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# AMATH 422/522 Problem set 1 solved

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## I Iterating Leslie Matrices and the Euler-lotka Formula

• Consider an age-structured population model as follows: maximum age A = 3. Also: p0 = 0.5,
p1 = .9, p2 = .95, f0 = 0, f1 = 1, f2 = 5, f3 = .5. Write a MATLAB model (feel free to modify
any code from class) that simulates the state vector n(t) of individuals of each age a = 0, 1, 2, 3.
Start with an initial population consisting of 100 individuals of each age.

Plot as functions of time
(1) the log of the total population size N(t) = n0(t) + n1(t) + n2(t) + n3(t) and (2) the fraction
of individuals in each age, wa(t) = na(t)/N(t), for a = 0, 1, 2, 3. Do this from t = 1 to t = Tmax,
where Tmax = 50. Use the polyfit (MATLAB) function to fit a first order polynomial to the log
N(t) and report the growth rate λ. Turn in the code you used for this, again OK if just modified
from class.

• Write down the Euler-lotka formula for this example, and solve it numerically (use .m files similar
to those from class) for the population growth rate λ. How close are your predictions of λ from
the Euler-lotka formulas and from the simulations above? Turn in the code you used for this.

## II (Taken with modifications from Ellner and Guckenheimer Ex 2.12).

According to Lande (1988), females
of the northern spotted owl begin breeding at age a=3 and are estimated to have an average of 0.24
female offspring until they die (fa = 0.24 for a ≥ 3). The survival probability from birth to age 3 is
estimated to be 0.0722, and the annual survival probability of adults (pa for age a = 3 to a = 49)
is 0.942. In our model we will take the maximum age A = 50. (These values refer to age-structured
conventions, so newborns are age 0).

The owl has been controversial in our region, because of the conflict of interest between the need for
old-growth forests as habitat, and the interest of logging companies in harvesting those forests.
• (a) We told you that I3 = p0p1p2 = 0.0722 but not the values of the individual p values. That
is because any choice of these individual p values with the same product will yield the same
long-term population growth rate (λ, from class). Why is this true?
• (b) Construct the projection matrix for the population.
• (c) Compute the long-term growth rate λ for the population.

• (d) [Note, you’ll probably want to save this problem for the very last of the whole HW, as it uses
the final lecture from the unit.] Compute the matrix of elasticities for your projection matrix. Is
the elasticity for fecundity values fa the same for all ages a? Is the elasticity for annual survival
probabilities values pa the same for all ages a? Give an intuitive explanation for your findings in
two to three sentences, and state one possible implication for management plans.

## III AMATH 522 ONLY: Ellner and Guckenheimer Exercise 2.15.

HOWEVER: modify the rule they state
there so that can never get a negative number of individuals: na(t+1) = [An(t)−h]a if [An(t)−h]a ≥ 0
and na(t+ 1) = 0 otherwise, where the subscript means take the a
th element. Note, you’ll need to look
at Ex. 2.13 to get the A matrix you need to get started.

One approach to this problem by numerical simulation of the stated dynamical rule and testing different
h values — one possibility is a for loop and logical operations to automate testing of a large number
of values.

### IV MATLAB programming – tools and tips:

Read through, and practice where necessary, the
provided sheet of MATLAB tools and tricks. Make sure you know how each one works. Write down,
explain, and turn in two additional tips or tools that you have found helpful.

### V What’s the point of all of this / project warmup:

Ellner and Guckenheimer Exercise 1.1. Note:
whatever paper you choose does NOT commit you to anything, project-wise. This is just to get us
started in thinking about dynamical modeling in biology problems that we personally care about!
NOTE – you need to turn in the writeup requested in this Exercise 1.1 of our book for this problem!

### Some suggestions as example papers (OK to choose one of these, too):

• Li and Anderson 2009, The Vitality model: a way to understand population survival and demographic heterogeneity, Theoretical Population Biology.
• Ma, Trusina, El-Samad, Lim, Tang. Defining Network Topologies that can achieve biochemical