## What did we learn?

After learning about the invasive spotted knapweed, and its ability to cause harm to natural environments, we were able to use mathematical tools to help us glean information about this biological system.

First, we wrote down a stage structured matrix model, dividing our population into seed, rosette, and flower stage classes.

Using linear algebra, we were able to find the dominant eigenvalue, and its corresponding right and left eigenvectors. These three results were interpreted biologically, and we obtained: the long-term growth rate, the stable stage structure, the reproductive values, and the elasticities for this model.

More specifically, We determined that the population is exploding, with a long-term growth rate of 5.22. Once the population reaches the stable stage structure, we know that 85.6% of the population will be in the seed stage, 13.5% in the rosette stage, and 0.9% in the flower stage. This tells us that relatively few flowers are needed to maintain the population, because the flowers produce so many viable seeds each year. This result is consistent with what we found about the reproductive value: that the flowers have the greatest effect on the future population size, so adding or removing a flower has the most significant effect on the long-term population change. Also, we found that the dominant eigenvalue is most (and equivalently) sensitive to three of the life history parameters: , the annual probability that a seed will germinate, , the annual probability that a rosette will become a flower, and , the fecundity.

### Recommendations for Spotted Knapweed control

From this analysis, we have determined that to control the spotted knapweed, control measures need to be taken to reduce the parameters which have the most significant effect on the long-term growth rate, so that this growth rate can be reduced to less than 1. This reduction in the long-term growth rate of the population will cause the population to go extinct in the long-term, and will allow the native habitats that have suffered die to the spotted knapweed invasion to be restored.