Population model

A population model is a type of mathematical model that is applied to the study of population dynamics.


Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change over time or in relation to each other. Many patterns can be noticed by using population modeling as a tool.[1]

Ecological population modeling is concerned with the changes in parameters such as population size and age distribution within a population. This might be due to interactions with the environment, individuals of their own species, or other species.[2]

Population models are used to determine maximum harvest for agriculturists, to understand the dynamics of biological invasions, and for environmental conservation. Population models are also used to understand the spread of parasites, viruses, and disease.[2]

Another way populations models are useful are when species become endangered. Population models can track the fragile species and work and curb the decline. [1]


Late 18th-century biologists began to develop techniques in population modeling in order to understand the dynamics of growing and shrinking of all populations of living organisms. Thomas Malthus was one of the first to note that populations grew with a geometric pattern while contemplating the fate of humankind.[3] One of the most basic and milestone models of population growth was the logistic model of population growth formulated by Pierre François Verhulst in 1838. The logistic model takes the shape of a sigmoid curve and describes the growth of a population as exponential, followed by a decrease in growth, and bound by a carrying capacity due to environmental pressures.[4]

Population modeling became of particular interest to biologists in the 20th century as pressure on limited means of sustenance due to increasing human populations in parts of Europe were noticed by biologist like Raymond Pearl. In 1921 Pearl invited physicist Alfred J. Lotka to assist him in his lab. Lotka developed paired differential equations that showed the effect of a parasite on its prey. Mathematician Vito Volterra equated the relationship between two species independent from Lotka. Together, Lotka and Volterra formed the Lotka–Volterra model for competition that applies the logistic equation to two species illustrating competition, predation, and parasitism interactions between species.[3] In 1939 contributions to population modeling were given by Patrick Leslie as he began work in biomathematics. Leslie emphasized the importance of constructing a life table in order to understand the effect that key life history strategies played in the dynamics of whole populations. Matrix algebra was used by Leslie in conjunction with life tables to extend the work of Lotka.[5] Matrix models of populations calculate the growth of a population with life history variables. Later, Robert MacArthur and E. O. Wilson characterized island biogeography. The equilibrium model of island biogeography describes the number of species on an island as an equilibrium of immigration and extinction. The logistic population model, the Lotka–Volterra model of community ecology, life table matrix modeling, the equilibrium model of island biogeography and variations thereof are the basis for ecological population modeling today.[6]


Logistic growth equation:

Lotka-Volterra equation:

Island biogeography:

Species–area relationship:

Examples of individual-based models

Logical deterministic individual-based cellular automata model of single species population growth
Logical deterministic individual-based cellular automata model of an ecosystem with one species. The model demonstrates a mechanism of S-shaped population growth.
Logical deterministic individual-based cellular automata model of interspecific competition for a single limited resource
Logical deterministic individual-based cellular automata model of interspecific competition for a single limited resource. A mechanism of competitive exclusion of one species by another.

See also


  1. ^ Worster, Donald (1994). Nature's Economy. Cambridge University Press. pp. 398–401.
  2. ^ a b Uyenoyama, Marcy (2004). Rama Singh (ed.). The Evolution of Population Biology. Cambridge University Press. pp. 1–19.
  3. ^ a b McIntosh, Robert (1985). The Background of Ecology. Cambridge University Press. pp. 171–198.
  4. ^ Renshaw, Eric (1991). Modeling Biological Populations in Space and Time. Cambridge University Press. pp. 6–9.
  5. ^ Kingsland, Sharon (1995). Modeling Nature: Episodes in the History of Population Ecology. University of Chicago Press. pp. 127–146.
  6. ^ Gotelli, Nicholas (2001). A Primer of Ecology. Sinauer.

External links

  • GreenBoxes code sharing network. Greenboxes (Beta) is a repository for open-source population modeling code. Greenboxes allows users an easy way to share their code and to search for others shared code.
Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Beverton–Holt model

The Beverton–Holt model is a classic discrete-time population model which gives the expected number n t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,

Here R0 is interpreted as the proliferation rate per generation and K = (R0 − 1) M is the carrying capacity of the environment. The Beverton–Holt model was introduced in the context of fisheries by Beverton & Holt (1957). Subsequent work has derived the model under other assumptions such as contest competition (Brännström & Sumpter 2005), within-year resource limited competition (Geritz & Kisdi 2004) or even as the outcome of a source-sink Malthusian patches linked by density-dependent dispersal (Bravo de la Parra et al. 2013). The Beverton–Holt model can be generalized to include scramble competition (see the Ricker model, the Hassell model and the Maynard Smith–Slatkin model). It is also possible to include a parameter reflecting the spatial clustering of individuals (see Brännström & Sumpter 2005).

Despite being nonlinear, the model can be solved explicitly, since it is in fact an inhomogeneous linear equation in 1/n. The solution is[citation needed]

Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is

and its solution is

Chicken (game)

The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while it is to both players’ benefit if one player yields, the other player's optimal choice depends on what their opponent is doing: if the player opponent yields, they should not, but if the opponent fails to yield, the player should.

The name "chicken" has its origins in a game in which two drivers drive towards each other on a collision course: one must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a "chicken", meaning a coward; this terminology is most prevalent in political science and economics. The name "hawk–dove" refers to a situation in which there is a competition for a shared resource and the contestants can choose either conciliation or conflict; this terminology is most commonly used in biology and evolutionary game theory. From a game-theoretic point of view, "chicken" and "hawk–dove" are identical; the different names stem from parallel development of the basic principles in different research areas. The game has also been used to describe the mutual assured destruction of nuclear warfare, especially the sort of brinkmanship involved in the Cuban Missile Crisis.

Effective population size

The effective population size is the number of individuals that an idealised population would need to have in order for some specified quantity of interest to be the same in the idealised population as in the real population. Idealised populations are based on unrealistic but convenient simplifications such as random mating, simultaneous birth of each new generation, constant population size, and equal numbers of children per parent. In some simple scenarios, the effective population size is the number of breeding individuals in the population. However, for most quantities of interest and most real populations, the census population size N of a real population is usually larger than the effective population size Ne. The same population may have multiple effective population sizes, for different properties of interest, including for different genetic loci.

The effective population size is most commonly measured with respect to the coalescence time. In an idealised diploid population with no selection at any locus, the expectation of the coalescence time in generations is equal to twice the census population size. The effective population size is measured as within-species genetic diversity divided by four times the mutation rate , because in such an idealised population, the heterozygosity is equal to . In a population with selection at many loci and abundant linkage disequilibrium, the coalescent effective population size may not reflect the census population size at all, or may reflect its logarithm.

The concept of effective population size was introduced in the field of population genetics in 1931 by the American geneticist Sewall Wright.

Empirical process

In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state.

For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state (without rescaling).

In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics.


Gwangju (Korean pronunciation: [kwaŋ.dʑu]) is the sixth-largest city in South Korea. It is a designated metropolitan city under the direct control of the central government's Home Minister. The city was also the capital of South Jeolla Province until the provincial office moved to the southern village of Namak in Muan County in 2005.

Its name is composed of the words Gwang (Korean: 광; Hanja: 光) meaning "light" and Ju (주; 州) meaning "province." Gwangju was historically recorded as Muju (무주; 武州), in which "Silla merged all of the land to establish the provinces of Gwangju, Ungju, Jeonju, Muju and various counties, plus the southern boundary of Goguryeo and the ancient territories of Silla" in the Samguk Sagi. In the heart of the agricultural Jeolla region, the city is also famous for its rich and diverse cuisine.

Idealised population

In population genetics an idealised population is one that can be described using a number of simplifying assumptions. Models of idealised populations are either used to make a general point, or they are fit to data on real populations for which the assumptions may not hold true. For example, coalescent theory is used to fit data to models of idealised populations. The most common idealized population in population genetics is described in the Wright-Fisher model after Sewall Wright and Ronald Fisher (1922, 1930) and (1931). Wright-Fisher populations have constant size, and their members can mate and reproduce with any other member. Another example is a Moran model, which has overlapping generations, rather than the non-overlapping generations of the Fisher-Wright model. The complexities of real populations can cause their behavior to match an idealised population with an effective population size that is very different from the census population size of the real population. For sexual diploids, idealized populations will have genotype frequencies related to the allele frequencies according to Hardy-Weinberg equilibrium.


Inolimomab is a mouse monoclonal antibody developed as an immunosuppressive drug against graft-versus-host disease. Its target is the alpha chain of the interleukin-2 receptor.

Kangal Shepherd Dog

The Kangal Shepherd Dog is a breed of large livestock guardian dog in Sivas. Historically the Anatolian Shepherd was treated as a separate breed by many canine registries, but this is now generally treated as part of the same breed population and the Turkish Kennel Club has renamed all Anatolian Shepherds as Kangal Shepherds. The breed is of an early Mastiff type with a solid pale tan or sabled coat, and a black mask. According to official Kangal Shepherd Dog organisations in Turkey, including the Cynology Federation of Turkey (Köpek Irkları ve Kinoloji Federasyonu, KIF) and the Ankara Kangal Association (Ankara Kangal Derneği, ANKADER), Kangals may also be brindle or feature a recessive black tan pattern; with or without a black mask; and/or with white markings.

While the Kangal Shepherd Dog is often referred to as a sheep dog, it is not a herding dog, but rather a flock guardian that lives with the flock of sheep to actively fend off predators of all sizes. Typically used as protection against wolves, bears, and jackals in its native Turkey, the breed has been exported to African countries like Namibia and Kenya in more recent years due to its intimidating size and capabilities as an effective guardian, where it successfully protects local herds from lions, cheetahs, and similar indigenous big cats, which has had the benefit of not only protecting livestock, but ensuring the continuity of endangered predators due to reduced cullings by local farmers.The Kangal Shepherd Dog's protectiveness, loyalty, and gentleness with small children and animals has led to its growing popularity as a guardian for families as well, as it regards people as its "flock" and guards them with extreme devotion.

Kow Swamp Archaeological Site

The Kow Swamp archaeological site comprises a series of late Pleistocene burials within the lunette of the eastern rim of a former lake known as Kow Swamp. The site is located 10 km south-east of Cohuna in the central Murray River valley, in northern Victoria, at 35.953553°S 144.318123°E / -35.953553; 144.318123. The site is significant for archaeological excavations by Alan Thorne between 1968 and 1972 which recovered the partial skeletal remains of more than 22 individuals.

Leslie matrix

In applied mathematics, the Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. It was invented by and named after Patrick H. Leslie. The Leslie matrix (also called the Leslie model) is one of the most well known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited environment, and where only one sex, usually the female, is considered.

The Leslie matrix is used in ecology to model the changes in a population of organisms over a period of time. In a Leslie model, the population is divided into groups based on age classes. A similar model which replaces age classes with ontogenetic stages is called a Lefkovitch matrix, whereby individuals can both remain in the same stage class or move on to the next one. At each time step, the population is represented by a vector with an element for each age class where each element indicates the number of individuals currently in that class.

The Leslie matrix is a square matrix with the same number of rows and columns as the population vector has elements. The (i,j)th cell in the matrix indicates how many individuals will be in the age class i at the next time step for each individual in stage j. At each time step, the population vector is multiplied by the Leslie matrix to generate the population vector for the subsequent time step.

To build a matrix, some information must be known from the population:

From the observations that at time t+1 is simply the sum of all offspring born from the previous time step and that the organisms surviving to time t+1 are the organisms at time t surviving at probability , one gets . This then implies the following matrix representation:

where is the maximum age attainable in the population.

This can be written as:


where is the population vector at time t and is the Leslie matrix. The dominant eigenvalue of , denoted , gives the population's asymptotic growth rate (growth rate at the stable age distribution). The corresponding eigenvector provides the stable age distribution, the proportion of individuals of each age within the population. Once the stable age distribution has been reached, a population undergoes exponential growth at rate .

The characteristic polynomial of the matrix is given by the Euler–Lotka equation.

The Leslie model is very similar to a discrete-time Markov chain. The main difference is that in a Markov model, one would have for each , while the Leslie model may have these sums greater or less than 1.


Manatees (family Trichechidae, genus Trichechus) are large, fully aquatic, mostly herbivorous marine mammals sometimes known as sea cows. There are three accepted living species of Trichechidae, representing three of the four living species in the order Sirenia: the Amazonian manatee (Trichechus inunguis), the West Indian manatee (Trichechus manatus), and the West African manatee (Trichechus senegalensis). They measure up to 4.0 metres (13.1 ft) long, weigh as much as 590 kilograms (1,300 lb), and have paddle-like flippers. The etymology of the name is dubious, with connections having been made to Latin "manus" (hand), and to a word sometimes cited as "manati" used by the Taíno, a pre-Columbian people of the Caribbean, meaning "breast". Manatees are occasionally called sea cows, as they are slow plant-eaters, peaceful and similar to cows on land. They often graze on water plants in tropical seas.

Maximum sustainable yield

In population ecology and economics, maximum sustainable yield (MSY) is theoretically, the largest yield (or catch) that can be taken from a species' stock over an indefinite period. Fundamental to the notion of sustainable harvest, the concept of MSY aims to maintain the population size at the point of maximum growth rate by harvesting the individuals that would normally be added to the population, allowing the population to continue to be productive indefinitely. Under the assumption of logistic growth, resource limitation does not constrain individuals' reproductive rates when populations are small, but because there are few individuals, the overall yield is small. At intermediate population densities, also represented by half the carrying capacity, individuals are able to breed to their maximum rate. At this point, called the maximum sustainable yield, there is a surplus of individuals that can be harvested because growth of the population is at its maximum point due to the large number of reproducing individuals. Above this point, density dependent factors increasingly limit breeding until the population reaches carrying capacity. At this point, there are no surplus individuals to be harvested and yield drops to zero. The maximum sustainable yield is usually higher than the optimum sustainable yield and maximum economic yield.

MSY is extensively used for fisheries management. Unlike the logistic (Schaefer) model, MSY has been refined in most modern fisheries models and occurs at around 30% of the unexploited population size. This fraction differs among populations depending on the life history of the species and the age-specific selectivity of the fishing method.

However, the approach has been widely criticized as ignoring several key factors involved in fisheries management and has led to the devastating collapse of many fisheries. As a simple calculation, it ignores the size and age of the animal being taken, its reproductive status, and it focuses solely on the species in question, ignoring the damage to the ecosystem caused by the designated level of exploitation and the issue of bycatch. Among conservation biologists it is widely regarded as dangerous and misused.


Nunavut ( (listen); French: [nynavy(t)]; Inuktitut syllabics ᓄᓇᕗᑦ [ˈnunavut]) is the newest, largest, and most northerly territory of Canada. It was separated officially from the Northwest Territories on April 1, 1999, via the Nunavut Act and the Nunavut Land Claims Agreement Act, though the boundaries had been drawn in 1993. The creation of Nunavut resulted in the first major change to Canada's political map since incorporating the province of Newfoundland in 1949.

Nunavut comprises a major portion of Northern Canada, and most of the Canadian Arctic Archipelago. Its vast territory makes it the fifth-largest country subdivision in the world, as well as North America's second-largest (after Greenland). The capital Iqaluit (formerly "Frobisher Bay"), on Baffin Island in the east, was chosen by the 1995 capital plebiscite. Other major communities include the regional centres of Rankin Inlet and Cambridge Bay.

Nunavut also includes Ellesmere Island to the far north, as well as the eastern and southern portions of Victoria Island in the west, and all islands in Hudson, James and Ungava Bays, including Akimiski Island far to the southeast of the rest of the territory. It is Canada's only geo-political region that is not connected to the rest of North America by highway.Nunavut is the largest in area and the second-least populous of Canada's provinces and territories. One of the world's most remote, sparsely settled regions, it has a population of 35,944, mostly Inuit, spread over a land area of just over 1,877,787 km2 (725,018 sq mi), or slightly smaller than Mexico (excluding water surface area). Nunavut is also home to the world's northernmost permanently inhabited place, Alert. Eureka, a weather station also on Ellesmere Island, has the lowest average annual temperature of any Canadian weather station.

Parapatric speciation

In parapatric speciation, two subpopulations of a species evolve reproductive isolation from one another while continuing to exchange genes. This mode of speciation has three distinguishing characteristics: 1) mating occurs non-randomly, 2) gene flow occurs unequally, and 3) populations exist in either continuous or discontinuous geographic ranges. This distribution pattern may be the result of unequal dispersal, incomplete geographical barriers, or divergent expressions of behavior, among other things. Parapatric speciation predicts that hybrid zones will often exist at the junction between the two populations.

In biogeography, the terms parapatric and parapatry are often used to describe the relationship between organisms whose ranges do not significantly overlap but are immediately adjacent to each other; they do not occur together except in a narrow contact zone. Parapatry is a geographical distribution opposed to sympatry (same area) and allopatry or peripatry (two similar cases of distinct areas).

Various "forms" of parapatry have been proposed and are discussed below. Coyne and Orr in Speciation categorise these forms into three groups: clinal (environmental gradients), "stepping-stone" (discrete populations), and stasipatric speciation in concordance with most of the parapatric speciation literature. Henceforth, the models are subdivided following a similar format.

Charles Darwin was the first to propose this mode of speciation. It was not until 1930 when Ronald Fisher published The Genetical Theory of Natural Selection where he outlined a verbal theoretical model of clinal speciation. In 1981, Joseph Felsenstein proposed an alternative, "discrete population" model (the "stepping-stone model). Since Darwin, a great deal of research has been conducted on parapatric speciation—concluding that its mechanisms are theoretically plausible, "and has most certainly occurred in nature".

Population and Development Review

Population and Development Review is a quarterly peer-reviewed academic journal published by Wiley-Blackwell on behalf of the Population Council. It was established in 1975 and the editor-in-chief is Landis MacKellar. The journal covers population studies, the relationships between population and economic, environmental, and social change, and related thinking on public policy. Content types are original research articles, commentaries, data and perspectives on statistics, archival documents on population issues, book reviews, and official documents from population agencies or related organizations.

According to the Journal Citation Reports, the journal has a 2017 impact factor of 3.547, ranking it first out of 28 journals in the category "Demography" and 6th out of 147 journals in the category "Sociology".

Population and Environment

Population and Environment is a peer-reviewed academic journal covering research on the bi-directional links between population, natural resources, and the natural environment. The editor-in-chief is Dr Elizabeth Fussell, associate professor of population studies and environment and society at Brown University. Former editors-in-chief of the journal include prominent racialists such as Virginia Abernethy and Kevin B. MacDonald; racialists appear to have since lost control of the journal since MacDonald's term as editor ended in 2004.

Population ecology

Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment. It is the study of how the population sizes of species change over time and space. The term population ecology is often used interchangeably with population biology or population dynamics.

The development of population ecology owes much to demography and actuarial life tables. Population ecology is important in conservation biology, especially in the development of population viability analysis (PVA) which makes it possible to predict the long-term probability of a species persisting in a given habitat patch. Although population ecology is a subfield of biology, it provides interesting problems for mathematicians and statisticians who work in population dynamics.

Ricker model

The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number N t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,

Here r is interpreted as an intrinsic growth rate and k as the carrying capacity of the environment. The Ricker model was introduced in 1954 by Ricker in the context of stock and recruitment in fisheries.

The model can be used to predict the number of fish that will be present in a fishery. Subsequent work has derived the model under other assumptions such as scramble competition, within-year resource limited competition or even as the outcome of a source-sink Malthusian patches linked by density-dependent dispersal. The Ricker model is a limiting case of the Hassell model which takes the form

When c = 1, the Hassell model is simply the Beverton–Holt model.

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