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| Population models are used in [[population ecology]] to model the dynamics of wildlife or human populations. '''Matrix population models''' are a specific type of population model that uses [[matrix algebra]]. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations.
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| All [[populations]] can be modeled by one simple equation:
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| :<math>N_{t+1}=N_{t}+B-D+I-E,</math>
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| where:
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| *N<sub>t+1</sub> = abundance at time t+1
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| *N<sub>t</sub> = abundance at time t
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| *B = number of births within the population between N<sub>t</sub> and N<sub>t+1</sub>
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| *D = number of deaths within the population between N<sub>t</sub> and N<sub>t+1</sub>
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| *I = number of individuals immigrating into the population between N<sub>t</sub> and N<sub>t+1</sub>
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| *E = number of individuals emigrating from the population between N<sub>t</sub> and N<sub>t+1</sub>
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| This equation is called a BIDE model (Birth, Immigration, Death, Emigration model).
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| 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, N<sub>t</sub>, often using some form of [[mark and recapture]] technique. Estimates of B might be obtained via a ratio of immatures to adults soon after the breeding season, R<sub>i</sub>. Number of deaths can be obtained by estimating annual survival probability, usually via [[mark and recapture]] methods, then multipling present abundance and [[survival rate]]. Often, immigration and emigration are ignored because they are so difficult to estimate.
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| 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.
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| The BIDE model can then be expressed as:
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| :<math>N_{t+1}=N_{t,a}\times S_{a}+N_{t,a}\times R_i\times S_i</math>
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| where:
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| * N<sub>t,a</sub> = number of adult females at time t
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| * N<sub>t,i</sub> = number of immature females at time t
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| * S<sub>a</sub> = annual survival of adult females from time t to time t+1
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| * S<sub>i</sub> = annual survival of immature females from time t to time t+1
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| * R<sub>i</sub> = ratio of surviving young females at the end of the breeding season per breeding female
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| In matrix notation this model can be expressed as:
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| :<math>
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| \begin{align}
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| \begin{pmatrix}
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| N_{t+l_i}\\
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| N_{t+l_a}
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| \end{pmatrix} &=
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| \begin{pmatrix}
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| S_iR_i & S_aR_i \\
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| S_i & S_a
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| \end{pmatrix}
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| \begin{pmatrix}
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| N_{t_i}\\
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| N_{t_a}
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| \end{pmatrix}
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| \end{align}.
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| </math>
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| 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: F<sub>1</sub>, F<sub>2</sub> and F<sub>3</sub>. Note, that F<sub>1</sub> = S<sub>i</sub>×R<sub>i</sub> in the matrix above. Since this species does not live to be 4 years old the matrix does not contain an S<sub>3</sub> term.
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| :<math> | |
| \begin{align}
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| \begin{pmatrix}
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| N_{t+l_1} \\
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| N_{t+l_2} \\
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| N_{t+l_3}
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| \end{pmatrix} &=
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| \begin{pmatrix}
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| F_1 & F_2 & F_3 \\
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| S_1 & 0 & 0 \\
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| 0 & S_2 & 0
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| \end{pmatrix}
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| \begin{pmatrix}
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| N_{t_1}\\
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| N_{t_2}\\
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| N_{t_3}
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| \end{pmatrix}
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| \end{align} .
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| </math>
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| These models can give rise to interesting cyclical or seemingly chaotic patterns in abundance over time when fertility rates are high.
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| The terms F<sub>i</sub> and S<sub>i</sub> 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.
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| ==See also==
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| *[[Population dynamics of fisheries]]
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| ==References==
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| *Caswell, H. 2001. Matrix population models: Construction, analysis and interpretation, 2nd Edition. Sinauer Associates, Sunderland, Massachusetts. ISBN 0-87893-096-5.
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| *[http://andrei1606.brinkster.net/MatrixPopulationModel.aspx Leslie Matrix Model demonstration (Silverlight)]
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| [[Category:Population ecology]]
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| [[Category:Population models]]
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