Age-structured and stage-structured
population models


Motivation


Birth and death rates usually depend on age.


Growth rates will differ between populations with different age structures.


Some age classes contribute more to population growth than others.


These facts have important management implications.

Human mortality risk

Human age distributions

Declining populations have relatively more old individuals than growing populations.

Human population data (Our World in Data)

Three age class example

Abundance of age class \(i\) in year \(t\) is denoted by \(n_{i,t}\).

The model for \(n_{i,t+1}\) depends on:

  • Age class survival \(s_{i}\), and
  • Age class fecundity \(f_{i}\)
  • Fecundity is the birth rate (\(b_i\)) multiplied by survival
    • In pre-reproductive surveys, \(f_i = b_i \times s_0\)
    • In post-reproductive surveys, \(f_i = b_i \times s_i\)

How does this population grow?

Age class Equation
1 \(n_{1,t+1} = n_{1,t} \times f_1 + n_{2,t} \times f_2 + n_{3,t} \times f_3\)
2 \(n_{2,t+1} = n_{1,t} \times s_{1}\)
3 \(n_{3,t+1} = n_{2,t} \times s_{2}\)


According to this particular age-structured model, all animals die after spending one time interval in the third age class.


In stage-structured models, animals can stay in a class for more than one time interval.

Age-class abundance and age distribution

  • Suppose initial abundance in the 3 age classes is:
    • \(n_{1,1} = 50\) (Age class 1, juveniles)
    • \(n_{2,1} = 40\) (Age class 2, subadults)
    • \(n_{3,1} = 10\) (Age class 3, adults)
    • \(N_1\) = 100

  • This implies an initial age distribution of:

    • \(c_{1,1} = n_{1,1}/N_1 = 0.5\)
    • \(c_{2,1} = n_{2,1}/N_1 = 0.4\)
    • \(c_{3,1} = n_{3,1}/N_1 = 0.1\)
An age distribution describes the proportion of individuals in each age class

Age distribution

Example

Let’s choose values of \(s_i\) and \(f_i\)


Age class Initial population size Survival probability Fecundity rate
\(n_{i,1}\) \(s_i\) \(f_i\)
1 50 0.5 0.0
2 40 0.6 0.8
3 10 0.0 1.7

Population size

Growth rates

Age distribution

Important things to note

Age distribution converges to a stable age distribution when survival and fecundity rates are constant.


Stable age distribution is the proportion of individuals in each age class when the population converges.


Growth rates of each age class differ at first, but converge once the stable age distribution is reached.


Asymptotic growth rate is \(\lambda\) (without subscript).


Growth rate at the stable age distribution is the same for all age classes, and it is geometric!

Matrix Models

Matrix models

Matrix models aren’t actually new models.


They are the same models we have been talking about.


But, they make it much easier to compute important quantities like \(\lambda\) and reproductive value.

Matrix models vs life tables

Matrix models

  • Age is discrete
  • Age class is denoted by \(i\)
  • Each age class can have its own vital rates

Life tables

  • Age is continuous
  • Actual age is denoted by \(x\)
  • Each age can have its own vital rates

What is a matrix?

Definition: A matrix is a rectangular array of numbers

  • Usually denoted by an uppercase, bold letter
  • Either square or rounded brackets are used
  • Usually, rows are indexed by \(i\), columns by \(j\)
Example of a \(3 \times 4\) matrix:

\[ { \bf A} = \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} \end{bmatrix} \]

Leslie matrix

What is it?

  • Square matrix
  • Fecundities on first row
  • Survival probs on lower off-diagonal

Example:

\[ {\bf A} = \begin{bmatrix} f_1 & f_2 & f_3 & f_4 \\ s_1 & 0 & 0 & 0 \\ 0 & s_2 & 0 & 0 \\ 0 & 0 & s_3 & 0 \end{bmatrix} \]

How do we use a Leslie matrix?

These two expressions are equivalent:

Age class Equation
1 \(n_{1,t+1} = n_{1,t} \times f_{1} + n_{2,t} \times f_{2} + n_{3,t} \times f_3\)
2 \(n_{2,t+1} = n_{1,t} \times s_{1}\)
3 \(n_{3,t+1} = n_{2,t} \times s_{2}\)

AND

\[{\bf n}_{t+1} = {\bf A}\times {\bf n}_{t}\]

Matrix multiplication


\[ \begin{bmatrix} aw + bx + cy + dz \\ ew + fx + gy + hz \\ iw + jx + ky + lz \\ mw + nx + oy + pz \end{bmatrix} = \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{bmatrix} \times \begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix} \]

Matrix multiplication and Leslie matrix

\[ \begin{bmatrix} n_{1,t+1} \\ n_{2,t+1} \\ n_{3,t+1} \\ n_{4,t+1} \end{bmatrix} = \begin{bmatrix} f_1 & f_2 & f_3 & f_4 \\ s_1 & 0 & 0 & 0 \\ 0 & s_2 & 0 & 0 \\ 0 & 0 & s_3 & 0 \end{bmatrix} \times \begin{bmatrix} n_{1,t} \\ n_{2,t} \\ n_{3,t} \\ n_{4,t} \end{bmatrix} \]

Reproductive value

Reproductive Value

Definition: The extent to which an individual in age class \(i\) will contribute to the ancestry of future generations.

\[ v_i = \sum_{j=i}^{I}\left(\prod_{h=i}^{j-1}s_h\right)f_j\lambda^{i-j-1} \]


Post-reproductive individuals have reproductive values of zero.


Depending on survival and fecundity rates, a first-year individual could have a higher or lower reproductive value than a second-year individual.

Reproductive value for previous example

Other important quantities

Net reproductive rate

The expected number of individuals produced by an individual over its lifetime:

\[ R_0 = \sum_{i=1}^I \left(\prod_{j=1}^{i-1} s_j\right)f_i \]

Generation time

The time required for a population to increase by a factor of \(R_0\).

\[ T = \frac{\log(R_0)}{\log(\lambda)} \]

Other properties of the Leslie matrix


The dominant eigenvalue of \({\bf A}\) is the growth rate \(\lambda\).


The right eigenvector gives the stable age distribution.


The left eigenvector gives the reproductive value.

Stage-structured models

Stage-structured population models

Age isn’t always the best way to think about population structure.

For some populations, it is much more useful to think about size structure or even spatial structure.

These “stage-structured” models differ from age-structured models in that individuals can remain in a stage class (with probability \(1-p_i\)) for multiple time periods.

Stage-structured population models

In stage-structured models, individuals transition from one stage to the next with probability \(p_i\).

We can add these transition probabilities to our projection matrix like this:

\[ {\bf A} = \begin{bmatrix} f_1 & f_2 & f_3 \\ s_1 & s_2(1-p_2) & 0 \\ 0 & s_2 p_2 & s_3 \end{bmatrix} \]

This example allows for individual to remain in stages 2 and 3 for more than one time period.

Summary

Vital rates (\(s\) and \(f\)) are usually age-specific.


Population growth will depend on age distribution.


If vital rates are constant, population will reach stable age distribution with constant growth rate \(\lambda\).


Reproductive value indicates which age class contributes the most to population growth.


Matrix models are a convenient method used to work with age-structured populations.