## Assumptions

To develop a model, we need to make some assumptions about the spotted knapweed. Assumptions are needed to simplify biological systems, so that we can translate them into problems that we can work with. This means that we won't be modeling our exact system, but a simplified version of the system. However, studying these simplified systems can provide insight into many different problems.

Since the spotted knapweed can be described by its life cycle stages, we say that this plant population exhibits stage structure. We would like to take advantage of this structure, and so we will use what is called a stage structured model to mathematically describe the spotted knapweed system. The stage structured model is one of many different types of models that can be used to describe a system which changes over time.

Definition: A stage structured model is one in which individuals can be classified according to different stages, where each stage has its own survival and reproductive characteristics.

To be able to use a stage structured model, we need to make several key assumptions. We assume that:

• Time is discrete: this means that we will "step" through time, so that the time variable will start at 0, and subsequently step through the values 1, 2, 3, etc. This will allow us to use recursion equations to determine the value of a variable at time , given information about the variable at the current time and previous times , , etc.

Since we know the spotted knapweed reproduces during the growing season, we can consider time to be discrete, with each time step corresponding to one year. However, by doing this, we are assuming that all growing, seed production, and germinating events happen simultaneously. We do not account for the fact that the growing season is several months long.

• All survival and reproductive events are constant throughout time: this means we ignore fluctuations in survival and repoduction events due to environmental change.
• There is no density dependence: this means that we assume that the number of individuals present does not have any effect on the survival and reproductive events.