Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years, 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.: 18 This principle provided the basis for the subsequent predictive theories, such as the demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.
A more general model formulation was proposed by F. J. Richards in 1959, 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, as well as the alternative Arditi–Ginzburg equations.
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 ('null hypothesis') of no external influence. Models can be more mathematically complex where "...several competing hypotheses are simultaneously confronted with the data." 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
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.
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.
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|
The doubling time (td) of a population is the time required for the population to grow to twice its size. 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.
The doubling time can be found by taking logarithms. For instance:
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.
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. However, both sets of constants share the mathematical relationship below.
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 Ronald Fisher in his 1930 article The Genetic Theory of Natural Selection. In 1973 John Maynard Smith formalised a central concept, the evolutionarily stable strategy.
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. Plant dynamics experience a higher degree of this seasonality than do mammals, birds, or bivoltine insects. When combined with perturbations due to disease, this often results in chaotic oscillations.
In popular culture
The computer game SimCity, Sim Earth and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.
- 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
- ^ Malthus, Thomas Robert. An Essay on the Principle of Population: Library of Economics
- ^ a b Turchin, P. (2001). "Does Population Ecology Have General Laws?". Oikos. John Wiley & Sons Ltd. (Nordic Society Oikos). 94 (1): 17–26. 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 New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society of London. 115: 513–585. doi:10.1098/rstl.1825.0026. S2CID 145157003.
- ^ Verhulst, P. H. (1838). "Notice sur la loi que la population poursuit dans son accroissement". Corresp. Mathématique et Physique. 10: 113–121.
- ^ Richards, F. J. (June 1959). "A Flexible Growth Function for Empirical Use". Journal of Experimental Botany. 10 (29): 290–300. doi:10.1093/jxb/10.2.290. JSTOR 23686557. Retrieved 16 November 2020.
- ^ Hoppensteadt, F. (2006). "Predator-prey model". Scholarpedia. 1 (10): 1563. Bibcode:2006SchpJ...1.1563H. doi:10.4249/scholarpedia.1563.
- ^ Lotka, A. J. (1910). "Contribution to the Theory of Periodic Reaction". J. Phys. Chem. 14 (3): 271–274. doi:10.1021/j150111a004.
- ^ Goel, N. S.; et al. (1971). On the Volterra and Other Non-Linear Models of Interacting Populations. Academic Press.
- ^ Lotka, A. J. (1925). Elements of Physical Biology. Williams and Wilkins.
- ^ Volterra, V. (1926). "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi". Mem. Acad. Lincei Roma. 2: 31–113.
- ^ Volterra, V. (1931). "Variations and fluctuations of the number of individuals in animal species living together". In Chapman, R. N. (ed.). Animal Ecology. McGraw–Hill.
- ^ Kingsland, S. (1995). Modeling Nature: Episodes in the History of Population Ecology. University of Chicago Press. ISBN 978-0-226-43728-6.
- ^ a b c Berryman, A. A. (1992). "The Origins and Evolution of Predator-Prey Theory" (PDF). Ecology. 73 (5): 1530–1535. doi:10.2307/1940005. JSTOR 1940005. Archived from the original (PDF) on 2010-05-31.
- ^ Arditi, R.; Ginzburg, L. R. (1989). "Coupling in predator-prey dynamics: ratio dependence" (PDF). Journal of Theoretical Biology. 139 (3): 311–326. Bibcode:1989JThBi.139..311A. doi:10.1016/s0022-5193(89)80211-5.
- ^ Abrams, P. A.; Ginzburg, L. R. (2000). "The nature of predation: prey dependent, ratio dependent or neither?". Trends in Ecology & Evolution. 15 (8): 337–341. doi:10.1016/s0169-5347(00)01908-x. PMID 10884706.
- ^ Johnson, J. B.; Omland, K. S. (2004). "Model selection in ecology and evolution" (PDF). Trends in Ecology and Evolution. 19 (2): 101–108. CiteSeerX 10.1.1.401.777. doi:10.1016/j.tree.2003.10.013. PMID 16701236. Archived from the original (PDF) on 2011-06-11. Retrieved 2010-01-25.
- ^ a b Vandermeer, J. H.; Goldberg, D. E. (2003). Population ecology: First principles. Woodstock, Oxfordshire: Princeton University Press. ISBN 978-0-691-11440-8.
- ^ Jahn, Gary C.; Almazan, Liberty P.; Pacia, Jocelyn B. (2005). "Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase of Hysteroneura setariae (Thomas) (Homoptera: Aphididae) on Rice (Oryza sativa L.)". Environmental Entomology. 34 (4): 938–43. doi:10.1603/0046-225X-34.4.938.
- ^ Hassell, Michael P. (June 1980). "Foraging Strategies, Population Models and Biological Control: A Case Study". The Journal of Animal Ecology. 49 (2): 603–628. doi:10.2307/4267. 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.
- ^ Chandler, M.; Bird, R.E.; Caro, L. (May 1975). "The replication time of the Escherichia coli K12 chromosome as a function of cell doubling time". Journal of Molecular Biology. 94 (1): 127–132. doi:10.1016/0022-2836(75)90410-6. PMID 1095767.
- ^ Tobiason, D. M.; Seifert, H. S. (19 February 2010). "Genomic Content of Neisseria Species". Journal of Bacteriology. 192 (8): 2160–2168. doi:10.1128/JB.01593-09. PMC 2849444. 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.
- ^ "Evolutionary Game Theory". Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information (CSLI), Stanford University. 19 July 2009. ISSN 1095-5054. Retrieved 16 November 2020.
- ^ Nanjundiah, V. (2005). "John Maynard Smith (1920–2004)" (PDF). Resonance. 10 (11): 70–78. doi:10.1007/BF02837646. 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. Ecology Letters. Blackwell Publishing Ltd (French National Centre for Scientific Research (CNRS)). 9 (4): 467–84. doi:10.1111/J.1461-0248.2005.00879.X. PMID 16623732. S2CID 12918683.
- 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.