Nonfluctuating Environments

As next simple case, consider a structured population in a constant environment. Then in discrete time p — ln(dominant eigenvalue of A) [9]

and in continuous time p — dominant eigenvalue of B [10]

For constant environments p is sometimes called Malthusian parameter, in which case it is often written as m, or intrinsic rate of increase, in which case it is usually written as r. The corresponding right eigenvector of A, or B, is called stationary h-state distribution, with usually some more germane expression substituted for 'h-state', while the components of the left eigenvector are called reproductive values.

Expressions [9] and [10] lead to useful algebraic procedures only when either the numbers of h-states are very small or when the matrices A or B have some special structure, as is for example the case when the h-state is age. Otherwise it is necessary to take recourse to numerics.

The most efficient numerical procedure for calculating the dominant eigenvalue of a large nonnegative matrix is by iteration:

• Start from some positive vector M(0) with 1TM(0) = 1.

• Successively calculate M(t) from

• w(t) then converges to the dominant eigenvalue of A: p = ln(limtw(t)).

Since for a population state N, 1TN corresponds to the total population size, w(t) can be interpreted as a per capita increase in population size. Therefore, this iterative procedure is equivalent to estimating an asymptotic population growth rate from a deterministic simulation.

The continuous time case can be dealt with numerically by first calculating the dominant eigenvalue of the nonnegative matrix B + ftl, with ft minus the most negative diagonal component of B, and subtracting ft from the result.

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