## Controlling the population

Since we are dealing with an invasive species which is detrimental to the ecosystems where it establishes itself, we would like to determine how to control the population. In particular, we know that if we decrease the long-term growth rate (the dominant eigenvalue), the population won't grow as quickly, or may even go extinct.

The question is: the long-term growth rate is most sensitive to which parameter in the spotted knapweed life cycle?

Once we have determined the answer to this question, we will have a better idea of how to focus our control efforts.

To answer this question, we need to use left eigenvectors.

### Left eigenvectors

Usually, when we refer eigenvalues and eigenvectors, we are referring to right eigenvalues and eigenvectors. That is, a scalar and vector which satisfy . There is an analogous definition for what are called left eigenvectors.

Definition: A left eigenvector of an matrix is a nonzero vector such that for some scalar . The scalar which satisfies this equation is called a left eigenvalue.

Note 1: The left eigenvalues of a matrix are the same as the right eigenvalues of .
Note 2: The left eigenvectors of are the right eigenvectors of the transpose of , .

Question: What is the left eigenvector associated with the dominant eigenvalue for the spotted knapweed?

The left-eigenvector associated with the dominant eigenvalue is found by solving for . The normalized solution is

How do these left eigenvectors help us determine how to focus our control efforts?

### Reproductive value

The left eigenvector can be interpreted as what is called the reproductive value of different stages.

The reproductive value determines the long-term payoff (in terms of long-term population growth) that will result when an individual is added to a given stage.

To determine the reproductive value, it is conventional to scale the left eigenvector so that the first entry is equal to 1. This is done so the eigenvector entries are reproductive values relative to that of a new member of the population.

Scaling our the left eigenvector for the spotted knapweed population, we obtain

This tells us that investing in the flower stage will have the greatest effect on the future population size. Investing in the Rosette stage will also have an effect, but much less significant than the flower class. This intuitively makes sense, since the flowers are the seed producers, and once a plant is in the flower stage, it produces seed annually until death.

#### Investigating the effect of adding one

Let's investigate the effect of adding an individual to each of the three different stage classes.

First, choose an initial condition . The numbers of individuals in each stage class at after 5 years is calculated and displayed in the left column.

Then, determine how adding an individual at the initial time to each of the stages affects the resulting number of individuals in each class after 5 years. To do this, click on the buttons at the top of each of the columns in the table.

The column with the button , uses the initial condition to determine how many additional individuals there are in each class after 5 years as a result of adding one more seed at time 0.

Similarly, the other two columns calculate the additional number of individuals in each stage class after 5 years as a result of adding one rosette, and lastly, one flower.

Does this agree with the results from the reproductive value?

### Elasticity

The left eigenvectors corresponding to the dominant eigenvalue helps us understand what is called eigenvalue elasticity. Eigenvalue elasticity tells us the effect of single matrix elements on the dominant eigenvalue. This is done by looking at the relative change in population growth rate given a relative change in the matrix element.

Definition: Eigenvalue elasticity refers to the change in the dominant eigenvalue given a proportional change in a matrix element.

If is the right eigenvector, and is the left eigenvector, both associated with the dominant eigenvalue , then the sensitivity of with respect to the entry in , denoted is given by

where is the dot product of the vectors and , and and refers to the -th and -th entries of and Once we have calculated the elasticities, we can write them in a matrix where is the entry in the row and column of the elasticity matrix.

Question: Find the elasticities for each of the entries of our projection matrix.

The elasticity matrix is

Once we've calculated the elasticity matrix, we look for the largest value(s), since these correspond to the parameter(s) that have the greatest impact on the dominant eigenvalue. Note that the largest value is 0.3074, and it appears in three different entries of the elasticity matrix. This means that the parameters corresponding to these largest elasticity values have the greatest impact on the dominant eigenvalue.

Question: Which spotted knapweed life cycle parameters should our control efforts be focused on?

The parameters corresponding to the largest elasticity value of 0.3074 are:
1. , which is the annual probability that a seed will germinate;
2. , which is the annual probability that a rosette will grow into a flower; and
3. , which is the fecundity.
Therefore, we should focus our control efforts on reducing these parameters.
Question: Which parameters in the life cycle have the least effect on the long-term growth rate?