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 = rRte

1−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 F

0

(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 ˆrLS. Be sure to verify it is a

least squares estimate.

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Q2. Consider the model from Q1 but slightly modified,

Rt+1 = Rte

ρ(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 = ztRte

ρ(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 ˆrMLE.

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Q3. In this question you will explore whether ˆrLS and ˆrMLE are biased.

(a) In R or a programming language of your choice, simulate time series data,

{R0, R1, …, Rn}, using the model

Rt+1 = ztrRte

1−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 ˆrLS, and ˆrMLE.

(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, rRte

1−Rt/k, with r = ˆrMLE and

r = ˆrLS. Which curve looks like it fits the data better?

(d) Determine which of your estimators r = ˆrMLE and r = ˆrLS 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).

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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) = θ

α−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/

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**Type of service : Math/Physic/Economic/Statistic ProblemsType Of Assignment : Math modelingSubject : Not definedPages / words : 4/1100Number Of Sources : 0Academic Level : Junior(college 3rd year)Paper Format : MLALine Spacing : DoubleLanguage Style : AU English**