Math model

Now let's convert the spotted knapweed life cycle graph to a set of math equations. We need to write down three equations, one for each of the seed, rosette, and flower stages. Each of these equations would determine the number of individuals at time based on the number of individuals at time . We will use the life cycle graph to help us.

life cycle diagram

Let's start by writing down an equation for the number of seeds at time . The number of seeds which will be in the seed stage at time is: the number of seeds from time which remained viable seeds, plus the new seeds produced by the flowers at time . We have

 

Question: What is the equation for , the number of rosettes at time ?
Answer: show hide

The number of rosettes at time is: the number of seeds from time which germinate into rosettes, plus the number of rosettes from time which remain rosettes. We have
Question: What is the equation for , the number of flowers at time ?
Answer: show hide

The number of flowers at time is: the number of rosettes from time which become flowers, plus the number of flowers from time which remain flowers. We have

We have a system of three equations, one for each of our stages. The system is

Notice that these equations are linear with respect to the variables , , and . This means we can write the systems of equations in matrix form as follows

Now we need to substitute the known values for the transition probabilities into the matrix equation. Recall that the life cycle diagram with these numerical values is

life cycle diagram
Therefore we have

We can write our model more succinctly if we define
Then our model can be written as

Here, the matrix is called the projection matrix since it maps to .

Note: Matrices that describe populations with stage or size structure are often called Lefkovitch matrices, after biologist L. P. Lefkovitch (1965).