Assignment 5: Lotka-Voterra Model

Assignment 5:
Instructions
Complete the following problem set showing your work. Problems may be worked out "by hand" or in "python" or with the
assistance of other analytical software (e.g., Mathematica, MatLab). You may use chatGPT to assist in coding.
Solutions must be type written (e.g., in Jupyter, markdown, or latex). Upload PDF solution by question to crowdmark (link will be
emailed to you) by 11:59pm on the Sunday of the corresponding week (see syllabus). 
All problems are equally weighted within an assignment. Students in 468 may or may not choose to attempt the challenge
question for a bonus pts. Students in 795 are required to complete the challenge question.
Problem Set
1. Lotka-Voterra Model
Consider the Lotka-Volterra Model of describing the number of prey (e.g., hares) and predators (e.g., lynx) as
described by the system of coupled differential equations with the harvesting of the prey and predators by the Hudson Bay
Company. Here we consider a semi-sustainable model of harvest where individuals are removed in proportion to their density to
the power , where . This ensures that the rate of harvest drops quickly as the number of individuals declines
Part A. Numerically integrate the ODEs above for , , and and
assuming we start with and .
Part B. Propose an analogous continuous time stochastic process for this model. Describe the 2D state space of this model.
Part C. Simulate the dynamics of the stochastic model you proposed and compare the simulated trajectories to your answer in
part 1.
Part D: Derive a system of master equations for the probability of having prey and predators at time in the absence of
harvesting . You do NOT need to solve them numerically.
2. SIS Model
Consider the model given in example 5.13: a stochastic SIS model where transmissions occur at a mass-action rate of
with , hosts recover (becoming susceptible again) at a rate and the total population size is .
*Our goal here is to model the mean and variance in the number of infections in this model using an ensemble moment
approximation.
Part A. What are the ODEs that describe the dynamics of and ? How do these equations depend on and why?
Part B. Implement moment closure assuming that the skew in the number of infections is small.
Part C. Numerically solve the dynamics for the mean and variance assuming and and no
variance for the first 10 units of time.
Part D. Challenge 795: Using your numerical solution in 3 comment on whether your approximation is valid for the time period
considered.
X(t) Y (t)
a N
a a > 1
= − −
dt
dX
Prey Birth
αX
Prey death
βXY
harvest
μX X1.5
= − −
dt
dY
Preditor birth
δXY
Preditor Birth
γY
harvest
μY Y
1.5
α = 1 β = 0.03 δ = 0.01 γ = 0.2 μX = μ = Y = 0.01
X(0) = 500 Y (0) = 200
n m t
μX = μY = 0
∗
κ
β S ∗ I
β = 0.5 γ = 0.1 κ = 100
⟨I⟩ ⟨I
2⟩ ⟨I
3⟩
β = 0.5, γ = 0.1, κ = 100 I(0) = 20
Part E. Challenge 795: Compare your solution to the EMA to the result you obtain from a simulation approach.
3. The Yule Process
In the lecture we derived the master equation for the size of a clade in the Yule model:
with the solution:
Part A. Challenge for 795: Show that this is the case
Part B: Plot the distribution of clade sizes at at for
Part C: Technically the number of lineages at time could be infinite, obviously we can make a plot in part 2 with an infinite
domain. How close is the approximation of the probability distribution you showed in part 2 to the truth? In other words how
much of the total probability are you missing by showing a plot on a finite domain?
Part D: Simulate 50 trees in the Yule model for these parameters. Do your simulations match the solutions from the master
equation?
4. The Birth-Death Process
Part A: Propose a system of Master Equations describing the size of a clade in the birth death model.
Part B: Numerically solve these equations assuming and and for .
Part C: How does clade size in this model compare to the Yule model above? Are you surprised they are the same/different?
= −λnP + λ(n − 1)P P (0) =
dt
dPn(t)
n n−1 n {
0
1
n = 1
n = 1
P(n,t) = e e − 1
−nλt (
λt )
(n−1)
λ = 1 T = 1.5 n = 1…20
T
λ = 1.5 μ = 0.5 t < T = 0.5