## 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.

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

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

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