We developed an exponential growth model showcasing knapweed spread to communicate our project's effectiveness. Being able to visualize actual numbers could help people better understand the true magnitude of spotted knapweed's reproduction scale and recognize the impact on environments such as Waterton, as well as our project's impact on said model.
Assumptions
To make this model work, we had to assume that all knapweed plants produce the same number of seeds per year. A knapweed plant was assumed to produce the standard 360 seeds in every year of its life (including the first year), and all seeds were assumed to successfully become knapweed plants the year after they are produced. Spotted knapweed plants were assumed to be essentially immortal. The equation that models the number of plants grown during each time interval based on the number of plants killed assumed the variable of calculated kill ratio of our herbicide is consistent and accurate as well.
While using an exponential growth model, the major assumption is that the growth rate remains constant regardless of population size and environmental capacity. By assuming this we can model the growth of spotted knapweed and the rate at which the specific population will die once our herbicide is applied.
Definitions of Parameters and Variables
= initial number of seeds
Most populations begin with one seed, so we will assume that there is initially only one spotted knapweed seed.
= maximum growth rate of the population
A typical knapweed plant produces 360 seeds per year provided it has not been affected by the carrying capacity of its environment.
= number of years
is the manipulated variable in our exponential growth equation. Because it represents a number of years, it has to be a real number. also has to be greater than or equal to 0, since has to be a whole number.
= population size
is the responding variable in our exponential growth equation. Because it represents the number of spotted knapweed plants in a population, it must be a whole number.
= siRNA kill ratio
We have not yet calculated our siRNA kill ratio; but because it is a ratio, it must be a number between 0 and 1.
Equation
[1]
Results
This graph shows the population of spotted knapweed as a function of the number of years. Because this is an exponential growth function with a large growth factor, it is somewhat difficult to find individual values on the graph. The only points which have truly been calculated are as follows, since our model only calculates the number of knapweed plants after a whole number of years. After 0 years there is 1 plant, after 1 year there are 361 plants, after 2 years there are 130 321 plants, after 3 years there are 47 045 881 plants, after 4 years there are approximately 17.0 billion plants, and after 5 years there are approximately 6.13 trillion plants.
Discussion
What does this tell us about our project?
The model reveals how important our project is based on the scale of the spread of spotted knapweed. We can also use this equation as a base and compare it to the equation below that incorporates our siRNA herbicide treatment. The comparison will show how different the two are and therefore how effective our project is.
How do we incorporate a factor for plants killed by siRNA treatment?
What are our next steps?
Our next step is to calculate the kill ratio of our herbicide so that we can input it into our equation and determine how effective our project is. We can also test our equation for the exponential growth of spotted knapweed on a model population to verify its accuracy.
Logistic Growth Model
Purpose
This model will tell us how fast knapweed grows, and it can then also be used to determine the total number of knapweed plants at any given time. A logistic growth model is more accurate than an exponential growth model, since it takes more factors into account. This model will help us to understand the importance of the spotted knapweed problem, and can also be used in conjunction with an equation on the use of our herbicide to determine how much of an effect our herbicide is having.
Assumptions
We have to assume that all spotted knapweed plants produce the same number of seeds as a typical plant. We also have to assume that spotted knapweed plants are similar enough to sunflowers that the optimal planting distance is similar, and that the optimal planting distance actually does reveal the carrying capacity. We assumed that the entirety of Waterton can support spotted knapweed. We also assumed that all knapweed seeds become plants that produce 360 seeds the year after they are created, assuming the carrying capacity does not affect them. We were forced to assume that an equation that assumes that we are compounding the number of plants continuously, even though we only compound yearly.
Definitions of Parameters and Variables
= population size
is either the manipulated or responding variable in equations it is a part of. Because it represents a number of plants, it has to be a whole number.
= number of years
is the manipulated variable in our logistic growth equation. Because it represents a number of years, it has to be a real number. also has to be greater than or equal to 0, since has to be a whole number.
= growth rate
A typical knapweed plant produces 360 seeds per year provided it has not been affected by the carrying capacity of its environment.
= carrying capacity
Both Centaurea (the knapweed genus) and Helianthus (the sunflower genus) belong to the same family: Asteraceae [2,3]. Since we could not find much information on knapweed, we used sunflowers in their place for calculating the carrying capacity. [4] says that the best way to get the maximum potential of sunflower plants is to grow 20 000 to 30 000 plants per hectare. Since we are looking for the maximum number of knapweed plants Waterton can sustain, we used the top end of the range (30 000). Spotted knapweed can make up more than 95% of the flora in an area [5], and Waterton Lakes National Park occupies 505 square kilometers [6], meaning that knapweed can occupy at least 479.75 square kilometers of Waterton (without taking significant figures into account). Using unit conversions, we found that 1 439 250 000 knapweed plants can be sustained by Waterton before significant figures. With significant figures, this number becomes 1.4 billion.
= initial number of seeds
Most populations begin with one seed, so we will assume that there is initially only one spotted knapweed seed.
= siRNA kill ratio
We have not yet calculated our siRNA kill ratio; but because it is a ratio, it must be a number between 0 and 1.
Equations
[7]
Results
This graph shows the rate that the spotted knapweed population is actually growing at after factoring in the growing capacity. This rate is allowed to be a negative number, because the number of knapweed plants could be diminishing if they exceed the carrying capacity.
This graph shows the number of spotted knapweed plants after a given number of years. After roughly 0.1 years, the population reaches the carrying capacity of the environment, so after that time it is always the same as the carrying capacity.
Discussion
What does this tell us about our project?
This model shows how quickly spotted knapweed spreads and therefore how important it is that our siRNA herbicide be effective in rapidly killing knapweed plants.
How do we incorporate a factor for plants killed by siRNA treatment?
What are our next steps?
Our next step is to find the effectiveness of our herbicide so that we can determine how effective it truly is in limiting the spread of knapweed. We should also test these equations on a model population to see if they are accurate.
This graph shows the population of spotted knapweed as a function of the number of years. Because this is an exponential growth function with a large growth factor, it is somewhat difficult to find individual values on the graph. The only points which have truly been calculated are as follows, since our model only calculates the number of knapweed plants after a whole number of years. After 0 years there is 1 plant, after 1 year there are 361 plants, after 2 years there are 130 321 plants, after 3 years there are 47 045 881 plants, after 4 years there are approximately 17.0 billion plants, and after 5 years there are approximately 6.13 trillion plants.
This graph shows the rate that the spotted knapweed population is actually growing at after factoring in the growing capacity. This rate is allowed to be a negative number, because the number of knapweed plants could be diminishing if they exceed the carrying capacity. 
This graph shows the number of spotted knapweed plants after a given number of years. After roughly 0.1 years, the population reaches the carrying capacity of the environment, so after that time it is always the same as the carrying capacity.