Iterating the model

We have written our model as , where , and is the projection matrix.

For some given initial number of seeds, rosettes, and flowers, we can write down an initial condition vector . This corresponds to the number of individuals in each stage at time zero, which is the starting time. Then, the number of individuals in each stage after one year is

What about after two years?
But we know , so we use this information to obtain
What about after three years?
Question: How do we determine , after years, using only and the initial condition ?
Answer: show hide

All we need is an initial condition

It is great news to know that if we have an initial condition, we can use our projection matrix to determine the number of individuals in each stage for any time.

Example: Let's explicitly calculate the number of individuals in each stage after two years given the initial conditions (so that we start with 400 seeds, 20 rosettes, and 10 flowers). After two years we have
After two years we have 6,026 seeds, 342 rosettes, and 112 flowers.

Plant counting

Can we use the model we've written down to figure out how many spotted knapweed plants, including rosettes and flowers, we would see on the ground if we went out and counted?

The answer is yes! The number of plants we would see on the ground at time is given by the number of rosettes at time plus the number of flowers at time , so

Example: How many plants are there after two years if we start with 400 seeds, 20 rosettes, and 10 flowers?

From the previous example we know that after two years we will have 6,026 seeds, 342 rosettes, and 112 flowers. Therefore, the number of plants we will have after two years is 342+112 = 453.