Purpose: To be introduced to the concept of modeling with differential equations
Purpose: To be introduced to the concept of modeling with differential equations
In this project, you’ll choose a well-known differential equation model from biology, try to understand the differential equations, and explore graphical solutions.
Choose one of the following topics:
- SIR ranavirus model. Ranavirus is a disease that affects reptiles, amphibians, and fish; “Ranavirus is believed to be the cause of several recent mass mortality events in amphibian populations across the globe” (link). Given a population, let [latex]S(t)[/latex] be the number of frogs suseptible to ranavirus, let [latex]I(t)[/latex] be the number of frogs currently infected with the disease, and let [latex]R(t)[/latex] be the number of frogs that have died. Note that at any time, [latex]S(t) + I(t) + R(t)[/latex] is the equal to the total population.Then
\begin{align*}
\frac{dS}{dt} & = -a S(t) \cdot I(t) \\
\frac{dI}{dt} & = a S(t) \cdot I(t) – b I(t) \\
\frac{dR}{dt} & = b I(t)
\end{align*}
Here, [latex]a[/latex] and [latex]b[/latex] are unknown parameters that affect the dynamics of this problem.Click here for the DiffEQ grapher for the Ranavirus SIR model.
- Lotka-Volterra equations for simple modeling of predator and prey dynamics, such as the moose and wolf populations in Isle Royale National Park (link). Given [latex]M[/latex] is a population of prey (Moose), and [latex]W[/latex] is the population of predators (wolves), we have
\begin{align*}
\frac{dM}{dt} & = a M(t) – b M(t) \cdot W(t) \\
\frac{dW}{dt} & = – c W(t) + d M(t) \cdot W(t)
\end{align*}
Here, [latex]a[/latex], [latex]b[/latex], [latex]c[/latex], and [latex]d[/latex] are unknown constants, but they are parameters that affect the interaction between the two species.Click here for the DiffEQ grapher for the Lotka-Volterra equations.
- Protozoa Competition: Paramecium aurelia and Paramecium caudatum are two species of single-celled protozoa, which were studied by G.F. Gause when he formulated his famous Competition exclusion principle (link). Let [latex]A(t)[/latex] be milligrams of Paramecium aurelia, and [latex]C(t)[/latex] be milligrams of Paramecium caudatum. Suppose [latex]A[/latex] and [latex]C[/latex] satisfy:
\begin{align*}
\frac{dA}{dt} & = a A(t) – b(A(t) + C(t)) A(t) \\
\frac{dC}{dt} & = c C(t) – d(A(t) + C(t)) C(t)
\end{align*}
Here, [latex]a, b, c, d[/latex] are the parameters that affect this problem.Click here for the DiffEQ grapher for the Protozoa equations.
Once you’ve chosen a topic, here is what to do:
- Follow the steps from the section on understanding differential equations to understand what the differential equation is saying.
- Give an example: what are some realistic numbers for the differential equation? What do they tell you?
- Using the supplied “DiffEQ” webpage, explore graphical solutions to the differential equation.
- What do the parameters [latex]a[/latex], [latex]b[/latex], etc., represent?
- What setting for the parameters create realistic looking graphs? How can you