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Population dynamics

From Wikipedia, the free encyclopedia
Mathematics of change in size and age
This article'slead sectionmay be too short to adequatelysummarize the key points. Please consider expanding the lead toprovide an accessible overview of all important aspects of the article.(January 2024)

Population dynamics is the type of mathematics used to model and study the size and age composition ofpopulations asdynamical systems. Population dynamics is a branch ofmathematical biology, and uses mathematical techniques such asdifferential equations to model behaviour. Population dynamics is also closely related to other mathematical biology fields such asepidemiology, and also uses techniques from evolutionary game theory in its modelling.

History

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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 ofMalthus, formulated as theMalthusian 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 thedemographic studies such as the work ofBenjamin Gompertz[3] andPierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.[4]

A more general model formulation was proposed byF. J. Richards in 1959,[5] further expanded bySimon Hopkins, in which the models of Gompertz, Verhulst and alsoLudwig von Bertalanffy are covered as special cases of the general formulation. TheLotka–Volterra predator-prey equations are another famous example,[6][7][8][9][10][11][12][13] as well as the alternativeArditi–Ginzburg equations.[14][15]

Logistic function

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Simplified population models usually start with four key variables (fourdemographic 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:dNdt=B−D=bN−dN=(b−d)N=rN,{\displaystyle {\mathrm {d} N \over \mathrm {d} t}=B-D=bN-dN=(b-d)N=rN,}whereN is the total number of individuals in the specific experimental population being studied,B is the number of births andD is the number of deaths per individual in a particular experiment or model. Thealgebraic symbolsb,d andr 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 (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 thelogistic equation:dNdt=rN(1−NK),{\displaystyle {\mathrm {d} N \over \mathrm {d} t}=rN\left(1-{N \over K}\right),}whereN is thepopulation size,r is the intrinsicrate of natural increase, andK is thecarrying capacity of the population. 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 thedemographics 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.[13][17]

Intrinsic rate of increase

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Main article:Rate of natural increase

The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as theintrinsic rate of increase. It isdNdt=rN{\displaystyle {\mathrm {d} N \over \mathrm {d} t}=rN}where thederivativedN/dt{\displaystyle dN/dt} is the rate of increase of the population,N is the population size, andr is the intrinsic rate of increase. Thusr 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 insectpopulation ecology ormanagement to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.[18]

Epidemiology

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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]

Geometric populations

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Operophtera brumata populations are geometric.[19]

The mathematical formula below is used to modelgeometric populations. Such populations grow in discrete reproductive periods between intervals ofabstinence, as opposed to populations which grow without designated periods for reproduction. Say that thenatural numbert is the index the generation (t=0 for the first generation,t=1 for the second generation, etc.). The lettert is used because the index of a generation is time. SayNt denotes, at generationt, the number of individuals of the population that will reproduce, i.e. the population size at generationt. The population at the next generation, which is the population at timet+1 is:[20]Nt+1=Nt+Bt−Dt+It−Et{\displaystyle N_{t+1}=N_{t}+B_{t}-D_{t}+I_{t}-E_{t}}where

  • Bt is the number of births in the population between generationst andt + 1,
  • Dt is the number of deaths between generationst andt + 1,
  • It is the number ofimmigrants added to the population between generationst andt + 1, and
  • Et is the number ofemigrants moving out of the population between generationst andt + 1.

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. Mathematically, it means that for allt,It =Et = 0. The previous equation becomes:Nt+1=Nt+Bt−Dt.{\displaystyle N_{t+1}=N_{t}+B_{t}-D_{t}.}

In general, the number of births and the number of deaths are approximately proportional to the population size. This remark motivates the following definitions.

  • The birth rate at timet is defined bybt =Bt /Nt.
  • The death rate at timet is defined bydt =Dt /Nt.

The previous equation can then be rewritten as:Nt+1=(1+bt−dt)Nt.{\displaystyle N_{t+1}=(1+b_{t}-d_{t})N_{t}.}

Then, we assume the birth and death rates do not depend on the timet (which is equivalent to assume that the number of births and deaths are effectively proportional to the population size). This is the core assumption for geometric populations, because with it we are going to obtain ageometric sequence. Then we define the geometric rate of increaseR =bt -dt to be the birth rate minus the death rate. The geometric rate of increase do not depend on timet, because both the birth rate minus the death rate do not, with our assumption. We obtain:Nt+1=(1+R)Nt.{\displaystyle {\begin{aligned}N_{t+1}&=\left(1+R\right)N_{t}.\end{aligned}}}This equation means that the sequence(Nt) isgeometric with first termN0 and common ratio1 +R, which we define to beλ.λ is also called the finite rate of increase.

Therefore, byinduction, we obtain the expression of the population size at timet:Nt=λtN0{\displaystyle N_{t}=\lambda ^{t}N_{0}}whereλt is the finite rate of increase raised to the power of the number of generations.This last expression is more convenient than the previous one, because it is explicit. For example, say one wants to calculate with a calculatorN10, the population at the tenth generation, knowingN0 the initial population andλ the finite rate of increase. With the last formula, the result is immediate by pluggingt = 10, whether with the previous one it is necessary to knowN9,N8, ...,N2 untilN1.

We can identify three cases:

  • Ifλ > 1, i.e. ifR > 0, i.e. (with the assumption that both birth and death rate do not depend on timet) ifb0 >d0, i.e. if the birth rate is strictly greater than the death rate, then the population size is increasing andtends to infinity. Of 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.
  • Ifλ < 1, i.e. ifR < 0, i.e. (with the assumption that both birth and death rate do not depend on timet) ifb0 <d0, i.e. if the birth rate is strictly smaller than the death rate, then the population size is decreasing andtends to0.
  • Ifλ = 1, i.e. ifR = 0, i.e. (with the assumption that both birth and death rate do not depend on timet) ifb0 =d0, i.e. if the birth rate is equal to the death rate, then the population size is constant, equal to the initial populationN0.

Doubling time

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G. stearothermophilus has a shorter doubling time (td) thanE. coli andN. meningitidis. Growth rates of 2bacterial species will differ by unexpected orders of magnitude if the doubling times of the 2 species differ by even as little as 10 minutes. Ineukaryotes 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, andN. meningitidis have 20 minute,[21] 30 minute,[22] and 40 minute[23] 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 ofG. stearothermophilus bacteria out of all the bacteria would approach 100% whilst the percentage ofE. coli andN. meningitidis combined out of all the bacteria would approach 0%. This graph is asimulation 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 isG. stearothermophilus
3044.4%
6053.3%
9064.9%
12072.7%
→∞100%
Time in minutes% that isE. coli
3029.6%
6026.7%
9021.6%
12018.2%
→∞0.00%
Time in minutes% that isN. meningitidis
3025.9%
6020.0%
9013.5%
1209.10%
→∞0.00%
Disclaimer: Bacterial populations arelogistic instead of geometric. Nevertheless, doubling times are applicable to both types of populations.

Thedoubling time (td) of a population is the time required for the population to grow to twice its size.[24] We can calculate the doubling time of a geometric population using the equation:Nt =λtN0 by exploiting our knowledge of the fact that the population (N) is twice its size (2N) after the doubling time.[20]

Ntd=λtdN02N0=λtdN0λtd=2{\displaystyle {\begin{aligned}N_{t_{d}}&=\lambda ^{t_{d}}N_{0}\\2N_{0}&=\lambda ^{t_{d}}N_{0}\\\lambda ^{t_{d}}&=2\end{aligned}}}

The doubling time can be found by takinglogarithms. For instance:tdlog2⁡(λ)=log2⁡(2)=1⟹td=1log2⁡(λ){\displaystyle t_{d}\log _{2}(\lambda )=\log _{2}(2)=1\implies t_{d}={1 \over \log _{2}(\lambda )}}Or:tdln⁡(λ)=ln⁡(2)⟹td=ln⁡(2)ln⁡(λ){\displaystyle t_{d}\ln(\lambda )=\ln(2)\implies t_{d}={\ln(2) \over \ln(\lambda )}}

Therefore:td=1log2⁡(λ)=0.693...ln⁡(λ){\displaystyle t_{d}={\frac {1}{\log _{2}(\lambda )}}={\frac {0.693...}{\ln(\lambda )}}}

Half-life of geometric populations

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Thehalf-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:Nt =λtN0 by exploiting our knowledge of the fact that the population (N) is half its size (0.5N) after a half-life.[20]

Nt1/2=λt1/2N0⟹12N0=λt1/2N0⟹λt1/2=12{\displaystyle N_{t_{1/2}}=\lambda ^{t_{1/2}}N_{0}\implies {\frac {1}{2}}N_{0}=\lambda ^{t_{1/2}}N_{0}\implies \lambda ^{t_{1/2}}={\frac {1}{2}}}wheret1/2 is the half-life.

The half-life can be calculated by takinglogarithms (see above).t1/2=1log0.5⁡(λ)=−ln⁡(2)ln⁡(λ){\displaystyle t_{1/2}={1 \over \log _{0.5}(\lambda )}=-{\ln(2) \over \ln(\lambda )}}

Note that as the population is assumed to decline,λ < 1, soln(λ) < 0.

Mathematical relationship between geometric and logistic populations

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In geometric populations,R andλ represent growth constants (see2 and2.3). In logistic populations however, the intrinsic growth rate, also known as intrinsic rate of increase (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.[25] However, both sets of constants share the mathematical relationship below.[20]

The growth equation for exponential populations isNt=N0ert{\displaystyle N_{t}=N_{0}e^{rt}} wheree isEuler's number, auniversal constant often applicable in logistic equations, andr is the intrinsic growth rate.

To find the relationship between a geometric population and a logistic population, we assume theNt is the same for both models, and we expand to the following equality:N0ert=N0λtert=λtrt=tln⁡(λ){\displaystyle {\begin{aligned}N_{0}e^{rt}&=N_{0}\lambda ^{t}\\e^{rt}&=\lambda ^{t}\\rt&=t\ln(\lambda )\end{aligned}}}Giving usr=ln⁡(λ){\displaystyle r=\ln(\lambda )} andλ=er.{\displaystyle \lambda =e^{r}.}

Evolutionary game theory

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Main article:Evolutionary game theory

Evolutionary game theory was first developed byRonald Fisher in his 1930 articleThe Genetic Theory of Natural Selection.[26] In 1973John Maynard Smith formalised a central concept, theevolutionarily stable strategy.[27]

Population dynamics have been used in severalcontrol 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

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Population size inplants experiences significantoscillation due to the annual environmental oscillation.[28] Plant dynamics experience a higher degree of thisseasonality than do mammals, birds, orbivoltine insects.[28] When combined withperturbations due todisease, this often results inchaotic oscillations.[28]

In popular culture

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Thecomputer gameSimCity,Sim Earth and theMMORPGUltima Online, among others, tried tosimulate some of these population dynamics.

See also

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References

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  1. ^Malthus, Thomas Robert.An Essay on the Principle of Population: Library of Economics
  2. ^abTurchin, P. (2001). "Does Population Ecology Have General Laws?".Oikos.94 (1).John Wiley & Sons Ltd. (Nordic Society Oikos):17–26.Bibcode:2001Oikos..94...17T.doi:10.1034/j.1600-0706.2001.11310.x.S2CID 27090414.
  3. ^Gompertz, Benjamin (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.doi:10.1098/rstl.1825.0026.S2CID 145157003.
  4. ^Verhulst, P. H. (1838)."Notice sur la loi que la population poursuit dans son accroissement".Corresp. Mathématique et Physique.10:113–121.
  5. ^Richards, F. J. (June 1959)."A Flexible Growth Function for Empirical Use".Journal of Experimental Botany.10 (29):290–300.doi:10.1093/jxb/10.2.290.JSTOR 23686557. Retrieved16 November 2020.
  6. ^Hoppensteadt, F. (2006)."Predator-prey model".Scholarpedia.1 (10): 1563.Bibcode:2006SchpJ...1.1563H.doi:10.4249/scholarpedia.1563.
  7. ^Lotka, A. J. (1910)."Contribution to the Theory of Periodic Reaction".J. Phys. Chem.14 (3):271–274.doi:10.1021/j150111a004.
  8. ^Goel, N. S.; et al. (1971).On the Volterra and Other Non-Linear Models of Interacting Populations.Academic Press.
  9. ^Lotka, A. J. (1925).Elements of Physical Biology.Williams and Wilkins.
  10. ^Volterra, V. (1926). "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi".Mem. Acad. Lincei Roma.2:31–113.
  11. ^Volterra, V. (1931). "Variations and fluctuations of the number of individuals in animal species living together". In Chapman, R. N. (ed.).Animal Ecology.McGraw–Hill.
  12. ^Kingsland, S. (1995).Modeling Nature: Episodes in the History of Population Ecology. University of Chicago Press.ISBN 978-0-226-43728-6.
  13. ^abcBerryman, A. A. (1992)."The Origins and Evolution of Predator-Prey Theory"(PDF).Ecology.73 (5):1530–1535.Bibcode:1992Ecol...73.1530B.doi:10.2307/1940005.JSTOR 1940005. Archived fromthe original(PDF) on 2010-05-31.
  14. ^Arditi, R.; Ginzburg, L. R. (1989)."Coupling in predator-prey dynamics: ratio dependence"(PDF).Journal of Theoretical Biology.139 (3):311–326.Bibcode:1989JThBi.139..311A.doi:10.1016/s0022-5193(89)80211-5. Archived fromthe original(PDF) on 2016-03-04. Retrieved2020-11-17.
  15. ^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.doi:10.1016/s0169-5347(00)01908-x.PMID 10884706.
  16. ^Johnson, J. B.; Omland, K. S. (2004)."Model selection in ecology and evolution"(PDF).Trends in Ecology and Evolution.19 (2):101–108.CiteSeerX 10.1.1.401.777.doi:10.1016/j.tree.2003.10.013.PMID 16701236. Archived fromthe original(PDF) on 2011-06-11. Retrieved2010-01-25.
  17. ^abVandermeer, J. H.; Goldberg, D. E. (2003).Population ecology: First principles.Woodstock, Oxfordshire:Princeton University Press.ISBN 978-0-691-11440-8.
  18. ^Jahn, Gary C.; Almazan, Liberty P.; Pacia, Jocelyn B. (2005)."Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase ofHysteroneura setariae (Thomas) (Homoptera: Aphididae) on Rice (Oryza sativa L.)".Environmental Entomology.34 (4):938–43.doi:10.1603/0046-225X-34.4.938.
  19. ^Hassell, Michael P. (June 1980). "Foraging Strategies, Population Models and Biological Control: A Case Study".The Journal of Animal Ecology.49 (2):603–628.Bibcode:1980JAnEc..49..603H.doi:10.2307/4267.JSTOR 4267.
  20. ^abcd"Geometric and Exponential Population Models"(PDF). Archived fromthe original(PDF) on 2015-04-21. Retrieved2015-08-17.
  21. ^"Bacillus stearothermophilus NEUF2011".Microbe wiki.
  22. ^Chandler, M.; Bird, R.E.; Caro, L. (May 1975). "The replication time of the Escherichia coli K12 chromosome as a function of cell doubling time".Journal of Molecular Biology.94 (1):127–132.doi:10.1016/0022-2836(75)90410-6.PMID 1095767.
  23. ^Tobiason, D. M.; Seifert, H. S. (19 February 2010)."Genomic Content ofNeisseria Species".Journal of Bacteriology.192 (8):2160–2168.doi:10.1128/JB.01593-09.PMC 2849444.PMID 20172999.
  24. ^Boucher, Lauren (24 March 2015)."What is Doubling Time and How is it Calculated?".Population Education.
  25. ^"Population Growth"(PDF).University of Alberta. Archived fromthe original(PDF) on 2018-02-18. Retrieved2020-11-16.
  26. ^"Evolutionary Game Theory".Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information (CSLI), Stanford University. 19 July 2009.ISSN 1095-5054. Retrieved16 November 2020.
  27. ^Nanjundiah, V. (2005)."John Maynard Smith (1920–2004)"(PDF).Resonance.10 (11):70–78.doi:10.1007/BF02837646.S2CID 82303195.
  28. ^abcAltizer, Sonia; Dobson, Andrew; Hosseini, Parviez; Hudson, Peter; Pascual, Mercedes; Rohani, Pejman (2006)."Seasonality and the dynamics of infectious diseases". Reviews and Syntheses.Ecology Letters.9 (4).Blackwell Publishing Ltd (French National Centre for Scientific Research (CNRS)):467–84.Bibcode:2006EcolL...9..467A.doi:10.1111/J.1461-0248.2005.00879.X.hdl:2027.42/73860.PMID 16623732.S2CID 12918683.

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