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l 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 edit 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 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 2 18 this principle provided the basis for the subsequent predictive theories such as the demographic studies such as the work of benjamin gompertz 3 and pierre françois verhulst in the early 19th century who refined and adjusted the malthusian demographic model 4 a more general model formulation was proposed by f j richards in 1959 5 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 6 7 8 9 10 11 12 13 as well as the alternative arditi ginzburg equations 14 15 logistic function edit 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 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 d n d t b d b n d n b d n r n displaystyle mathrm d n over mathrm d t b d bn dn b d n rn where 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 the algebraic symbols 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 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 the logistic equation d n d t r n 1 n k displaystyle mathrm d n over mathrm d t rn left 1 n over k right where n is the population size r is the intrinsic rate of natural increase and k is the carrying 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 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 13 17 intrinsic rate of increase edit 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 the intrinsic rate of increase it is d n d t r n displaystyle mathrm d n over mathrm d t rn where the derivative d n d t displaystyle dn dt is the rate of increase of the population n is the population size and 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 18 epidemiology edit 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 edit operophtera brumata populations are geometric 19 the mathematical formula below is used to model 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 t is the index the generation t 0 for the first generation t 1 for the second generation etc the letter t is used because the index of a generation is time say n t 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 20 n t 1 n t b t d t i t e t displaystyle n_ t 1 n_ t b_ t d_ t i_ t e_ t where b t is the number of births in the population between generations t and t 1 d t is the number of deaths between generations t and t 1 i t is the number of immigrants added to the population between generations t and t 1 and e t is the number of emigrants moving out of the population between generations t and 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 t i t e t 0 the previous equation becomes n t 1 n t b t d t 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 time t is defined by b t b t n t the death rate at time t is defined by d t d t n t the previous equation can then be rewritten as n t 1 1 b t d t n t displaystyle n_ t 1 1 b_ t d_ t n_ t then 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 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 r b t d t to be the birth rate minus the death rate the geometric rate of increase do not depend on time t because both the birth rate minus the death rate do not with our assumption we obtain n t 1 1 r n t displaystyle begin aligned n_ t 1 left 1 r right n_ t end aligned this equation means that the sequence n t is geometric with first term n 0 and common ratio 1 r which we define to be λ λ is also called the finite rate of increase therefore by induction we obtain the expression of the population size at time t n t λ t n 0 displaystyle n_ t lambda t n_ 0 where λ t 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 n 10 the population at the tenth generation knowing n 0 the initial population and λ the finite rate of increase with the last formula the result is immediate by plugging t 10 whether with the previous one it is necessary to know n 9 n 8 n 2 until n 1 we can identify three cases if λ 1 i e if r 0 i e with the assumption that both birth and death rate do not depend on time t if b 0 d 0 i e if the birth rate is strictly greater than the death rate then the population size is increasing and 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 λ 1 i e if r 0 i e with the assumption that both birth and death rate do not depend on time t if b 0 d 0 i e if the birth rate is strictly smaller than the death rate then the population size is decreasing and tends to 0 if λ 1 i e if r 0 i e with the assumption that both birth and death rate do not depend on time t if b 0 d 0 i e if the birth rate is equal to the death rate then the population size is constant equal to the initial population n 0 doubling time edit 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 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 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 disclaimer bacterial populations are logistic instead of geometric nevertheless doubling times are applicable to both types of populations the doubling time t d 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 n t λ t n 0 by exploiting our knowledge of the fact that the population n is twice its size 2 n after the doubling time 20 n t d λ t d n 0 2 n 0 λ t d n 0 λ t d 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 taking logarithms for instance t d log 2 λ log 2 2 1 t d 1 log 2 λ displaystyle t_ d log _ 2 lambda log _ 2 2 1 implies t_ d 1 over log _ 2 lambda or t d ln λ ln 2 t d ln 2 ln λ displaystyle t_ d ln lambda ln 2 implies t_ d ln 2 over ln lambda therefore t d 1 log 2 λ 0 693 ln λ displaystyle t_ d frac 1 log _ 2 lambda frac 0 693 ln lambda half life of geometric populations edit 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 n t λ t n 0 by exploiting our knowledge of the fact that the population n is half its size 0 5 n after a half life 20 n t 1 2 λ t 1 2 n 0 1 2 n 0 λ t 1 2 n 0 λ t 1 2 1 2 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 where t 1 2 is the half life the half life can be calculated by taking logarithms see above t 1 2 1 log 0 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 so ln λ 0 mathematical relationship between geometric and logistic populations edit in geometric populations r and λ represent growth constants see 2 and 2 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 is n t n 0 e r t displaystyle n_ t n_ 0 e rt where e is euler s number a universal constant often applicable in logistic equations and r is the intrinsic growth rate to find the relationship between a geometric population and a logistic population we assume the n t is the same for both models and we expand to the following equality n 0 e r t n 0 λ t e r t λ t r t t ln λ displaystyle begin aligned n_ 0 e rt n_ 0 lambda t e rt lambda t rt t ln lambda end aligned giving us r ln λ displaystyle r ln lambda and λ e r displaystyle lambda e r evolutionary game theory edit main article evolutionary game theory evolutionary game theory was first developed by ronald fisher in his 1930 article the genetic theory of natural selection 26 in 1973 john maynard smith formalised a central concept the evolutionarily stable strategy 27 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 edit population size in plants experiences significant oscillation due to the annual environmental oscillation 28 plant dynamics experience a higher degree of this seasonality than do mammals birds or bivoltine insects 28 when combined with perturbations due to disease this often results in chaotic oscillations 28 in popular culture edit the computer game simcity sim earth and the mmorpg ultima online among others tried to simulate some of these population dynamics see also edit 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 references edit malthus thomas robert an essay on the principle of population library of economics 1 2 turchin 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 gompertz benjamin 1825 on the nature of the function expressive of the law of human mortality and on a ne...