Population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of
History
Population dynamics has traditionally been the dominant branch of
The beginning of population dynamics is widely regarded as the work of
A more general model formulation was proposed by
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:
Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation:
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
Epidemiology
Population dynamics overlap with another active area of research in mathematical biology:
Geometric populations
The mathematical formula below can used to model
When there is no migration to or from the population,
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.
At t + 1 | Nt+1 = λNt |
At t + 2 | Nt+2 = λNt+1 = λλNt = λ2Nt |
At t + 3 | Nt+3 = λNt+2 = λλ2Nt = λ3 Nt |
Therefore:
Doubling time
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% |
The doubling 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 = λt N0 by exploiting our knowledge of the fact that the population (N) is twice its size (2N) after the doubling time.[20]
The doubling time can be found by taking logarithms. For instance:
Therefore:
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: Nt = λt N0 by exploiting our knowledge of the fact that the population (N) is half its size (0.5N) after a half-life.[20]
The half-life can be calculated by taking logarithms (see above).
Geometric (R) growth constant
Finite (λ) growth constant
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
To find the relationship between a geometric population and a logistic population, we assume the Nt is the same for both models, and we expand to the following equality:
Evolutionary game theory
Evolutionary game theory was first developed by
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
In popular culture
The
See also
- 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
- ^ Malthus, Thomas Robert. An Essay on the Principle of Population: Library of Economics
- ^ S2CID 27090414.
- S2CID 145157003.
- ^ Verhulst, P. H. (1838). "Notice sur la loi que la population poursuit dans son accroissement". Corresp. Mathématique et Physique. 10: 113–121.
- JSTOR 23686557. Retrieved 16 November 2020.
- .
- .
- ^ Goel, N. S.; et al. (1971). On the Volterra and Other Non-Linear Models of Interacting Populations. Academic Press.
- Williams and Wilkins.
- ^ Volterra, V. (1926). "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi". Mem. Acad. Lincei Roma. 2: 31–113.
- McGraw–Hill.
- ISBN 978-0-226-43728-6.
- ^ JSTOR 1940005. Archived from the original(PDF) on 2010-05-31.
- doi:10.1016/s0022-5193(89)80211-5. Archived from the original(PDF) on 2016-03-04. Retrieved 2020-11-17.
- PMID 10884706.
- PMID 16701236. Archived from the original(PDF) on 2011-06-11. Retrieved 2010-01-25.
- ^ ISBN 978-0-691-11440-8.
- .
- JSTOR 4267.
- ^ a b c d "Geometric and Exponential Population Models" (PDF). Archived from the original (PDF) on 2015-04-21. Retrieved 2015-08-17.
- ^ "Bacillus stearothermophilus NEUF2011". Microbe wiki.
- PMID 1095767.
- PMID 20172999.
- ^ Boucher, Lauren (24 March 2015). "What is Doubling Time and How is it Calculated?". Population Education.
- ^ "Population Growth" (PDF). University of Alberta. Archived from the original (PDF) on 2018-02-18. Retrieved 2020-11-16.
- ISSN 1095-5054. Retrieved 16 November 2020.
- S2CID 82303195.
- ^ a b c
Altizer, Sonia; Dobson, Andrew; Hosseini, Parviez; Hudson, Peter; Pascual, Mercedes; Rohani, Pejman (2006). "Seasonality and the dynamics of infectious diseases". Reviews and Syntheses. S2CID 12918683.
Further reading
- 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". JSTOR 1931519.
External links
- 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.