Modeling Linear Population Growth with Matrices.

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Populations that undergo a fixed set of age transitions can be modeled as a system of linear equations, from which age distributions can be calculated.

As a brief review of linear algebra, a system of linear equations can be represented as a matrix equation of the form Ax = b. In this form, ‘A is the set of coefficients as some m × n matrix, ‘x is some input vector in n, and ‘b’ is the product of ‘A’ and ‘x’ as an output vector in m, shown below:

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=

Consider a population of rabbits with the following characteristics:

  1. Half of the rabbits survive their first year. Of those, half survive their second year, and their maximum lifespan is 3 years.
  2. During the first year, the rabbits don’t produce any offspring. During the second year, the average number of offspring produced is 6, and during the third year, the average number of offspring produced is 8.

If this population starts with 24 rabbits in their first year, 20 rabbits in their second year, and 12 rabbits in their third year, how many rabbits will there be of each age after two years?

This is quickly solved with a matrix equation of the form Ax = b, where the input vector represents current ages and the output vector represents ages after 1 year:

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After 1 year, there will be rabbits in their 1st year, rabbits in their 2nd year, and rabbits in their 3rd year.
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After 2 years, there will be rabbits in their 1st year, rabbits in their 2nd year, and rabbits in their 3rd year.

Suppose we want to find a stable age distribution for this system, one in which the ratio of all age groups remains the same. For example, if the ratio among different age groups is 5:3:2, then it will remain 5:3:2 indefinitely — as long as the birth rates and death rates remain the same.

This stable age distribution requires a special kind of input vector called an eigenvector. An eigenvector is a vector that is scaled by a linear transformation but retains it’s component ratio.

We can find these eigenvectors by finding all eigenvalues (scalars denoted by λ) that are solutions to the the equation: Ax = λx. Notice from this equation how the output vector is a scalar multiple of the input vector.

We start by finding the determinant of A - λI and solving for the roots of its polynomial expression, which will be the eigenvalues:

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-
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by Laplace expansion
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Solving above= and = (with algebraic multiplicity 2)
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Choosing the positive eigenvalue, substitute =
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Reduced row echelon form
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The eigenvector for =is
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All input vectors of the form will have output vectors with a ratio of ::
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For = ,theoutput vector still has a ratio ::

Starting with a population of 16 rabbits in their first year, 4 rabbits in their second year, and 1 rabbit in it’s third year, we have a stable age distribution of our age transition system.

For a more detailed explanation of some of these concepts, this series explains them in an intuitive way and with brilliant illustrations.

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