Math 142 - Homework 3: Due in class October 24th 2018
M. Roper
October 17th, 2018
Exercise 1. Demographic modeling with Leslie matrices In class we discussed a model for
the growth of a population of insects, in which the insect life history could be broken into two
stages: 1. a larval form, and 2. a mature form. Denote by N(l)(t) the number of larvae at time t
and N(m)(t) the number of mature insects. Assume the following data:
1. In one unit of time a fraction ml of larvae die, and a fraction mm of mature insects die.
2. In one unit of time, a fraction p of larvae metamorphose into mature insects.
3. In one unit of time, a fraction F of mature adults lay eggs. You may assume that these eggs
hatch into larval insects immediately. For each mature adult that lays eggs, a number K of
o spring will be produced
(a) Show how to represent all of these data as a Leslie matrix model that takes a vector N(t)
of subpopulation sizes at time t to a vector of subpopulation sizes N(t + t), time t later. You
may assume that the interval between predictions t is small enough that only one e ect needs
to be considered for each insect at each time interval (that is, you don’t have to worry e.g. that
an insect might metamorphose and then lay eggs in the same interval, or lay eggs and then die etc.).
(b) Let’s assume that you are able to measure all of the parameters in the model except for the
number of o spring per adult K. You deduce:
N(l)(t + t)
N(m)(t + t)
=
0:8 0:002K
0:1 0:85
N(l)(t)
N(m)(t)
What is the minimum value of K that ensures that the population does not become extinct over
time? (Hint: As before, you may nd it useful to look at det(L I), but you should explain why
you do so.).
(c) In reality insect eggs do not hatch immediately { the egg sits around for a certain amount
of time before it hatches (and during this time, it may be eaten or otherwise destroyed, so that
it doesn’t hatch at all). Explain how you could incorporate this extra information into a Leslie
matrix model. What extra parameters would you need to measure?
Exercise 2. Demographic modeling with Leslie matrices Consider a discrete time model
for the growth of two linked geographic subpopulations on two islands:
with initial conditions N0 =
(a) Without implementing the recurrence equation (i.e. just by looking at the Leslie matrix),
explain whether the population will (generally) go extinct, or grow exponentially.
(b) Is it possible that for any choice of starting population (i.e. any N0 that has both entries
non-negative) that the population can decay over time?
(c) For the given initial condition calculate an explicit expression for Nk. Calculate, also, the stable
distribution of organisms between the two subpopulations, as k!1.
Exercise 3. Stochastic simulations of population growth Modify the Matlab code that we
wrote to simulation population growth: with a birth rate b = 0:5.
(a) Starting with a step size t = 0:001, and initial population size of N(0) = 1, modify the code
to run 10000 replicate simulations, and to calculate the distribution of population sizes (i.e. the
fraction of populations with n = 1, n = 2, . . . individuals, at time t = 5. In class we will show that
the probability that there are exactly n organisms in the population is: Pn(t) = e bt(1 e bt)n 1.
Con rm this result by your simulations, by making a plot showing Pn(5) as a function of n, and
the estimates coming from your simulations.
(b) (Importance of step-size). We argued in class that the simulations should be run with a small
step-size. However, provided that the step-size is small enough, it does not a ect the results.
Con rm this by re-running the simulations from part (a) using a step size of t = 0:005.
What happens if you run the simulation with a step size of t = 1. Do your simulations still
reproduce the predicted distribution of population sizes? Can you explain why not?
(c) Now consider what happens if in addition to cells dividing, the cells in the population may also
die. That is, in each interval t, the likelihood of a given cell dying is m t. Explain in words
how you would need to modify your stochastic simulation algorithm to include cell death. (Hint:
Consider the number of cells that are born and the number that die separately. Explain why you
don’t need to worry, so long as t is small, about the probability of a cell both dividing and dying
in the same time interval).
(d) Implement your stochastic simulation from part (c). By running replicate stochastic simulations,
make a plot of how the average population size changes as a function of time. Assume that m = 0:3.
Characterize the average growth (i.e. is it exponential? If so, what is the rate of growth?).
Exercise 4. Population genetics The Moran process is a model that describes how the diversity
of populations changes with time. Consider a population of N = 10 cells. These cells have some
form. of diversity (i.e. di erent genes): to keep the biology simple, let’s imagine that initially half
of the cells are colored red and the other half are colored green. At each time step, exactly one cell
divides. We pick this cell at random, and make a copy of it (if the original cell is red, then its child
will also be red etc.). Because of over-crowding, the overall number of cells must remain constant
(i.e. equal to N). Each time a cell divides, one must be pushed out of the population. So pick one
of the original N cells and remove it from the population (in particular, the cell that is removed
could be the same or di erent as the one that divides).
(a) Do you expect the average number of red cells (initially 5) to change over time?
(b) Simulate a single Moran process for a population of N = 10 cells. Show that after some nite
time, your simulated population becomes only red, or only green.
(c) Simulate 100 populations of N = 10 cells. Show that they all, eventually become homogeneous,
in the same way as in part (b).
(d) How do you reconcile your answer from part (c) with your answer from part (a). Can you
descrie in words, why the populations eventually homogenize?
(e) Use simulations with di erent values of N to show that the average amount of time for the
population to become homogeneous is proportional to N2. Calculate the average time for the
population to become homogeneous for each N, by running replicate stochastic simulations. How
can you plot these data in a way that makes it clear the the time to become homogeneous is
proportional to N2? (f) Humans don’t reproduce by dividing themselves. But everyone inherits
their mitochondrial DNA (this is the DNA in the mitochondria, which power our cells] only from
their mother. So from the point of view of mitochondrial DNA, reproduction is the same as just
dividing a woman into two. Brie y explain whether you think it is realistic to model the diversity
of human mitochondrial populations using a Moran process. Would you expect us to all have the
same mitochondrial DNA?