R TUTORIAL. Part 1:1: Direction Fields

Introduction
You must have wondered, can I use Rstudio to plot direction fields? The answer is, “YES! you can.” In fact, you can do so much more with Rstudio for different ODE’s, including plotting nullclines and phase portraits to name only few. Let’s once again use the autonomous mice population problem and plot the direction field at our equilibrium solution.
Plotting Direction fields
We present a couple of examples to plot direction fields for the first order differential equation.

Example: Consider a population of field mice who inhabit a certain rural area. Suppose that several owls live in the same neighborhood and they kill 15 mice per day. Then the mouse population P(t), t is measured in months, satisfies the differential equation

\[ \frac{{\text d}P}{{\text d}t} = \frac{P}{3} - 360. \]
Plot a direction field for the given differential equation using R software package. Provide the codes of your plot or state what resources did you use. Based on the slope field, determine the behavior of P(t) as t → ∞.

Solution: In order to solve this problem and first find equilibrium solutions where slope is zero. Upon equating islope to zero, \( P/3 - 360 =0, \) we find that P = 1080 is the critical point, also called the stationary solution. To plot our solution around P = 1080 on Rstudio, we need to import R’s phaseR package. Please note that to solve your equation, you do still need the deSolve package. Next, we write the code, as below: 

library(phaseR)

apma1 <- function(t, y, parameters){
    a <- parameters[1]
    dy <- a*((y/3) - 360)
    list(dy)
} 

apma1.flowField <- flowField(apma1, xlim = c(0, 10),
                             ylim    = c(1070, 1090), parameters = c(1),
                             points = 9, system = "one.dim",
                             add = FALSE, xlab = "time", ylab = "P",
                             main = "Mice Population")

grid()

There are few new elements in the code above, namely the flowField command, which plots one or two dimensional autonomous ODE systems and is in phaseR, thus our system in our code is one dimensional. Next, we see new assignments for the limits on x and y. In this code, I put our t, or time, ranging from 0 to 10, while our y, or P, is either 10 units above or below 1080, our equilibrium solution. The add command sets our equation to FALSE because if TRUE our plot is added to an existing plot, while FALSE logically implies the creation of a new plot. Moreover, I set the label for the x-axis as time as while I set the y-axis as P. I set the points to 9 because that is the number of directions fields drawn in the plotting. You can experiment and see the graph that appeals to you the most, but I chose a reasonable but random 9 in this case. Lastly, grid() adds rectangular grids to our plot. Below is our coding in Rstudio and the subsequent direction field plot.

Plot of mice population

We plot the slope field for the differential equation

\[ \frac{{\text d}y}{{\text d}x} = \frac{y + e^{x/2}}{x + e^{x/2}} . \]
We therefore see how our solution behaves above and below 1080, where it, as time increases indefinitely, approaches if P> 1080, and approaches \( -\infty \) if P<1080. Thus, we learn that our equilibrium solution is asymptotically unstable.