Projection exercises

Stable age distribution and population growth rate

Reproductive value of different ages

Not all matrices yield a stable age distribution concentration of

reproduction in the last age

oscillations

Eigenvalues, eigenvectors and the projection equation

« Eigen »

A natural direction of the dynamics of the matrix

Such that if the system starts on this direction, it will stay in the same direction

Solution of the projection equation

A matrix has many eigenvalues (speeds) A matrix has many eigenvectors (directions)

Right eigenvectors

These are columns Matrix * column = a

new column of the same shape

Left eigenvectors

These are transpose of rows

Row * Matrix = a new row of the same shape

Population projection matrix

Eggs /hatchlings

Small juveniles

Large juveniles Subadults Adults

Eggs/hatchlings 0 0 0 4.665 61.896

Small juveniles 0.675 0.703 0 0 0

Large Juveniles 0 0.047 0.657 0 0

Subadults 0 0 0.019 0.682 0Adults 0 0 0 0.061 0.8091

Stage at time tStage at time t+1

Carreta carreta (Crowder et al. 1994)

Try it (we will use mat2valvecs)

Multiply the matrix by the column Multiply the row by the matrix

Solution of the projection equation

A matrix has many eigenvalues (speeds) A matrix has many eigenvectors (directions) The number of individuals in a each stage at

some time in the future will depend upon ALL these

Eigenvalues are raised to the « tth » power (see 4.49, Caswell 2001, p. 75)

To find out what will happen

We need to know how fast the system is moving in each direction

In other words, the relative impact of lambdas after time (ie after raising them to the « tth » power

Raising lambdas to powers

Fractional or not? Positive or negative? Absolute value = 1? Real or complex If complex, the absolute magnitude?

Raising lambdas to powers

Exponential increase Exponential decrease Oscillations Spiraling through the complex plane

Directions are weighted

The direction associated with a lambda that increases (or decreases) exponentially when raised to successively higher powers will dominate over other directions

Some analytical entities

Dominant eigenvalue Dominant right eigenvector (ssd) Dominant left eigenvector (rv)

Sensitivity: What ’s important to population growth?

A bad question! Good questions are more specific Prospective vs. retrospective questions A parameter which does not vary can not

contribute to variation in population growth no matter how high its sensitivity is!