is used for managing wildlife stocks and setting harvest quotas.\n\n# Intrinsic rate of increase\nThe 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.\nIt is\nwhere the [derivative](https://bliptext.com/articles/derivative) dN/dt is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase.\nThus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate.\nThe concept is commonly used in insect [population ecology](https://bliptext.com/articles/population-ecology) or [management](https://bliptext.com/articles/integrated-pest-management) to determine how environmental factors affect the rate at which pest populations increase.\nSee also exponential population growth and logistic population growth.\n\n# Epidemiology\nPopulation dynamics overlap with another active area of research in mathematical biology: [mathematical epidemiology](https://bliptext.com/articles/mathematical-modelling-in-epidemiology), the study of infectious disease affecting populations.\nVarious models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.\n\n# Geometric populations\nThe mathematical formula below is used to model [geometric](https://bliptext.com/articles/geometric-progression) populations.\nSuch populations grow in discrete reproductive periods between intervals of [abstinence](https://bliptext.com/articles/abstinence), as opposed to populations which grow without designated periods for reproduction.\nSay that the [natural number](https://bliptext.com/articles/natural-number) t is the index the generation (t=0 for the first generation, t=1 for the second generation, etc.).\nThe letter t is used because the index of a generation is time.\nSay Nt denotes, at generation t, the number of individuals of the population that will reproduce, i.e. the population size at generation t. The population at the next generation, which is the population at time t+1 is:\nwhere\nBt\nis the number of births in the population between generations t and\nt + 1\n,\nDt\nis the number of deaths between generations t and\nt + 1\n,\nIt\nis the number of [immigrants](https://bliptext.com/articles/immigrants) added to the population between generations t and\nt + 1\n, and\nEt\nis the number of [emigrants](https://bliptext.com/articles/emigrants) moving out of the population between generations t and\nt + 1\n.\nFor 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.\nMathematically, it means that for all t, It = Et = 0.\nThe previous equation becomes:\nIn general, the number of births and the number of deaths are approximately proportional to the population size.\nThis remark motivates the following definitions.\nbt = Bt / Nt\n.\ndt = Dt / Nt\n. The previous equation can then be rewritten as:\nThen, 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).\nThis is the core assumption for geometric populations, because with it we are going to obtain a [geometric sequence](https://bliptext.com/articles/geometric-sequence).\nThen we define the geometric rate of increase R = bt - dt to be the birth rate minus the death rate.\nThe geometric rate of increase do not depend on time t, because both the birth rate minus the death rate do not, with our assumption.\nWe obtain:\nThis equation means that the sequence\n(Nt)\nis [geometric](https://bliptext.com/articles/geometric-sequence) with first term\nN0\nand common ratio\n1 + R\n, which we define to be\nλ\n.\nλ\nis also called the finite rate of increase.\nTherefore, by [induction](https://bliptext.com/articles/mathematical-induction), we obtain the expression of the population size at time t:\nwhere\nλt\nis the finite rate of increase raised to the power of the number of generations.\nThis last expression is more convenient than the previous one, because it is explicit.\nFor example, say one wants to calculate with a calculator\nN10\n, the population at the tenth generation, knowing\nN0\nthe initial population and\nλ\nthe finite rate of increase.\nWith the last formula, the result is immediate by plugging\nt = 10\n, whether with the previous one it is necessary to know\nN9\n,\nN8\n, ...,\nN2\nuntil\nN1\n.\nWe can identify three cases:\nλ > 1\n, i.e. if\nR > 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 > d0\n, i.e. if the birth rate is strictly greater than the death rate, then the population size is increasing and [tends](https://bliptext.com/articles/limit-mathematics) to infinity.\nOf course, in real life, a population cannot grow indefinitely: at some point the population lacks resources and so the death rate increases, which invalidates our core assumption because the death rate now depends on time.\nλ < 1\n, i.e. if\nR < 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 < d0\n, i.e. if the birth rate is strictly smaller than the death rate, then the population size is decreasing and [tends](https://bliptext.com/articles/limit-mathematics) to\n0\n.\nλ = 1\n, i.e. if\nR = 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 = d0\n, i.e. if the birth rate is equal to the death rate, then the population size is constant, equal to the initial population\nN0\n.\n# [Operophtera brumata populations are geometric. | upload.wikimedia.org/wikipedia/commons/7/73/Operophtera.brumata.6961.jpg]\n\n## Doubling time\n[[File:[G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) has a shorter doubling time (td) than [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis).png|400px|thumbnail|right|[G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) has a shorter doubling time (td) than [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis). Growth rates of 2 [[bacteria]]l species will differ by unexpected orders of magnitude if the doubling times of the 2 species differ by even as little as 10 minutes. In [eukaryotes](https://bliptext.com/articles/eukaryote) such as animals, fungi, plants, and protists, doubling times are much longer than in bacteria. This reduces the growth rates of [eukaryotes](https://bliptext.com/articles/eukaryote) in comparison to Bacteria. [G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus), [E. coli](https://bliptext.com/articles/escherichia-coli), and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis) have 20 minute, 30 minute, and 40 minute doubling times under optimal conditions respectively. If bacterial populations could grow indefinitely (which they do not) then the number of bacteria in each species would approach infinity (∞). However, the percentage of [G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) bacteria out of all the bacteria would approach 100% whilst the percentage of [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis) combined out of all the bacteria would approach 0%. This graph is a [simulation](https://bliptext.com/articles/simulation) of this hypothetical scenario. In reality, bacterial populations do not grow indefinitely in size and the 3 species require different optimal conditions to bring their doubling times to minima.\n''Disclaimer: Bacterial populations are [logistic](https://bliptext.com/articles/logistic-function) instead of geometric.\nNevertheless, doubling times are applicable to both types of populations.''\n]]\nThe [doubling time](https://bliptext.com/articles/doubling-time) (\ntd\n) 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:\nNt = λt N0\nby exploiting our knowledge of the fact that the population (N) is twice its size (\n2N\n) after the doubling time.\nThe doubling time can be found by taking [logarithms](https://bliptext.com/articles/logarithms).\nFor instance:\nOr:\nTherefore:\n\n## Half-life of geometric populations\nThe [half-life](https://bliptext.com/articles/half-life) of a population is the time taken for the population to decline to half its size.\nWe can calculate the half-life of a geometric population using the equation:\nNt = λt N0\nby exploiting our knowledge of the fact that the population (N) is half its size (\n0.5N\n) after a half-life.\nwhere\nt1/2\nis the half-life.\nThe half-life can be calculated by taking [logarithms](https://bliptext.com/articles/logarithms) (see above).\nNote that as the population is assumed to decline,\nλ < 1\n, so\nln(λ) < 0\n.\n\n## Mathematical relationship between geometric and logistic populations\nIn geometric populations, R and λ represent growth constants (see [2](https://bliptext.com/articles/population-ecology) and [2](https://bliptext.com/articles/population-ecology).3).\nIn logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase (r) is the relevant growth constant.\nSince 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.\nHowever, both sets of constants share the mathematical relationship below.\nThe growth equation for exponential populations is where\ne\nis [Euler's number](https://bliptext.com/articles/euler-s-number), a [universal constant](https://bliptext.com/articles/universal-constant) often applicable in logistic equations, and\nr\nis the intrinsic growth rate.\nTo find the relationship between a geometric population and a logistic population, we assume the\nNt\nis the same for both models, and we expand to the following equality:\nGiving us and\n\n# Evolutionary game theory\nEvolutionary game theory was first developed by [Ronald Fisher](https://bliptext.com/articles/ronald-fisher) in his 1930 article [The Genetic Theory of Natural Selection](https://bliptext.com/articles/the-genetic-theory-of-natural-selection).\nIn 1973 [John Maynard Smith](https://bliptext.com/articles/john-maynard-smith) formalised a central concept, the [evolutionarily stable strategy](https://bliptext.com/articles/evolutionarily-stable-strategy).\nPopulation dynamics have been used in several [control theory](https://bliptext.com/articles/control-theory) applications.\nEvolutionary game theory can be used in different industrial or other contexts.\nIndustrially, it is mostly used in multiple-input-multiple-output ([MIMO](https://bliptext.com/articles/mimo)) systems, although it can be adapted for use in single-input-single-output ([SISO](https://bliptext.com/articles/single-input-single-output-system)) systems.\nSome other examples of applications are military campaigns, [water distribution](https://bliptext.com/articles/water-supply-network), [dispatch of distributed generators](https://bliptext.com/articles/dispatchable-generation), lab experiments, transport problems, communication problems, among others.\n\n# Oscillatory\nPopulation size in [plants](https://bliptext.com/articles/plant) experiences significant [oscillation](https://bliptext.com/articles/oscillation) due to the annual environmental [oscillation](https://bliptext.com/articles/oscillation).\nPlant dynamics experience a higher degree of this [seasonality](https://bliptext.com/articles/seasonality) than do mammals, birds, or [bivoltine](https://bliptext.com/articles/bivoltine) insects.\nWhen combined with [perturbations](https://bliptext.com/articles/perturbation-biology) due to [disease](https://bliptext.com/articles/plant-disease), this often results in [chaotic oscillations](https://bliptext.com/articles/chaotic-oscillation).\n\n# In popular culture\nThe [computer game](https://bliptext.com/articles/computer-game) [SimCity](https://bliptext.com/articles/simcity-1989-video-game), [Sim Earth](https://bliptext.com/articles/sim-earth) and the [MMORPG](https://bliptext.com/articles/mmorpg) [Ultima Online](https://bliptext.com/articles/ultima-online), among others, tried to [simulate](https://bliptext.com/articles/computer-simulation) some of these population dynamics.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:50:02.141Z","updated_at":"2025-01-25T19:50:02.141Z","search_vector":"'dynam':2 'popul':1","search_content":" \n \n\n# \n**Population dynamics** is the type of mathematics used to model and study the size and age composition of [populations](https://bliptext.com/articles/population) as [dynamical systems](https://bliptext.com/articles/dynamical-systems).\n\n# History\nPopulation dynamics has traditionally been the dominant branch of [mathematical biology](https://bliptext.com/articles/mathematical-biology), which has a history of more than 220 years, although over the last century the scope of [mathematical biology](https://bliptext.com/articles/mathematical-biology) has greatly expanded.\nThe beginning of population dynamics is widely regarded as the work of [Malthus](https://bliptext.com/articles/malthus), formulated as the [Malthus](https://bliptext.com/articles/malthus)ian growth model.\nAccording to Malthus, assuming that the conditions (the environment) remain constant ([ceteris paribus](https://bliptext.com/articles/ceteris-paribus)), a population will grow (or decline) [exponentially](https://bliptext.com/articles/exponential-growth).\nThis principle provided the basis for the subsequent predictive theories, such as the [demographic](https://bliptext.com/articles/demography) studies such as the work of [Benjamin Gompertz](https://bliptext.com/articles/benjamin-gompertz) and [Pierre François Verhulst](https://bliptext.com/articles/pierre-fran-ois-verhulst) in the early 19th century, who refined and adjusted the Malthusian [demographic](https://bliptext.com/articles/demography) model.\nA more general model formulation was proposed by [F. J. Richards](https://bliptext.com/articles/f-j-richards) in 1959, further expanded by [Simon Hopkins](https://bliptext.com/articles/simon-hopkins), in which the models of Gompertz, Verhulst and also [Ludwig von Bertalanffy](https://bliptext.com/articles/ludwig-von-bertalanffy) are covered as special cases of the general formulation.\nThe [Lotka–Volterra predator-prey equations](https://bliptext.com/articles/lotka-volterra-equation) are another famous example, as well as the alternative [Arditi–Ginzburg equations](https://bliptext.com/articles/arditi-ginzburg-equations).\n\n# Logistic function\nSimplified population models usually start with four key variables (four **demographic processes**) including death, birth, immigration, and emigration.\nMathematical models used to calculate changes in population demographics and evolution hold the assumption of no external influence.\nModels can be more mathematically complex where \"...several competing hypotheses are simultaneously confronted with the data.\"\nFor 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:\nwhere N is the total number of individuals in the specific experimental population being studied, B is the number of births and D is the number of deaths per individual in a particular experiment or model.\nThe [algebraic symbols](https://bliptext.com/articles/algebra) b, d and r stand for the rates of birth, death, and the rate of change per individual in the general population, the intrinsic rate of increase.\nThis formula can be read as the rate of change in the population (\ndN/dt\n) is equal to births minus deaths (\nB − D\n).\nUsing these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the [logistic equation](https://bliptext.com/articles/logistic-function):\nwhere N is the [population size](https://bliptext.com/articles/population-size), r is the intrinsic [rate of natural increase](https://bliptext.com/articles/rate-of-natural-increase), and K is the [carrying capacity](https://bliptext.com/articles/carrying-capacity) of the population.\nThe formula can be read as follows: the rate of change in the population (\ndN/dt\n) is equal to growth (\nrN\n) that is limited by carrying capacity\n(1 − N/K)\n. From these basic mathematical principles the discipline of population ecology expands into a field of investigation that queries the [demographics](https://bliptext.com/articles/demographics) of real populations and tests these results against the statistical models.\nThe field of population ecology often uses data on life history and matrix algebra to develop projection matrices on fecundity and survivorship.\nThis information is used for managing wildlife stocks and setting harvest quotas.\n\n# Intrinsic rate of increase\nThe 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.\nIt is\nwhere the [derivative](https://bliptext.com/articles/derivative) dN/dt is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase.\nThus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate.\nThe concept is commonly used in insect [population ecology](https://bliptext.com/articles/population-ecology) or [management](https://bliptext.com/articles/integrated-pest-management) to determine how environmental factors affect the rate at which pest populations increase.\nSee also exponential population growth and logistic population growth.\n\n# Epidemiology\nPopulation dynamics overlap with another active area of research in mathematical biology: [mathematical epidemiology](https://bliptext.com/articles/mathematical-modelling-in-epidemiology), the study of infectious disease affecting populations.\nVarious models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.\n\n# Geometric populations\nThe mathematical formula below is used to model [geometric](https://bliptext.com/articles/geometric-progression) populations.\nSuch populations grow in discrete reproductive periods between intervals of [abstinence](https://bliptext.com/articles/abstinence), as opposed to populations which grow without designated periods for reproduction.\nSay that the [natural number](https://bliptext.com/articles/natural-number) t is the index the generation (t=0 for the first generation, t=1 for the second generation, etc.).\nThe letter t is used because the index of a generation is time.\nSay Nt denotes, at generation t, the number of individuals of the population that will reproduce, i.e. the population size at generation t. The population at the next generation, which is the population at time t+1 is:\nwhere\nBt\nis the number of births in the population between generations t and\nt + 1\n,\nDt\nis the number of deaths between generations t and\nt + 1\n,\nIt\nis the number of [immigrants](https://bliptext.com/articles/immigrants) added to the population between generations t and\nt + 1\n, and\nEt\nis the number of [emigrants](https://bliptext.com/articles/emigrants) moving out of the population between generations t and\nt + 1\n.\nFor 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.\nMathematically, it means that for all t, It = Et = 0.\nThe previous equation becomes:\nIn general, the number of births and the number of deaths are approximately proportional to the population size.\nThis remark motivates the following definitions.\nbt = Bt / Nt\n.\ndt = Dt / Nt\n. The previous equation can then be rewritten as:\nThen, 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).\nThis is the core assumption for geometric populations, because with it we are going to obtain a [geometric sequence](https://bliptext.com/articles/geometric-sequence).\nThen we define the geometric rate of increase R = bt - dt to be the birth rate minus the death rate.\nThe geometric rate of increase do not depend on time t, because both the birth rate minus the death rate do not, with our assumption.\nWe obtain:\nThis equation means that the sequence\n(Nt)\nis [geometric](https://bliptext.com/articles/geometric-sequence) with first term\nN0\nand common ratio\n1 + R\n, which we define to be\nλ\n.\nλ\nis also called the finite rate of increase.\nTherefore, by [induction](https://bliptext.com/articles/mathematical-induction), we obtain the expression of the population size at time t:\nwhere\nλt\nis the finite rate of increase raised to the power of the number of generations.\nThis last expression is more convenient than the previous one, because it is explicit.\nFor example, say one wants to calculate with a calculator\nN10\n, the population at the tenth generation, knowing\nN0\nthe initial population and\nλ\nthe finite rate of increase.\nWith the last formula, the result is immediate by plugging\nt = 10\n, whether with the previous one it is necessary to know\nN9\n,\nN8\n, ...,\nN2\nuntil\nN1\n.\nWe can identify three cases:\nλ > 1\n, i.e. if\nR > 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 > d0\n, i.e. if the birth rate is strictly greater than the death rate, then the population size is increasing and [tends](https://bliptext.com/articles/limit-mathematics) to infinity.\nOf course, in real life, a population cannot grow indefinitely: at some point the population lacks resources and so the death rate increases, which invalidates our core assumption because the death rate now depends on time.\nλ < 1\n, i.e. if\nR < 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 < d0\n, i.e. if the birth rate is strictly smaller than the death rate, then the population size is decreasing and [tends](https://bliptext.com/articles/limit-mathematics) to\n0\n.\nλ = 1\n, i.e. if\nR = 0\n, i.e. (with the assumption that both birth and death rate do not depend on time\nt\n) if\nb0 = d0\n, i.e. if the birth rate is equal to the death rate, then the population size is constant, equal to the initial population\nN0\n.\n# [Operophtera brumata populations are geometric. | upload.wikimedia.org/wikipedia/commons/7/73/Operophtera.brumata.6961.jpg]\n\n## Doubling time\n[[File:[G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) has a shorter doubling time (td) than [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis).png|400px|thumbnail|right|[G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) has a shorter doubling time (td) than [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis). Growth rates of 2 [[bacteria]]l species will differ by unexpected orders of magnitude if the doubling times of the 2 species differ by even as little as 10 minutes. In [eukaryotes](https://bliptext.com/articles/eukaryote) such as animals, fungi, plants, and protists, doubling times are much longer than in bacteria. This reduces the growth rates of [eukaryotes](https://bliptext.com/articles/eukaryote) in comparison to Bacteria. [G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus), [E. coli](https://bliptext.com/articles/escherichia-coli), and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis) have 20 minute, 30 minute, and 40 minute doubling times under optimal conditions respectively. If bacterial populations could grow indefinitely (which they do not) then the number of bacteria in each species would approach infinity (∞). However, the percentage of [G. stearothermophilus](https://bliptext.com/articles/bacillus-stearothermophilus) bacteria out of all the bacteria would approach 100% whilst the percentage of [E. coli](https://bliptext.com/articles/escherichia-coli) and [N. meningitidis](https://bliptext.com/articles/neisseria-meningitidis) combined out of all the bacteria would approach 0%. This graph is a [simulation](https://bliptext.com/articles/simulation) of this hypothetical scenario. In reality, bacterial populations do not grow indefinitely in size and the 3 species require different optimal conditions to bring their doubling times to minima.\n''Disclaimer: Bacterial populations are [logistic](https://bliptext.com/articles/logistic-function) instead of geometric.\nNevertheless, doubling times are applicable to both types of populations.''\n]]\nThe [doubling time](https://bliptext.com/articles/doubling-time) (\ntd\n) 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:\nNt = λt N0\nby exploiting our knowledge of the fact that the population (N) is twice its size (\n2N\n) after the doubling time.\nThe doubling time can be found by taking [logarithms](https://bliptext.com/articles/logarithms).\nFor instance:\nOr:\nTherefore:\n\n## Half-life of geometric populations\nThe [half-life](https://bliptext.com/articles/half-life) of a population is the time taken for the population to decline to half its size.\nWe can calculate the half-life of a geometric population using the equation:\nNt = λt N0\nby exploiting our knowledge of the fact that the population (N) is half its size (\n0.5N\n) after a half-life.\nwhere\nt1/2\nis the half-life.\nThe half-life can be calculated by taking [logarithms](https://bliptext.com/articles/logarithms) (see above).\nNote that as the population is assumed to decline,\nλ < 1\n, so\nln(λ) < 0\n.\n\n## Mathematical relationship between geometric and logistic populations\nIn geometric populations, R and λ represent growth constants (see [2](https://bliptext.com/articles/population-ecology) and [2](https://bliptext.com/articles/population-ecology).3).\nIn logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase (r) is the relevant growth constant.\nSince 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.\nHowever, both sets of constants share the mathematical relationship below.\nThe growth equation for exponential populations is where\ne\nis [Euler's number](https://bliptext.com/articles/euler-s-number), a [universal constant](https://bliptext.com/articles/universal-constant) often applicable in logistic equations, and\nr\nis the intrinsic growth rate.\nTo find the relationship between a geometric population and a logistic population, we assume the\nNt\nis the same for both models, and we expand to the following equality:\nGiving us and\n\n# Evolutionary game theory\nEvolutionary game theory was first developed by [Ronald Fisher](https://bliptext.com/articles/ronald-fisher) in his 1930 article [The Genetic Theory of Natural Selection](https://bliptext.com/articles/the-genetic-theory-of-natural-selection).\nIn 1973 [John Maynard Smith](https://bliptext.com/articles/john-maynard-smith) formalised a central concept, the [evolutionarily stable strategy](https://bliptext.com/articles/evolutionarily-stable-strategy).\nPopulation dynamics have been used in several [control theory](https://bliptext.com/articles/control-theory) applications.\nEvolutionary game theory can be used in different industrial or other contexts.\nIndustrially, it is mostly used in multiple-input-multiple-output ([MIMO](https://bliptext.com/articles/mimo)) systems, although it can be adapted for use in single-input-single-output ([SISO](https://bliptext.com/articles/single-input-single-output-system)) systems.\nSome other examples of applications are military campaigns, [water distribution](https://bliptext.com/articles/water-supply-network), [dispatch of distributed generators](https://bliptext.com/articles/dispatchable-generation), lab experiments, transport problems, communication problems, among others.\n\n# Oscillatory\nPopulation size in [plants](https://bliptext.com/articles/plant) experiences significant [oscillation](https://bliptext.com/articles/oscillation) due to the annual environmental [oscillation](https://bliptext.com/articles/oscillation).\nPlant dynamics experience a higher degree of this [seasonality](https://bliptext.com/articles/seasonality) than do mammals, birds, or [bivoltine](https://bliptext.com/articles/bivoltine) insects.\nWhen combined with [perturbations](https://bliptext.com/articles/perturbation-biology) due to [disease](https://bliptext.com/articles/plant-disease), this often results in [chaotic oscillations](https://bliptext.com/articles/chaotic-oscillation).\n\n# In popular culture\nThe [computer game](https://bliptext.com/articles/computer-game) [SimCity](https://bliptext.com/articles/simcity-1989-video-game), [Sim Earth](https://bliptext.com/articles/sim-earth) and the [MMORPG](https://bliptext.com/articles/mmorpg) [Ultima Online](https://bliptext.com/articles/ultima-online), among others, tried to [simulate](https://bliptext.com/articles/computer-simulation) some of these population dynamics.","revisionCount":0},"hasMore":false,"nextCursor":null}