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{{Short description|Mathematics of change in size and age}}{{Lead too short|date=January 2024}}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,Malthus, Thomas Robert. An Essay on the Principle of Population: Library of Economics although over the last century the scope of mathematical biology has greatly expanded.{{cn|date=April 2023}}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.JOURNAL, Turchin, P., Does Population Ecology Have General Laws?, Oikos (journal), Oikos, John Wiley & Sons Ltd. (Nordic Society Oikos), 94, 1, 17–26, 2001, 10.1034/j.1600-0706.2001.11310.x, 2001Oikos..94...17T, 27090414, {{rp|18}} This principle provided the basis for the subsequent predictive theories, such as the demographic studies such as the work of Benjamin GompertzJOURNAL, 1825, On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies, Philosophical Transactions of the Royal Society of London, 115, 513–585,visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55920, 10.1098/rstl.1825.0026, Gompertz, Benjamin, 145157003, and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.JOURNAL, Verhulst, P. H.,books.google.com/books?id=8GsEAAAAYAAJ, Notice sur la loi que la population poursuit dans son accroissement, Corresp. Mathématique et Physique, 10, 113–121, 1838, A more general model formulation was proposed by F. J. Richards in 1959,JOURNAL, Richards, F. J., June 1959, A Flexible Growth Function for Empirical Use,www.jstor.org/stable/23686557, Journal of Experimental Botany, 10, 29, 290–300, 10.1093/jxb/10.2.290, 23686557, 16 November 2020, 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,JOURNAL, Hoppensteadt, F., Predator-prey model, Scholarpedia, 1, 10, 1563, 2006, 10.4249/scholarpedia.1563, 2006SchpJ...1.1563H, free, JOURNAL, Lotka, A. J., Contribution to the Theory of Periodic Reaction, Journal of Physical Chemistry A, J. Phys. Chem., 14, 3, 271–274, 1910, 10.1021/j150111a004,zenodo.org/record/1428768, BOOK, Goel, N. S., etal, On the Volterra and Other Non-Linear Models of Interacting Populations, Academic Press, 1971, BOOK, Lotka, A. J., Elements of Physical Biology, Williams and Wilkins, 1925, JOURNAL, Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Accademia dei Lincei, Mem. Acad. Lincei Roma, 2, 31–113, 1926, BOOK, Volterra, V., Variations and fluctuations of the number of individuals in animal species living together, Animal Ecology, Chapman, R. N., McGraw–Hill, 1931, BOOK, Kingsland, S., Modeling Nature: Episodes in the History of Population Ecology, University of Chicago Press, 1995, 978-0-226-43728-6, JOURNAL, Berryman, A. A.,entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf, The Origins and Evolution of Predator-Prey Theory, Ecology (journal), Ecology, 73, 5, 1530–1535, 1992, dead,entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf," title="web.archive.org/web/20100531204042entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf,">web.archive.org/web/20100531204042entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf, 2010-05-31, 10.2307/1940005, 1940005, as well as the alternative Arditi–Ginzburg equations.JOURNAL, Arditi, R., Ginzburg, L. R., 1989,life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf, Coupling in predator-prey dynamics: ratio dependence, Journal of Theoretical Biology, 139, 3, 311–326, 10.1016/s0022-5193(89)80211-5, 1989JThBi.139..311A, 2020-11-17, 2016-03-04,life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf," title="web.archive.org/web/20160304053545life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf,">web.archive.org/web/20160304053545life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf, dead, JOURNAL, Abrams, P. A., Ginzburg, L. R., 2000, The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15, 8, 337–341, 10.1016/s0169-5347(00)01908-x, 10884706,

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 of no external influence. Models can be more mathematically complex where “...several competing hypotheses are simultaneously confronted with the data.“JOURNAL, Johnson, J. B., Omland, K. S., Model selection in ecology and evolution., Trends in Ecology and Evolution, 19, 2, 101–108, 2004,www.usm.maine.edu/bio/courses/bio621/model_selection.pdf, 10.1016/j.tree.2003.10.013, 16701236, 10.1.1.401.777, 2010-01-25,www.usm.maine.edu/bio/courses/bio621/model_selection.pdf," title="web.archive.org/web/20110611094158www.usm.maine.edu/bio/courses/bio621/model_selection.pdf,">web.archive.org/web/20110611094158www.usm.maine.edu/bio/courses/bio621/model_selection.pdf, 2011-06-11, dead, 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:frac{dN}{dT} = B - D = bN - dN = (b - d)N = rN,where {{mvar|N}} is the total number of individuals in the specific experimental population being studied, {{mvar|B}} is the number of births and D is the number of deaths per individual in a particular experiment or model. The algebraic symbols {{mvar|b}}, {{mvar|d}} and {{mvar|r}} 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 ({{math|dN/dT}}) is equal to births minus deaths ({{math|B − D}}).BOOK, Vandermeer, J. H., Goldberg, D. E., Population ecology: First principles, Woodstock, Oxfordshire, Princeton University Press, 2003, 978-0-691-11440-8, Using these techniques, Malthus’ population principle of growth was later transformed into a mathematical model known as the (Logistic function#In ecology: modeling population growth|logistic equation):frac{dN}{dT} = aN left( 1 - frac{N}{K} right),where {{mvar|N}} is the biomass density, {{mvar|a}} is the maximum per-capita rate of change, and {{mvar|K}} is the carrying capacity of the population. The formula can be read as follows: the rate of change in the population ({{math|dN/dT}}) is equal to growth ({{math|aN}}) that is limited by carrying capacity {{math|(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.

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 isfrac{dN}{dt} = r Nwhere the derivative dN / dt is the rate of increase of the population, {{mvar|N}} is the population size, and {{mvar|r}} 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.JOURNAL, 10.1603/0046-225X-34.4.938, Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase of Hysteroneura setariae (Thomas) (Homoptera: Aphididae) on Rice (Oryza sativa L.), Environmental Entomology, 34, 4, 938–43, 2005, Jahn, Gary C., Almazan, Liberty P., Pacia, Jocelyn B., free,

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.{{cn|date=April 2023}}

Geometric populations

File:Operophtera.brumata.6961.jpg|thumbnail|right|(Operophtera brumata]] populations are geometric.JOURNAL, Hassell, Michael P., Foraging Strategies, Population Models and Biological Control: A Case Study, The Journal of Animal Ecology, June 1980, 49, 2, 603–628, 10.2307/4267, 4267, 1980JAnEc..49..603H, )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.WEB, Geometric and Exponential Population Models,www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf, 2015-08-17,www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf," title="web.archive.org/web/20150421081753www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf,">web.archive.org/web/20150421081753www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf, 2015-04-21, dead, N_{t+1} = N_t + B_t + I_t - D_t - E_twhere {{math|Nt}} is the population size in generation {{mvar|t}}, and {{math|Nt+1}} is the population size in the generation directly after {{math|Nt}}; {{math|Bt}} is the sum of births in the population between generations {{mvar|t}} and {{math|t + 1}} (i.e. the birth rate); {{math|It}} is the sum of immigrants added to the population between generations; {{math|Dt}} is the sum of deaths between generations (death rate); and {{math|Et}} is the sum of emigrants moving out of the population between generations.When there is no migration to or from the population, N_{t+1} = N_t + B_t - D_t.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.begin{align}N_{t+1} &= N_t + R N_t N_{t+1} &= left(1 + Rright) N_t N_{t+1} &= lambda N_tend{align}where {{math|1=λ = 1 + R}} is the finite rate of increase.{| class=“wikitable”

t + 1}}	{{math|1=N’t+1 = λN’t}}

t + 2}}	{{math|1=N’t+2 = λN’t+1 = λλN’t = λ2N’t}}

t + 3}}	{{math|1=N’t+3 = λN’t+2 = λλ2N’t = λ3 N’t}}

Therefore: N_t = lambda^t N_0 where {{math|λt}} is the Finite rate of increase raised to the power of the number of generations (e.g. for {{math|t + 2}} [two generations] → {{math|λ2}}, for {{math|t + 1}} [one generation] → {{math|1=λ1 = λ}}, and for {{mvar|t}} [before any generations - at time zero] → {{math|1=λ0 = 1}}

Doubling time

File:G. stearothermophilus has a shorter doubling time (td) than E. coli and N. meningitidis.png|400px|thumbnail|right|G. stearothermophilus has a shorter doubling time (td) than E. coli and N. meningitidis. Growth rates of 2 bacterial 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 such as animals, fungi, plants, and protists, doubling times are much longer than in bacteria. This reduces the growth rates of eukaryotes in comparison to Bacteria. G. stearothermophilus, E. coli, and N. meningitidis have 20 minute,WEB, Bacillus stearothermophilus NEUF2011,microbewiki.kenyon.edu/index.php/Bacillus_stearothermophilus_NEUF2011, Microbe wiki, 30 minute,JOURNAL, Chandler, M., Bird, R.E., Caro, L., The replication time of the Escherichia coli K12 chromosome as a function of cell doubling time, Journal of Molecular Biology, May 1975, 94, 1, 127–132, 10.1016/0022-2836(75)90410-6, 1095767, and 40 minuteJOURNAL, Tobiason, D. M., Seifert, H. S., Genomic Content of Neisseria Species, Journal of Bacteriology, 19 February 2010, 192, 8, 2160–2168, 10.1128/JB.01593-09, 20172999, 2849444, 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 bacteria out of all the bacteria would approach 100% whilst the percentage of E. coli and N. meningitidis combined out of all the bacteria would approach 0%. This graph is a 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.{|! Time in minutes !! % that is G. stearothermophilus

| 44.4%

| 53.3%

| 64.9%

| 72.7%

| 100%

{|! Time in minutes !! % that is E. coli

| 29.6%

| 26.7%

| 21.6%

| 18.2%

| 0.00%

{|! Time in minutes !! % that is N. meningitidis

| 25.9%

| 20.0%

| 13.5%

| 9.10%

| 0.00%

Disclaimer: Bacterial populations are logistic instead of geometric. Nevertheless, doubling times are applicable to both types of populations.]]The doubling time ({{math|1=td}}) of a population is the time required for the population to grow to twice its size.WEB, What is Doubling Time and How is it Calculated?, Lauren, Boucher, 24 March 2015, Population Education,www.populationeducation.org/content/what-doubling-time-and-how-it-calculated, We can calculate the doubling time of a geometric population using the equation: {{math|1=N’t = λ’t N0}} by exploiting our knowledge of the fact that the population ({{mvar|N}}) is twice its size ({{math|2N}}) after the doubling time.begin{align}N_{t_d} &= lambda_{t_d} N_0 2 N_0 &= lambda_{t_d} N_0 lambda_{t_d} &= 2end{align}The doubling time can be found by taking logarithms. For instance:t_d log_2(lambda) = log_2(2) = 1implies t_d = frac{1}{log_2(lambda)}Or:t_d ln(lambda) = ln(2)implies t_d = frac{ln(2)}{ln(lambda)} = frac{0.693...}{ln(lambda)}Therefore: t_d = frac{1}{log_2(lambda)} = frac{0.693...}{ln(lambda)}

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: {{math|1=N’t = λ’t N0}} by exploiting our knowledge of the fact that the population ({{mvar|N}}) is half its size ({{math|0.5N}}) after a half-life.N_{t_{1/2}} = lambda^{t_{1/2}} N_0implies frac{1}{2} N_0 = lambda^{t_{1/2}} N_0implies lambda^{t_{1/2}} = frac{1}{2}where {{math|t1/2}} is the half-life.The half-life can be calculated by taking logarithms (see above).t_{1/2} = frac{1}{log_{0.5}(lambda)} = frac{ln(0.5)}{ln(lambda)}

Geometric (R) growth constant

R = b - dbegin{align}N_{t+1} &= N_t + R N_t N_{t+1} - N_t &= R N_t N_{t+1} - N_t &= Delta Nend{align}where {{math|ΔN}} is the change in population size between two generations (between generation {{math|t + 1}} and {{mvar|t}}).Delta N = R N_t implies frac{Delta N}{N_t} = Tλ) growth constant“>

Finite (λ) growth constant

1 + R = lambdaN_{t+1} = lambda N_t implies lambda = frac{N_{t+1}}{N_t}

Mathematical relationship between geometric and logistic populations

In geometric populations, {{mvar|R}} and {{mvar|λ}} represent growth constants (see 2 and 2.3). In logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase ({{mvar|r}}) 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.WEB, Population Growth, University of Alberta,www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf, 2020-11-16, 2018-02-18,www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf," title="web.archive.org/web/20180218231304www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf,">web.archive.org/web/20180218231304www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf, dead, However, both sets of constants share the mathematical relationship below.The growth equation for exponential populations is N_t = N_0 e^{rt} where {{math|e}} is Euler’s number, a universal constant often applicable in logistic equations, and {{math|r}} is the intrinsic growth rate.To find the relationship between a geometric population and a logistic population, we assume the {{math|Nt}} is the same for both models, and we expand to the following equality:begin{align}N_0 e^{rt} &= N_0 lambda^t e^{rt} &= lambda^t rt &= t ln(lambda)end{align}Giving us r = ln(lambda) and lambda = e^r.

Evolutionary game theory

Evolutionary game theory was first developed by Ronald Fisher in his 1930 article The Genetic Theory of Natural Selection.WEB,plato.stanford.edu/entries/game-evolutionary/, Evolutionary Game Theory, 19 July 2009, Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab, Center for the Study of Language and Information (CSLI), Stanford University, 1095-5054, 16 November 2020, In 1973 John Maynard Smith formalised a central concept, the evolutionarily stable strategy.JOURNAL, Nanjundiah, V., John Maynard Smith (1920–2004), 10.1007/BF02837646, Resonance (journal), Resonance, 10, 11, 70–78, 2005, 82303195,www.ias.ac.in/resonance/Nov2005/pdf/Nov2005p70-78.pdf, 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.

Oscillatory

Population size in plants experiences significant oscillation due to the annual environmental oscillation. Plant dynamics experience a higher degree of this seasonality than do mammals, birds, or bivoltine insects. When combined with perturbations due to disease, this often results in chaotic oscillations.

In popular culture

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

See also

{{div col|colwidth=22em}}

• Delayed density dependence
• Lotka-Volterra equations
• Minimum viable population
• Maximum sustainable yield
• Nicholson–Bailey model
• Pest insect population dynamics
• Population cycle
• Population dynamics of fisheries
• Population ecology
• Population genetics
• Population modeling
• Ricker model
• r/K selection theory
• System dynamics
• Random generalized Lotka–Volterra model
• Consumer-resource model{{div col end}}

References

{{Reflist|30em}}

Further reading

• Andrey Korotayev, Artemy Malkov, and Daria Khaltourina. Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth. {{ISBN|5-484-00414-4}}
• Turchin, P. 2003. Complex Population Dynamics: a Theoretical/Empirical Synthesis. Princeton, NJ: Princeton University Press.
• JOURNAL, Smith, Frederick E., Experimental methods in population dynamics: a critique, Ecology (journal), Ecology, 33, 4, 441–450, 1952, 10.2307/1931519, 1931519, 1952Ecol...33..441S,

External links

• The Virtual Handbook on Population Dynamics. An online compilation of state-of-the-art basic tools for the analysis of population dynamics with emphasis on benthic invertebrates.

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