MATH3070: Natural Resource Mathematics ASSIGNMENT 4 – SEMESTER 2, 2021
Attempt all questions. This assignment is due on September X, 4:00pm. Make sure that you show clearly the reasoning you use to solve the problems. It is also advisable to keep a photocopy of the assignment you hand in. You are free to use any reference you wish as long as you cite your source. Each question is worth 25 points.
Q1. Consider a fish population that eats kelp. It grows according to the model
Rt+1 = rRte1−Rt/k, (1) where Rt is the biomass at time t, for t ∈ {0,1,…,n}, with carrying capacity k and a parameter related to the growth of the population, r. Assume that k has been estimated independently by biologists, who know exactly how much kelp is in the ocean. So you can consider k as fixed.
(a) What is proliferation in this model? In other words, compute F0(0). (b) Given data, {R0,R1,…,Rn}, for population biomass in years t = 0,1,…n. Derive the least squares estimate of r. Call it ˆ rLS. Be sure to verify it is a least squares estimate.
1
Q2. Consider the model from Q1 but slightly modified,
Rt+1 = Rteρ(1−Rt/k). (2)
(a) Write down an expression for ρ in terms of r, such that the model from Q1 is mathematically equivalent to the model above.
(b) Now we introduce noise, in the following way. Consider
Rt+1 = ztRteρ(1−Rt/k), (3) where zt is a random variable such that log(zt) ∼ N(0,σ2). Assume that k is fixed. Find the maximum likelihood estimate of ρ, in terms of k and the given time series data, {R0,R1,…,Rn}. (c) From parts (a) and (b), write down an estimate for r. Call it ˆ rMLE.
2
Q3. In this question you will explore whether ˆ rLS and ˆ rMLE are biased.
(a) In R or a programming language of your choice, simulate time series data, {R0,R1,…,Rn}, using the model Rt+1 = ztrRte1−Rt/k, (4)
with r = 1.3, k = 100, σ = 0.1, starting at R0 = 5 and n = 50, where zt is described as in Q2. Plot your time series data with t on the x-axis and Rt on the y-axis
(b) Given your time series data, compute the values of ˆ rLS, and ˆ rMLE.
(c) Plot your data with Rt on the x-axis and Rt+1 on the y-axis. On top of this data plot two curves, the Ricker function, rRte1−Rt/k, with r = ˆ rMLE and r = ˆ rLS. Which curve looks like it fits the data better?
(d) Determine which of your estimators r = ˆ rMLE and r = ˆ rLS has greater bias? To do this you need to simulate 10,000 sets of time series (e.g. repeat part(a) 10,000 times with new values of zt each time. You can store the time series in a 10,000 by n matrix. You may want to consult the week 5 practical.
(e) Which estimator is more biased? Can you guess in words why one is more biased than the other? Note you can take a guess at this question even if you were unable to do the programming for parts (a-d).
3
Q4. Consider a species where individuals survive each year with probability θ. Say you monitored n1 species and discovered that n2 were alive at the end of the year.
(a) Compute the posterior distribution of θ given your data n1, n2. Assume a Beta distributed prior for θ, with parameters α and β. That is assume the prior,
p(θ) =
θα−1(1−θ)β−1 B(α,β)
. (5)
(b) Let α = 1, and β = 1. Plot the pdf of the prior. In words, what are we assuming about θ in the absence of data? Plot the pdf of the posterior if n2 = 8 and n2 = 10. How did observing 8 surviving individuals out of 10 change our belief in the survival probability? You can use R to do the plots (it is good practice with R) but if you are in a rush you can also screenshot this app for the plots. http://eurekastatistics.com/beta-distribution-pdf-grapher/
4
