MAT 124: Homework Assignment #1 Due Wednesday, April 8 at 10 AM Upload to Gradescope (Tab in Canvas page) 5 problems, 100 total points Problem 1.1 (25 points) Consider a population of fish modeled by the differential equation dP dt = a ( 1− P K ) P − δP where P (t) is the population of fish at time t (in years), K is defined as the carrying capacity, a > 0 is the maximum growth rate (units of fish/year), and δ > 0 is the death rate. The carrying capacity gives the maximum population that the resources can naturally sustain. This is a form of a logistical growth model. 1. Interpret the logistical growth rate a ( 1− PK ) . For what values of the population is this rate positive and negative? 2. Conduct a complete qualitative analysis of the logistic growth model. That is, determine all equilibria and their stability, sketch representative solutions P (t) as a function of t, and indicate where solutions change convexity. 3. For what values of δ does the population die out? Problem 1.2 (20 points) Consider the constant-coefficient system of differential equations given by X ′ = AX with A ∈ R2 and X = ( x1 x2 ) . Draw the associated phase planes for each of the given eigenvalue and eigenvectors of A. 1. λ1 = −2, v1 = ( 3 1 ) , λ2 = −4, v2 = (−1 2 ) 2. λ1 = 1, v1 = ( 0 1 ) , λ2 = −5, v2 = ( 1 3 ) Problem 1.3 (15 points) For each of the eigenvalue-eigenvectors pairs given in Problem 1.2, draw the approximate solutions x1(t) and x2(t) that begin with the initial condition X(0) = ( x1(0) x2(0) ) = ( 1 5 ) . Your solutions do not (and should not) be exact, but need to capture the overall trends. 1 Problem 1.4 (25 points) Some diseases have a latency period between when an individual is first exposed and begins to show symptoms. Models of this type are called SEIR for Susceptible-Exposed-Infected-Recovered. We will make the following assumptions: – Exposed individuals are not contageous – Each infected individual spreads pathogens to b individuals per unit time – The mean latency time is 1/α – The mean residence time is 1/γ – The disease confers life-long immunity – No births and deaths 1. Are the assumptions realistic? 2. Construct a compartment model diagram for the SEIR model. Hint: There are many simi- larities to the SIR model. 3. Write a system of differential equations for the SEIR model. Problem 1.5 (15 points) Consider a generic scalar autonomous ODE of the form dx dt = f(x). Is it possible to have two stable or two unstable equilibrium points next to each other? Justify your answer. 2
