Growth or extinction?

Will the spotted knapweed population grow, or will it become extinct?

This is one of the most basic questions we can ask about our population of interest. Intuitively, from reading about the biology of the spotted knapweed, and knowing that it is an invasive species, we might guess that the population will grow in the long run.

But how do we use math to figure out what will happen to our population in the long run?

The answer depends on the properties of the projection matrix . In particular, the eigenvalues and the eigenvectors of will provide us with the information we need to figure out the long-term behaviour of the spotted knapweed.

Writing the solution in terms of eigenvalues and eigenvectors

Definition: An eigenvector of an matrix is a nonzero vector such that for some scalar . A scalar is called an eigenvalue of if there is a nontrivial solution of . We call the eigenvector corresponding to .

An matrix must have eigenvalues, but they may not all be different. However, the typical situation is to have distinct eigenvalues each with a corresponding eigenvector.

Theorem: If an matrix has distinct eigenvalues, then the set of the corresponding eigenvectors is linearly independent.

This means that if we have distinct eigenvalues, we can write any vector with real entries as a linear combination of these eigenvectors. Let's say we have an matrix , with distinct eigenvalues , and corresponding eigenvectors . Since the matrix model initial condition is a vector with real entries, we can write

for some constants , where we assume is not zero.


Question: How do we write in terms of the eigenvalues and eigenvectors of ?
Answer: show hide

The second iteration is

And the solution is...

The solution of the stage structured matrix model, in terms of the eigenvalues and eigenvectors of the projection matrix, is

But why did we need to write our solution in this way? Remember, we are trying to determine how the spotted knapweed population will grow (or shrink) in time.

Long term behaviour

Now that we have an expression for the solution in terms of the eigenvalues and eigenvectors of , we need to figure out how to use this expression to determine the long-term population behaviour. Let's use a fact from linear algebra, which says that if we find a particular eigenvalue and its eigenvector, we know the long-term behaviour of the solution.

Fact: The long-term behaviour of the population is determined by the dominant eigenvalue and its eigenvector.

What does it mean for an eigenvalue to be dominant?

Definition: An eigenvalue is called dominant if for all . Note: For real numbers means take the absolute value of . If is complex, and .

So, if is dominant, then in the long-term (as ) we have , where means looks like for large time. This means that in the long-term, the total population size changes exponentially at a rate . So our dominant eigenvalue determines the long-term growth rate.

To figure out what will happen to our population, we look at . Since and are constant and non-zero, the limit depends only on the magnitude of .

Question: What is for
  1. ,
  2. , or
  3. .
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  1. If , then ,
  2. If , then , or
  3. If , then .
Question: What does this tell us about the population numbers in the long-term?
Answer: show hide

  1. If , then the population will become extinct.
  2. If , then the population will remain constant.
  3. If , then the population will grow.

Apply the theory

Now back to the spotted knapweed problem. First, let's find the eigenvalues of our projection matrix . To find the eigenvalues of we solve the characteristic equation

for to obtain the three solutions

Question: Which one is the largest in magnitude?
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The magnitudes of the eigenvalues are
Therefore, the eigenvalue with the largest magnitude is .
Question: Based on the magnitude of the largest eignvalues, does the population grow or become extinct?
Answer: show hide

The magnitude of the largest eigenvalue is 5.2236, which is greater than 1, so the population grows exponentially with growth rate .

Annual growth rate

Another way we can calculate the annual growth rate at time , call it , is to take the number of individuals at time , and divide by the number of individuals at time . Mathematically we write

where is the total population.

Annual growth rate for the first few years

Example: Let's calculate the annual growth rate for the first four years, starting with 400 seeds, 40 rosettes, and 20 flowers. We have

Earlier, we found the long-term growth rate to be 5.22. Notice that the annual growth rate for the first three years is different from this long-term growth rate. Actually, this annual growth rate is approaching the long-term growth rate over time.

Annual and long-term growth rates

How long does it take until the annual growth rate converges to the long-term growth rate?

Let's answer this question graphically. Input an initial number of seeds, rosettes, and flowers. Also, choose an end time between 5 and 50 years. The graph generated depicts the annual growth rate against time. We want to know when does the annual growth rate become a constant, equal to the dominant eigenvalue.

Once the annual growth rate does converge to the long-term growth rate, the population will have reached its stable stage structure.