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,^[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

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:

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

Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation:

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

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 is

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

${\displaystyle dN/dt}$

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.^{[citation needed]}

Geometric populations

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]

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

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.

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

Therefore:

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

Doubling time

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:

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

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

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]

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

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

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

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

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

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

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

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:

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

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

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

Evolutionary game theory

Evolutionary game theory was first developed by Ronald Fisher in his 1930 article The Genetic Theory of Natural Selection.^[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

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

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