LS50B Problem Set #6
Due Friday, March 11, 2016 at 5 PM
Problem 1: A puddle of snails
A population of snails lives in a puddle that can contain no more than K snails. We will model their
population dynamics by a Markov process, where each snail dies with rate δ, and gives birth with rate β
only if there are less than K snails in the puddle.
1. Given that today there are n snails in the puddle, what is the probability that this population will go
extinct? Feel free to reuse results obtained in class, as long as they are relevant here.
2. If we let wn to be the mean time to extinction (in cases when extinction is inevitable), show that
Pm−1
P∞
w1 = B1 and wm = B1 + k=1 ρk Bk+1 , where Bm = j=m bi1ρi . This is done (more or less) in the
handouts, so your job is to complete the missing steps, and justify all the steps in words (especially
the one that determines w1 ).
3. Calculate and plot the mean time to extinction for w1 of a snail population, with β = 2/day, δ = 1/day,
and K = 100. Next, calculate w for arbitrary m. This calculation cannot be done analytically, so carry
out the infinite sums numerically (e.g. in MATLAB, WolframAlpha, etc.).
Problem 2: Logistic snails (or, les escargots logistiques)
A more realistic model of a snail population is perhaps the logistic, in which each snail gives birth with a rate
β(K − n)/K for n ≤ K. In this model, the birth rate linearly declines with the distance of the population
size from the carrying capacity, K. For this problem, your job is to simulate this model using the Gillespie
algorithm in MATLAB.
1. In MATLAB, write down, in the form of comments, how you would create this simulation as though
we had given you a template in which to fill in code.
2. Within this commented scaffold, write a model of this system with the Gillespie algorithm. Your job
now is to get this model to run correctly. To show that it runs, produce a plot of n(t) vs. t. On the
same axes, plot at least one simulation with K = 50 and at least one simulation with K = 200. Use
β = 2/day, δ = 1/day, and n0 = 15. You should simulate until extinction or 60 simulated days elapse,
whichever is shorter.
3. With your model, find the mean first passage time (i.e. time to extinction) for K = 20 and K = 30
when β = 5/day, δ = 2.5/day, and n0 = 10. Hint: In order to find the mean passage time, you have to
run the simulation a large number of times. Report the mean and standard deviation of the passage
time for each of the values for K you simulated above.
1
4. Describe how you could determine whether the average time to extinction was significantly different
depending on whether K = 20 or K = 30. (No need to use the word ”p-value” in your answer.) How
might your ability to detect a difference between the mean passage times for two different values of K
depend on how many trials you run?
Problem 3: Snail rain (or, il pleut des escargots! )
Seagulls access the soft parts of their shelled prey by dropping them from the sky onto hard surfaces (video
link). Seagulls can eat snails, but they need to knock them out first so they can take their time extracting
their soft parts. A snail can survive an impact with the ground if it hits with a force ≤ 50N ; otherwise, it
gets a concussion and is readily eaten by the gull. Seagulls fly and drop their snails at a mean height of 20m,
but this height varies approximately according to a Gaussian distribution with standard deviation of 2m.
1. Calculate the probability that a 5g snail gets concussed (and is thus eaten) upon impact after being
dropped from the sky by a seagull. For this calculation, assume the air provides no resistance during
the free-fall of the snail. Assume that transfer of momentum with the ground occurs over a time
interval of 2 milliseconds. Approximate g as 10m/s2 . (Hint: For this calculation, pay attention to
units and their magnitudes!).
2. What is the largest mass a snail can have to survive a drop from very, very high? Now also assume
that there is a pressure drag force from the air F~d equal to 21 cD ρv 2 A on the falling body, where the
density of air ρ = 1.2922kg/m3 and the drag coefficient for a spherical body cD = 0.47, assumed to be
constants in this case. A is the cross-sectional area of a snail treated as a sphere. Assume also that
mass (in grams) scales with αr2 , where α = 4 and r is the snail’s radius in cm.
2