Editing Population dynamics

{{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 [[population]]s as [[dynamical systems]]. Population dynamics is a branch of [[Mathematical and theoretical biology|mathematical biology]], and uses mathematical techniques such as [[Differential equation|differential equations]] to model behaviour. Population dynamics is also closely related to other mathematical biology fields such as [[epidemiology]], and also uses techniques from evolutionary game theory in its modelling. 

== History ==
Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years,<ref name=malthus>Malthus, Thomas Robert. [[An Essay on the Principle of Population]]: Library of Economics</ref> 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) [[Exponential growth|exponentially]].<ref name="Turchin01">{{Cite journal | last = Turchin | first = P. | title = Does Population Ecology Have General Laws? | journal = [[Oikos (journal)|Oikos]] | publisher=[[John Wiley & Sons Ltd.]] ([[Nordic Society Oikos]]) |volume = 94 | issue = 1 | pages = 17–26 | year = 2001 | doi = 10.1034/j.1600-0706.2001.11310.x | bibcode = 2001Oikos..94...17T | s2cid = 27090414 }}</ref>{{rp|18}} This principle provided the basis for the subsequent predictive theories, such as the [[demography|demographic]] studies such as the work of [[Benjamin Gompertz]]<ref>{{cite journal |year=1825 |title=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 |journal=[[Philosophical Transactions of the Royal Society of London]] |volume=115 |pages=513–585 |url=http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55920 |doi= 10.1098/rstl.1825.0026 |last=Gompertz|first=Benjamin|s2cid=145157003 |doi-access=free }}</ref> and [[Pierre François Verhulst]] in the early 19th century, who refined and adjusted the Malthusian demographic model.<ref>{{cite journal|last=Verhulst|first=P. H.|url=https://books.google.com/books?id=8GsEAAAAYAAJ|title=Notice sur la loi que la population poursuit dans son accroissement|journal=[[Corresp. Mathématique et Physique]]|volume=10|pages=113–121|year=1838}}</ref>

A more general model formulation was proposed by [[F. J. Richards]] in 1959,<ref>{{cite journal |last1=Richards |first1=F. J. |date=June 1959 |title=A Flexible Growth Function for Empirical Use |url=https://www.jstor.org/stable/23686557 |journal=[[Journal of Experimental Botany]] |volume=10 |issue=29 |pages=290–300 |doi=10.1093/jxb/10.2.290 |jstor=23686557 |access-date=16 November 2020|url-access=subscription }}</ref> 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 equation|Lotka–Volterra predator-prey equations]] are another famous example,<ref name="scholarpedia">{{cite journal |last=Hoppensteadt |first=F. |title=Predator-prey model |journal=[[Scholarpedia]] |volume=1 |issue=10 |page=1563 |year=2006|doi=10.4249/scholarpedia.1563 |bibcode=2006SchpJ...1.1563H |doi-access=free }}</ref><ref>{{cite journal|last=Lotka|first=A. J.|title=Contribution to the Theory of Periodic Reaction|journal=[[Journal of Physical Chemistry A|J. Phys. Chem.]]|volume=14|issue=3|pages=271–274|year=1910|doi=10.1021/j150111a004|url=https://zenodo.org/record/1428768}}</ref><ref name="Goelmany">{{cite book|last=Goel|first=N. S.|display-authors=etal|title=On the Volterra and Other Non-Linear Models of Interacting Populations|publisher=[[Academic Press]]|year=1971}}</ref><ref>{{cite book|last=Lotka|first=A. J.|title=Elements of Physical Biology|publisher=[[Williams and Wilkins]]|year=1925}}</ref><ref>{{cite journal|last=Volterra|first=V.|title=Variazioni e fluttuazioni del numero d'individui in specie animali conviventi|journal=[[Accademia dei Lincei|Mem. Acad. Lincei Roma]]|volume=2|pages=31–113|year=1926}}</ref><ref>{{cite book|last=Volterra|first=V.|chapter=Variations and fluctuations of the number of individuals in animal species living together|title=Animal Ecology|editor-last=Chapman|editor-first=R. N.|publisher=[[McGraw–Hill]]|year=1931}}</ref><ref>{{cite book|last=Kingsland|first=S.|title=Modeling Nature: Episodes in the History of Population Ecology|publisher=University of Chicago Press|year=1995|isbn=978-0-226-43728-6}}</ref><ref name=Berryman1992>{{cite journal|last=Berryman|first=A. A.|url=http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|title=The Origins and Evolution of Predator-Prey Theory|journal=[[Ecology (journal)|Ecology]]|volume=73|issue=5|pages=1530–1535 |year=1992|url-status=dead|archive-url=https://web.archive.org/web/20100531204042/http://entomology.wsu.edu/profiles/06BerrymanWeb/Berryman%2892%29Origins.pdf|archive-date=2010-05-31|doi=10.2307/1940005|jstor=1940005|bibcode=1992Ecol...73.1530B }}</ref> as well as the alternative [[Arditi–Ginzburg equations]].<ref>{{cite journal|last1=Arditi|first1=R.|last2=Ginzburg|first2=L. R.|year=1989|url=http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf|title=Coupling in predator-prey dynamics: ratio dependence|journal=[[Journal of Theoretical Biology]]|volume=139|issue=3|pages=311–326|doi=10.1016/s0022-5193(89)80211-5|bibcode=1989JThBi.139..311A|access-date=2020-11-17|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304053545/http://life.bio.sunysb.edu/ee/ginzburglab/Coupling%20in%20Predator-Prey%20Dynamics%20-%20Arditi%20and%20Ginzburg,%201989.pdf|url-status=dead}}</ref><ref>{{cite journal|last1=Abrams|first1=P. A.|last2=Ginzburg|first2=L. R. |year=2000|title=The nature of predation: prey dependent, ratio dependent or neither?|journal=[[Trends in Ecology & Evolution]]| volume=15|issue=8|pages=337–341|doi=10.1016/s0169-5347(00)01908-x|pmid=10884706}}</ref>

== 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."<ref name="Johnson04">{{Cite journal | last1 = Johnson | first1 = J. B. | last2 = Omland | first2 = K. S. | title = Model selection in ecology and evolution. | journal = [[Trends in Ecology and Evolution]] | volume = 19 | issue = 2 | pages = 101–108 | year = 2004 | url = http://www.usm.maine.edu/bio/courses/bio621/model_selection.pdf | doi = 10.1016/j.tree.2003.10.013 | pmid = 16701236 | citeseerx = 10.1.1.401.777 | access-date = 2010-01-25 | archive-url = https://web.archive.org/web/20110611094158/http://www.usm.maine.edu/bio/courses/bio621/model_selection.pdf | archive-date = 2011-06-11 | url-status = dead }}</ref> 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:
<math display="block">{\mathrm{d}N\over\mathrm{d}t} = B - D = bN - dN = (b - d)N = rN,</math>
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 [[Algebra|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''}}).<ref name="Turchin01" /><ref name=Berryman1992/><ref name="Vandermeer03">{{Cite book |last1=Vandermeer |first1=J. H. |last2=Goldberg |first2=D. E. |title=Population ecology: First principles |place=[[Woodstock, Oxfordshire]] |publisher=[[Princeton University Press]] |year=2003 |isbn=978-0-691-11440-8}}</ref>

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]]:
<math display="block">{\mathrm{d}N\over\mathrm{d}t} = rN \left(1 - {N\over K}\right),</math>
where {{mvar|N}} is the [[population size]], {{mvar|r}} is the intrinsic [[rate of natural increase]], 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|''rN''}}) 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.<ref name=Berryman1992/><ref name="Vandermeer03" />

== Intrinsic rate of increase ==
{{Main|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 the ''intrinsic rate of increase''. It is
<math display="block">{\mathrm{d}N\over\mathrm{d}t} = r N</math>
where the [[derivative]] <math>dN/dt </math> 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 [[Integrated pest management|management]] to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.<ref name="Jahn2005">{{cite journal | doi=10.1603/0046-225X-34.4.938 |title=Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase of ''Hysteroneura'' setariae (Thomas) (Homoptera: Aphididae) on Rice (''Oryza sativa'' L.) |journal=[[Environmental Entomology]] |volume=34 |issue=4 | pages=938–43 |year=2005 |last1=Jahn |first1=Gary C. |last2=Almazan |first2=Liberty P. |last3=Pacia |first3=Jocelyn B. |doi-access=free }}</ref>

==Epidemiology==
Population dynamics overlap with another active area of research in mathematical biology: [[Mathematical modelling in epidemiology|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.<ref>{{cite journal |last1=Hassell|first1=Michael P.|title=Foraging Strategies, Population Models and Biological Control: A Case Study|journal=The Journal of Animal Ecology|date=June 1980|volume=49|issue=2|pages=603–628|doi=10.2307/4267|jstor=4267|bibcode=1980JAnEc..49..603H }}</ref>]]
The mathematical formula below is used to model [[Geometric progression|geometric]] populations. Such populations grow in discrete reproductive periods between intervals of [[abstinence]], as opposed to populations which grow without designated periods for reproduction. Say that the [[natural number]] {{mvar|t}} is the index the generation ({{mvar|1=t=0}} for the first generation, {{mvar|1=t=1}} for the second generation, etc.). The letter {{mvar|t}} is used because the index of a generation is time. Say {{mvar|N<sub>t</sub>}} denotes, at generation {{mvar|t}}, the number of individuals of the population that will reproduce, i.e. the population size at generation {{mvar|t}}. The population at the next generation,  which is the population at time {{mvar|t+1}} is:<ref name="afrc.uamont.edu">{{cite web|title=Geometric and Exponential Population Models|url=http://www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf|access-date=2015-08-17|archive-url=https://web.archive.org/web/20150421081753/http://www.afrc.uamont.edu/whited/Geometric%20and%20exponential%20population%20models.pdf|archive-date=2015-04-21|url-status=dead}}</ref>
<math display="block">
N_{t+1} = N_t + B_t - D_t + I_t - E_t
</math>
where 
*{{math|''B<sub>t</sub>''}} is the number of births in the population between generations {{mvar|t}} and {{math|''t'' + 1}},
*{{math|''D<sub>t</sub>''}} is the number of deaths between generations {{mvar|t}} and {{math|''t'' + 1}},
*{{math|''I<sub>t</sub>''}} is the number of [[immigrants]] added to the population between generations {{mvar|t}} and {{math|''t'' + 1}}, and 
*{{math|''E<sub>t</sub>''}} is the number of [[emigrants]] moving out of the population between generations {{mvar|t}} and {{math|''t'' + 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 all {{mvar|t}}, {{mvar|1=''I<sub>t</sub>'' =  ''E<sub>t</sub>'' = 0}}. The previous equation becomes:
<math display="block">N_{t+1} = N_t + B_t - D_t.</math>

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 time {{mvar|t}} is defined by {{math|1=''b<sub>t</sub>'' = ''B<sub>t</sub>'' / ''N<sub>t</sub>''}}.
* The death rate at time {{mvar|t}} is defined by {{math|1=''d<sub>t</sub>'' = ''D<sub>t</sub>'' / ''N<sub>t</sub>''}}.
The previous equation can then be rewritten as:
<math display="block">N_{t+1} = (1 + b_t - d_t)N_t.</math>

Then, we assume the birth and death rates do not depend on the time {{mvar|t}} (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 a [[geometric sequence]]. Then we define the geometric rate of increase {{mvar|1=''R'' = ''b<sub>t</sub>'' - ''d<sub>t</sub>''}} to be the birth rate minus the death rate. The geometric rate of increase do not depend on time {{mvar|t}}, because both the birth rate minus the death rate do not, with our assumption. We obtain:
<math display="block">\begin{align}
N_{t+1} &= \left(1 + R\right) N_t.
\end{align}</math>
This equation means that the sequence {{math|''(N<sub>t</sub>)''}} is [[Geometric sequence|geometric]] with first term {{math|''N<sub>0</sub>''}} and common ratio {{math|1=1 + ''R''}}, which we define to be {{math|''λ''}}. {{math|''λ''}} is also called the finite rate of increase.

Therefore, by [[Mathematical induction|induction]], we obtain the expression of the population size at time {{mvar|t}}:
<math display="block">N_t = \lambda^t N_0</math>
where {{math|''λ''<sup>''t''</sup>}} 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 calculator {{math|1=''N''<sub>10</sub>}}, the population at the tenth generation, knowing {{math|''N''<sub>0</sub>}} the initial population and {{math|''λ''}} the finite rate of increase. With the last formula, the result is immediate by plugging {{math|1=''t'' = 10}}, whether with the previous one it is necessary to know {{math|''N''<sub>9</sub>}}, {{math|''N''<sub>8</sub>}}, ..., {{math|''N''<sub>2</sub>}} until {{math|''N''<sub>1</sub>}}.

We can identify three cases:
* If {{math|1=''λ'' > 1}}, i.e. if {{math|1=''R'' > 0}}, i.e. (with the assumption that both birth and death rate do not depend on time {{math|''t''}}) if {{math|1=''b<sub>0</sub>'' > ''d<sub>0</sub>''}}, i.e. if the birth rate is strictly greater than the death rate, then the population size is increasing and [[Limit (mathematics)|tends]] 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.
* If {{math|1=''λ'' < 1}}, i.e. if {{math|1=''R'' < 0}}, i.e. (with the assumption that both birth and death rate do not depend on time {{math|''t''}}) if {{math|1=''b<sub>0</sub>'' < ''d<sub>0</sub>''}}, i.e. if the birth rate is strictly smaller than the death rate, then the population size is decreasing and [[Limit (mathematics)|tends]] to {{math|1=0}}.
* If {{math|1=''λ'' = 1}}, i.e. if {{math|1=''R'' = 0}}, i.e. (with the assumption that both birth and death rate do not depend on time {{math|''t''}}) if {{math|1=''b<sub>0</sub>'' = ''d<sub>0</sub>''}}, i.e. if the birth rate is equal to the death rate, then the population size is constant, equal to the initial population {{math|1=''N''<sub>0</sub>}}.

=== 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 [[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 [[eukaryote]]s 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. ''[[Bacillus stearothermophilus|G. stearothermophilus]]'', ''[[Escherichia coli|E. coli]]'', and ''[[Neisseria meningitidis|N. meningitidis]]'' have 20 minute,<ref>{{cite web|title=Bacillus stearothermophilus NEUF2011|url=https://microbewiki.kenyon.edu/index.php/Bacillus_stearothermophilus_NEUF2011|website=Microbe wiki}}</ref> 30 minute,<ref>{{cite journal|last1=Chandler|first1=M.| last2=Bird|first2=R.E.|last3=Caro|first3=L.|title=The replication time of the Escherichia coli K12 chromosome as a function of cell doubling time|journal=[[Journal of Molecular Biology]]|date=May 1975|volume=94|issue=1|pages=127–132| doi=10.1016/0022-2836(75)90410-6| pmid=1095767}}</ref> and 40 minute<ref>{{cite journal|last1=Tobiason|first1=D. M.| last2=Seifert|first2=H. S.|title=Genomic Content of ''Neisseria'' Species|journal=[[Journal of Bacteriology]]|date=19 February 2010| volume=192|issue=8|pages=2160–2168|doi=10.1128/JB.01593-09|pmid=20172999|pmc=2849444}}</ref> 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''
|-
| 30 || 44.4%
|-
| 60 || 53.3%
|-
| 90 || 64.9%
|-
| 120 || 72.7%
|-
| →∞ || 100%
|}

{|
! Time in minutes !! % that is ''E. coli''
|-
| 30 || 29.6%
|-
| 60 || 26.7%
|-
| 90 || 21.6%
|-
| 120 || 18.2%
|-
| →∞ || 0.00%
|}

{|
! Time in minutes !! % that is ''N. meningitidis''
|-
| 30 || 25.9%
|-
| 60 || 20.0%
|-
| 90 || 13.5%
|-
| 120 || 9.10%
|-
| →∞ || 0.00%
|}

<small>''Disclaimer: Bacterial populations are [[logistic function|logistic]] instead of geometric. Nevertheless, doubling times are applicable to both types of populations.''</small>
]]

The [[doubling time]] ({{math|1=''t<sub>d</sub>''}}) of a population is the time required for the population to grow to twice its size.<ref>{{cite web|title=What is Doubling Time and How is it Calculated?|first=Lauren|last=Boucher|date=24 March 2015| website=Population Education|url=https://www.populationeducation.org/content/what-doubling-time-and-how-it-calculated}}</ref> We can calculate the doubling time of a geometric population using the equation: {{math|1=''N''<sub>''t''</sub> = ''λ''<sup>''t''</sup> ''N''<sub>0</sub>}} by exploiting our knowledge of the fact that the population ({{mvar|N}}) is twice its size ({{math|2''N''}}) after the doubling time.<ref name="afrc.uamont.edu"/>

<math display="block">\begin{align}
N_{t_d} &= \lambda^{t_d} N_0 \\
2 N_0 &= \lambda^{t_d} N_0 \\
\lambda^{t_d} &= 2
\end{align}</math>

The doubling time can be found by taking [[logarithms]]. For instance:
<math display="block">t_d \log_2(\lambda) = \log_2(2) = 1
\implies t_d = {1\over\log_2(\lambda)}</math>
Or:
<math display="block">t_d \ln(\lambda) = \ln(2)
\implies t_d = {\ln(2)\over\ln(\lambda)}</math>

Therefore: <math display="block">t_d = \frac{1}{\log_2(\lambda)} = \frac{0.693...}{\ln(\lambda)}</math>

=== 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''<sub>''t''</sub> = ''λ''<sup>''t''</sup> ''N''<sub>0</sub>}} by exploiting our knowledge of the fact that the population ({{mvar|N}}) is half its size ({{math|0.5''N''}}) after a half-life.<ref name="afrc.uamont.edu"/>

<math display="block">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}</math>
where {{math|''t''<sub>1/2</sub>}} is the half-life.

The half-life can be calculated by taking [[logarithms]] (see above).
<math display="block">t_{1/2} = {1\over\log_{0.5}(\lambda)} = -{\ln(2)\over\ln(\lambda)}</math>

Note that as the population is assumed to decline, {{math|1=''λ'' < 1}}, so {{math|1=ln(''λ'') < 0}}.

===Mathematical relationship between geometric and logistic populations===

In geometric populations, {{mvar|R}} and {{mvar|λ}} represent growth constants (see [[Population ecology#Geometric populations|2]] and [[Population ecology#Geometric (R) and finite (λ) growth constants|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.<ref>{{cite web|title=Population Growth|website=[[University of Alberta]]|url=http://www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf|access-date=2020-11-16|archive-date=2018-02-18|archive-url=https://web.archive.org/web/20180218231304/http://www.biology.ualberta.ca/courses.hp/bio208/Lecture_Notes_files/Biol208_Lecture19_PopulationGrowth.pdf|url-status=dead}}</ref> However, both sets of constants share the mathematical relationship below.<ref name="afrc.uamont.edu"/>

The growth equation for exponential populations is <math display="block">N_t = N_0 e^{rt}</math> 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|''N''<sub>''t''</sub>}} is the same for both models, and we expand to the following equality:
<math display="block">\begin{align}
N_0 e^{rt} &= N_0 \lambda^t \\
e^{rt} &= \lambda^t \\
rt &= t \ln(\lambda)
\end{align}</math>
Giving us <math display="block">r = \ln(\lambda)</math> and <math display="block">\lambda = e^r.</math>

== Evolutionary game theory ==
{{main|Evolutionary game theory}}
Evolutionary game theory was first developed by [[Ronald Fisher]] in his 1930 article ''[[The Genetic Theory of Natural Selection]]''.<ref>{{cite web |url=https://plato.stanford.edu/entries/game-evolutionary/ |title=Evolutionary Game Theory |author=<!--Not stated--> |date=19 July 2009 |website=Stanford Encyclopedia of Philosophy |publisher=The Metaphysics Research Lab, Center for the Study of Language and Information (CSLI), Stanford University |issn=1095-5054 |access-date=16 November 2020}}</ref> In 1973 [[John Maynard Smith]] formalised a central concept, the [[evolutionarily stable strategy]].<ref>{{Cite journal | last1 = Nanjundiah | first1 = V. | title = John Maynard Smith (1920–2004) | doi = 10.1007/BF02837646 | journal = [[Resonance (journal)|Resonance]] | volume = 10 | issue = 11 | pages = 70–78 | year = 2005 | s2cid = 82303195 | url = http://www.ias.ac.in/resonance/Nov2005/pdf/Nov2005p70-78.pdf}}</ref>

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 ([[Single-input single-output system|SISO]]) systems. Some other examples of applications are military campaigns, [[Water supply network|water distribution]], [[Dispatchable generation|dispatch of distributed generators]], lab experiments, transport problems, communication problems, among others.

== Oscillatory ==
Population size in [[plant]]s experiences significant [[oscillation]] due to the annual environmental oscillation.<ref name = "Infectious-Diseases" /> Plant dynamics experience a higher degree of this [[seasonality]] than do mammals, birds, or [[bivoltine]] insects.<ref name = "Infectious-Diseases" /> When combined with [[perturbation (biology)|perturbation]]s due to [[plant disease|disease]], this often results in [[chaotic oscillation]]s.<ref name = "Infectious-Diseases" >
 {{ Cite journal |
           pmid=16623732|
            doi=10.1111/J.1461-0248.2005.00879.X|
          title=Seasonality and the dynamics of infectious diseases|
          s2cid=12918683|
          last1=Altizer|
         first1=Sonia|
          last2=Dobson|
         first2=Andrew|
          last3=Hosseini|
         first3=Parviez|
          last4=Hudson|
         first4=Peter|
          last5=Pascual|
         first5=Mercedes|
          last6=Rohani|
         first6=Pejman|
      publisher=[[Blackwell Publishing Ltd]] ([[French National Centre for Scientific Research]] (CNRS))|
           year=2006|
         volume=9|
          issue=4|
          pages=467–84|
     department=Reviews and Syntheses|
        journal=[[Ecology Letters]]|
       language=en|doi-access=free|
 bibcode=2006EcolL...9..467A|hdl=2027.42/73860|hdl-access=free}}
</ref>

==In popular culture==
The [[computer game]] ''[[SimCity (1989 video game)|SimCity]]'', ''[[Sim Earth]]'' and the [[MMORPG]] [[Ultima Online]], among others, tried to [[computer simulation|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|''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}}
* [[Peter Turchin|Turchin, P.]] 2003. ''Complex Population Dynamics: a Theoretical/Empirical Synthesis''. Princeton, NJ: Princeton University Press.
* {{cite journal|last =Smith|first=Frederick E.|title=Experimental methods in population dynamics: a critique|journal =[[Ecology (journal)|Ecology]]|volume=33|issue=4|pages=441–450|year=1952|doi=10.2307/1931519|jstor =1931519|bibcode=1952Ecol...33..441S }}

== External links ==
* [http://www.thomas-brey.de/science/virtualhandbook 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.

{{biological organisation}}
{{Population}}

{{Authority control}}

[[Category:Fisheries science]]
[[Category:Population dynamics]]