Math 3MB3 Assignment 1
Due: by 11:59pm on Monday, January 29, 2024 via Crowdmark
Instructions: Please read the following background information and answer all ques- tions. You may handwrite or type (e.g., Microsoft Word, LaTeX) your solutions, but if they are handwritten please make sure they are legible. You are welcome to discuss this assignment with others in the class, but you must write up your own solutions; if your work too closely matches the work of another student, this will be treated as an act of academic dishonesty. All work for this assignment is due by 11:59pm on Monday, January 29, 2024 through Crowdmark.
Late work is subject to the late work policy described in the official course outline.
Final Submission: Your final submission will be done through Crowdmark. You will need to upload a pdf file of your solutions for each question individually, either by scanning or taking a picture of your work. You will also need to upload pdf files of each of the plots you make for Problem 4 and screenshots of the R code used to generate those plots.
Background: Imagine we want to model the growth of a population of fish living in a lake every month. Based on observed data, we know that the fish grow roughly logistically with an intrinsic growth rate r > 0, but each month some fraction 0 < h < 1 of the population is harvested by local fishers. Let pt represent the number of fish in the current month t as a fraction of the carrying capacity (the maximum number of fish the lake can sustain) and let pt+1 represent the number of fish in the next month t + 1, t = 1, 2, 3, . . . , as a fraction of the carrying capacity. This means 0 ≤ pt , pt+1 ≤ 1. We can describe this scenario with the following model (modified from the logistic growth model we discussed in class):
pt+1 = pt + r(1 − pt )pt − hpt. (1)
Problems
Problem 1 [3 points]: This model contains three terms: pt , r(1 − pt )pt , and hpt. 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.
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 p* = 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 “r = xx” where “xx” 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 “Save image as...” to save the plot to your computer. Please also submit screenshots of the code used to create each of these plots.
Problem 5 [4 points]: Write a one-paragraph summary of the results of your model analysis. This should not be a restatement of the equilibrium and stability conditions you found above, but instead should bean explanation of what your model predicts could happen to the population of fish. This should include a description of all possible outcomes for the fish population and what factors influence which outcome will actually be realized. It should be written in plain (non-technical) language and say things like “we predict the fish population will go extinct if the intrinsic growth rate is lower than this particular value” .
Problem 6 [2 points]: Suppose after delivering the above summary of your modelling efforts to a team of biologists and policy makers studying the fish population, they tell you that they have collected data that suggests the population has an intrinsic growth rate of approximately r = 2.4. They then ask you, based on your modelling, what level of harvesting you would recommend to maintain a stable non-zero population. In other words, what fraction h should the fishers be allowed to catch each month to maintain a stable fish population? Write a short (one or two sentences) response to the research team indicating what level of harvesting should be allowed and giving them an estimate of the number of fish (as a fraction of the carrying capacity) that will be maintained in the lake given this level of harvesting. Your answer for the level of harvesting and the associated fraction of fish should be phrased as a range of values (e.g., a < h < b for constants a and b).