Population dynamics

Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.

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Voiceover: Population dynamics looks at how the population of a country or a region or even the world changes. It takes into account the factors that increase a population and the factors that decrease population to create a total growth rate. There are three factors that contribute to this total growth rate. These are fertility, migration, and mortality. Fertility is the natural ability of human beings to have babies. And babies, obviously, add to the population. Migration looks at the number of people who are moving into and out of countries. It doesn't change the total number of people living on the planet. But it does change the number of people living in a specific country or region. This doesn't mean that when you go on vacation to a different country, that you are migrating there. Migration means that you are moving somewhere permanently to live and work and eventually die. Which brings us to our third factor, mortality. Which is the fact that everyone eventually will die. This obviously decreases the population. In order to measure these three factors, we use rates to measure the number of people who are born. The number of people that move out of or into a different country, and the number of people who die in a certain period of time. Usually, we measure birth migration and death rates over a year's worth of population change. Because it is enough time that an obvious change is visible, but not so much time that we miss trends in how the population changes. And as human beings, we tend to like nice, round measurements like one year. We also like to compare rates that are actually comparable. So all these rates are measured per 1,000 people. This way, we can compare the different rates equally because they are all scaled to that same value. A worldwide birth rate of about 18 point nine births per 1,000 people is much easier to grasp than looking at the around 134 million total births worldwide. And it's easier to compare country birth rates when they are scaled, rather than looking at the total births. Take Mali, for example, where about 700 thousand people will be born this year. This doesn't seem like a lot compared to the 134 million births in the world. But when scaled, the birth rate in Mali is about 46 births per 1,000 people. Which is more than twice the world average. That's a pretty big difference. All right, so now that we've got some of the background and terminology out of the way, we can look at what it is that affects population changes. For this, I'm going to just look at the population growth within a single country. There are two factors that will increase a population of a country, births and immigrations. Immigration is the movement of a person into a country. Conversely, movement out of the country's called emigration. But that decreases the population, so I'll get to that in a minute. As we've already seen, birth rate is the number of births per 1,000 people. But another way to look at it is the total fertility rate, which is the number of kids a woman is predicted to give birth to in her childbearing years. That's why you hear of the typical family with two point one kids. On average, a woman in the U.S. is predicted to give birth to two point one children in her life. This means that the total population of the U.S. will slowly increase. A total fertility rate of exactly two neither adds nor subtracts from the population. This is because the woman is giving birth to the number of people that created the children. This rate only looks at the number of children a woman will bear. So the two children exactly replace the biological parents. And finally, a fertility rate of less than two will decrease the population. Immigration, as I mentioned before, is the movement of people into a country. This number is also scaled and, again, is measured by the number of people immigrating per 1,000 people in the country. If you want to look at the total population increase, take the number of births plus the number of immigrations per 1,000 people, and there you go! That is the rate at which people were added to the population. Now, we can look at the factors that decrease a country's population And you guessed it, death and emigration. When we look at the number of people who have died, we often refer to mortality rates, which are the number of deaths per 1,000 people. You have to be kind of careful when looking at mortality rates, especially when comparing the mortality rates of different countries. Just because the country has a high mortality rate, doesn't mean there are a lot of young or unnatural deaths. For this, we can take a look at population pyramids. A population pyramid graphs the age and sex distribution of a population. On either side, you have males and females. And along the vertical axis, you have increasing age. If there are a lot of elderly people living in one country, who are living a long time and dying of old age, you get either a stationary or constrictive pyramid. These kinds of population pyramids usually indicate low birth rates and low death rates throughout the population. A constrictive pyramid indicates that there are fewer young people than old and is generally seen for very developed countries. So the death rate of this country with lots of old people doesn't compare well to a country where, perhaps, people are dying young from disease. This country's population can be modeled by an expansive population pyramid, which indicates high birth rates and high death rates. A better comparison is to look at age-specific mortality rates, so you can see how many people are dying within a specific age range. Now, you can compare just elderly mortality rates between different countries or just the mortality rates for any specific age or age range. Say, 20 to 24-year-olds, or 45 to 49-year-olds. There's a lot more information you can glean from age-specific comparisons. When you split up the mortality rates by age, you can get what's called a life table or a mortality table. This table tells you the probability that someone will die given their age, which can vary from country to country. But when looking at the population of a single country, an all-encompassing mortality rate is sufficient. The second factor that decreases a population is the emigration of people to other countries. Just like all the other rates I've mentioned, emigration rate is measured by the number of people emigrating per 1,000 people in the country. So if you take the number of deaths plus the number of emigrations per 1,000 people, you can get the rate at which people are removed from the country. Sometimes, it's interesting to look at just the migration statistics. The net migration is the difference between the number of people entering the country and the number of people leaving the country. There are many reasons for people to migrate from the country of their birth to a new country. Worldwide, the trend is generally that people are moving to the more industrialized countries because of the potential for a better life. Some people leave their homes for political reasons and become refugees in a new country. Some people migrate for their jobs, or because they just want to live somewhere foreign. For many of these same reasons, people will often move within their own country, which is called internal migration. While this doesn't change the total population of the country, it can affect the economics or cultures of a country. Often, internal migration is a large factor in urbanization, as people move from rural areas to urban areas. Fertility, migration, and mortality all contribute to the growth rate of a country. But remember that growth rate is not always a positive number. If we want to look at this as an equation, you have the initial population, plus the number of births, minus the number of deaths, plus the number of people emigrating into the country, minus the number of people emigrating out of the country. So now, you have your current population. And if this current population is less than the initial population, then you get a negative growth rate for that country. So while our world population continues to grow, the growth rate of some countries is, in fact, negative, as more people die and leave that country than are born and move into it. So as you can see, there's a lot to consider when looking at how a population changes. You can't just look at births or deaths, alone. You have to take into account that people move to different countries. And even if you take into account every factor you can think of, it is still only an estimate. We can't go around to get a headcount of all seven billion-plus people in the world. We can try with surveys and records. But in some countries, even that just isn't feasible. So we extrapolate and estimate as well as we can. So we can study how our populations are changing.

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:

{\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:

{\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]

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

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

where the derivative

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

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

*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]

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,

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.

{\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.

At $t + 1$	$N t +1 = λN t$
At $t + 2$	$N t +2 = λN t +1 = λλN t = λ 2 N t$
At $t + 3$	$N t +3 = λN t +2 = λλ 2 N t = λ 3 N t$

Therefore:

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] →

λ 2

, for

t + 1

[one generation] →

λ 1 = λ

, and for

t

[before any generations - at time zero] →

λ 0 = 1

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 ( $t d$ ) 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]

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

Or:

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

Therefore:

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: $N t = λ t N 0$ 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}}=\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}={\frac {1}{\log _{0.5}(\lambda )}}={\frac {\ln(0.5)}{\ln(\lambda )}}

Geometric (R) growth constant

R=b-d

{\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

Δ N

is the change in population size between two generations (between generation

t + 1

and

t

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

Finite (λ) growth constant

1+R=\lambda

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

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:

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

Giving us

r=\ln(\lambda )

and

\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.

References

^ 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. 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 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. Archived from the original (PDF) on 2016-03-04. Retrieved 2020-11-17.
^ 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. Bibcode:1980JAnEc..49..603H. 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. Bibcode:2006EcolL...9..467A. doi:10.1111/J.1461-0248.2005.00879.X. hdl:2027.42/73860. PMID 16623732. S2CID 12918683.

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.

v t e Hierarchy of life
Biosphere > Biome > Ecosystem > Biocoenosis > Population > Organism > Organ system > Organ > Tissue > Cell > Organelle > Biomolecular complex > Macromolecule > Biomolecule

This page was last edited on 21 March 2024, at 02:02

From Wikipedia, the free encyclopedia

YouTube Encyclopedic

Transcription

History

Logistic function

Intrinsic rate of increase

Epidemiology

Geometric populations

Doubling time

Half-life of geometric populations

Geometric (R) growth constant

Finite (λ) growth constant

Mathematical relationship between geometric and logistic populations

Evolutionary game theory

Oscillatory

In popular culture

See also

References

Further reading

External links