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Matrix population models are a specific type of population model that uses matrix algebra. Population models are used in
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, such as birth and death rates, and by immigration and emigration. The discipline is import ...
to model the dynamics of wildlife or human populations. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations. All
populations Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
can be modeled :N_=N_+B-D+I-E, where: *Nt+1 = abundance at time t+1 *Nt = abundance at time t *B = number of births within the population between Nt and Nt+1 *D = number of deaths within the population between Nt and Nt+1 *I = number of individuals immigrating into the population between Nt and Nt+1 *E = number of individuals emigrating from the population between Nt and Nt+1 This equation is called a BIDE model (Birth, Immigration, Death, Emigration model). Although BIDE models are conceptually simple, reliable estimates of the 5 variables contained therein (N, B, D, I and E) are often difficult to obtain. Usually a researcher attempts to estimate current abundance, Nt, often using some form of
mark and recapture Mark and recapture is a method commonly used in ecology to estimate an animal population's size where it is impractical to count every individual. A portion of the population is captured, marked, and released. Later, another portion will be captur ...
technique. Estimates of B might be obtained via a ratio of immatures to adults soon after the breeding season, Ri. Number of deaths can be obtained by estimating annual survival probability, usually via
mark and recapture Mark and recapture is a method commonly used in ecology to estimate an animal population's size where it is impractical to count every individual. A portion of the population is captured, marked, and released. Later, another portion will be captur ...
methods, then multiplying present abundance and survival rate. Often, immigration and emigration are ignored because they are so difficult to estimate. For added simplicity it may help to think of time t as the end of the breeding season in year t and to imagine that one is studying a species that has only one discrete breeding season per year. The BIDE model can then be expressed as: :N_=N_\times S_+N_\times R_i\times S_i where: * Nt,a = number of adult females at time t * Nt,i = number of immature females at time t * Sa = annual survival of adult females from time t to time t+1 * Si = annual survival of immature females from time t to time t+1 * Ri = ratio of surviving young females at the end of the breeding season per breeding female In matrix notation this model can be expressed as: : \begin \begin N_\\ N_ \end &= \begin S_iR_i & S_aR_i \\ S_i & S_a \end \begin N_\\ N_ \end \end. Suppose that you are studying a species with a maximum lifespan of 4 years. The following is an age-based Leslie matrix for this species. Each row in the first and third matrices corresponds to animals within a given age range (0–1 years, 1–2 years and 2–3 years). In a Leslie matrix the top row of the middle matrix consists of age-specific fertilities: F1, F2 and F3. Note, that F1 = Si×Ri in the matrix above. Since this species does not live to be 4 years old the matrix does not contain an S3 term. : \begin \begin N_ \\ N_ \\ N_ \end &= \begin F_1 & F_2 & F_3 \\ S_1 & 0 & 0 \\ 0 & S_2 & 0 \end \begin N_\\ N_\\ N_ \end \end . These models can give rise to interesting cyclical or seemingly chaotic patterns in abundance over time when fertility rates are high. The terms Fi and Si can be constants or they can be functions of environment, such as habitat or population size. Randomness can also be incorporated into the environmental component.


See also

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Population dynamics of fisheries A fishery is an area with an associated fish or aquatic population which is harvested for its commercial or recreational value. Fisheries can be wild or farmed. Population dynamics describes the ways in which a given population grows and shrinks ...


References

*Caswell, H. 2001. Matrix population models: Construction, analysis and interpretation, 2nd Edition. Sinauer Associates, Sunderland, Massachusetts. {{isbn, 0-87893-096-5.
Leslie Matrix Model demonstration (Silverlight)
Population dynamics Population models