Population dynamics

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.

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]

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:

$${\displaystyle {\frac {dN}{dT}}=B-D=bN-dN=(b-d)N=rN,}$$

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:

$${\displaystyle {\frac {dN}{dT}}=aN\left(1-{\frac {N}{K}}\right),}$$

where N is the biomass density, a is the maximum per-capita rate of change, 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 (aN) 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]

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

$${\displaystyle {\frac {dN}{dt}}=rN}$$

where the derivative ${\displaystyle dN/dt}$ 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]

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.

Operophtera brumata populations are geometric.[19]

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.[20]

$${\displaystyle N_{t+1}=N_{t}+B_{t}+I_{t}-D_{t}-E_{t}}$$

where N_t is the population size in generation t, and N_t+1 is the population size in the generation directly after N_t; B_t is the sum of births in the population between generations t and t + 1 (i.e. the birth rate); I_t is the sum of immigrants added to the population between generations; D_t is the sum of deaths between generations (death rate); and E_t is the sum of emigrants moving out of the population between generations.

When there is no migration to or from the population,

$${\displaystyle N_{t+1}=N_{t}+B_{t}-D_{t}.}$$

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.

$${\displaystyle {\begin{aligned}N_{t+1}&=N_{t}+RN_{t}\\N_{t+1}&=\left(1+R\right)N_{t}\\N_{t+1}&=\lambda N_{t}\end{aligned}}}$$

where λ = 1 + R is the finite rate of increase.

Therefore:

$${\displaystyle N_{t}=\lambda ^{t}N_{0}}$$

where λ^t is the Finite rate of increase raised to the power of the number of generations (e.g. for t + 2 [two generations] → λ², for t + 1 [one generation] → λ¹ = λ, and for t [before any generations - at time zero] → λ⁰ = 1

Doubling time

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 N₀ by exploiting our knowledge of the fact that the population (N) is twice its size (2N) after the doubling time.[20]

$${\displaystyle {\begin{aligned}N_{t_{d}}&=\lambda _{t_{d}}N_{0}\\2N_{0}&=\lambda _{t_{d}}N_{0}\\\lambda _{t_{d}}&=2\end{aligned}}}$$

The doubling time can be found by taking logarithms. For instance:

$${\displaystyle t_{d}\log _{2}(\lambda )=\log _{2}(2)=1\implies t_{d}={\frac {1}{\log _{2}(\lambda )}}}$$

Or:

$${\displaystyle t_{d}\ln(\lambda )=\ln(2)\implies t_{d}={\frac {\ln(2)}{\ln(\lambda )}}={\frac {0.693...}{\ln(\lambda )}}}$$

Therefore:

$${\displaystyle t_{d}={\frac {1}{\log _{2}(\lambda )}}={\frac {0.693...}{\ln(\lambda )}}}$$

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: N_t = λ^t N₀ by exploiting our knowledge of the fact that the population (N) is half its size (0.5N) after a half-life.[20]

$${\displaystyle N_{t_{1/2}}=\lambda ^{t_{1/2}}N_{0}\implies {\frac {1}{2}}N_{0}=\lambda ^{t_{1/2}}N_{0}\implies \lambda ^{t_{1/2}}={\frac {1}{2}}}$$

where t_1/2 is the half-life.

The half-life can be calculated by taking logarithms (see above).

$${\displaystyle t_{1/2}={\frac {1}{\log _{0.5}(\lambda )}}={\frac {\ln(0.5)}{\ln(\lambda )}}}$$

Geometric (R) growth constant

$${\displaystyle R=b-d}$$

$${\displaystyle {\begin{aligned}N_{t+1}&=N_{t}+RN_{t}\\N_{t+1}-N_{t}&=RN_{t}\\N_{t+1}-N_{t}&=\Delta N\end{aligned}}}$$

where ΔN is the change in population size between two generations (between generation t + 1 and t).

$${\displaystyle \Delta N=RN_{t}\implies {\frac {\Delta N}{N_{t}}}=T}$$

Finite (λ) growth constant

$${\displaystyle 1+R=\lambda }$$

$${\displaystyle N_{t+1}=\lambda N_{t}\implies \lambda ={\frac {N_{t+1}}{N_{t}}}}$$

Mathematical relationship between geometric and logistic populations

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

$${\displaystyle N_{t}=N_{0}e^{rt}}$$

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:

$${\displaystyle {\begin{aligned}N_{0}e^{rt}&=N_{0}\lambda ^{t}\\e^{rt}&=\lambda ^{t}\\rt&=t\ln(\lambda )\end{aligned}}}$$

Giving us

$${\displaystyle r=\ln(\lambda )}$$

and

$${\displaystyle \lambda =e^{r}.}$$

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.

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]

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

• 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.
• Smith, Frederick E. (1952). "Experimental methods in population dynamics: a critique". Ecology. 33 (4): 441–450. doi:10.2307/1931519. JSTOR 1931519.

• 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.