p_(l+1)=p_(l)+r(1-p_(l))p_(l)-hp_(l).
Problem 4 [9 points]: Your analysis in the previous question should have shown that all
equilibria are unstable if r > h + 2. Use R to simulate your model and create a plot of the
population size (as a fraction of the carrying capacity) versus month (see the Lab 2 activity
for help). Set h = 0.3 and simulate until time t = 5. Create three separate plots: one where
r is set so that pā = 0 is stable, one where r is set so that the non-zero equilibrium is stable,
and one where r = 3 so that neither equilibrium is stable. Be sure to set appropriate axis
labels and set the title of each plot to be ār = xxā where āxxā is the value you used for r
in creating that plot. You should save each plot as a separate pdf file and upload these files
with your submission on Crowdmark. To save plots in RStudio, click on the Zoom button in
the Plots panel and then right click and choose āSave image as. . . ā to save the plot to your
computer. Please also submit screenshots of the code used to create each of these plots.
P+1=p+r1-pp-hp
(1)
Problems
Problem 1 [3 points]: This model contains three terms: pt, r(1 - pi)pt, and hp. Briefly one sentence each) describe what each of these terms represents in the context of the real world system we're trying to model.
Problem 2 [3 points]: Find all equilibrium points of the model. Be sure to show your work.
Problem 3 [14 points]: Analytically determine conditions under which each equilibrium is stable. These conditions should be written in terms of bounds on the parameter r. Be sure to show your work