Math 5110/6830
Instructor: Alla Borisyuk
Homework 2.2
1. Recall the model of the segmental organism we considered in class. Consider a slightly simplified version (nex-to-terminal segments that grow
laeral branches are not considered next-to-terminal anymore). The number of terminal segments and the number of the next-to terminal segments
satisfy
an+1 = pan + 2qan + rbn
bn+1 = an
a) Reduce this system to a single equation for an , also use the fact that
p+q =1
b) If initially there was just one segment, how many terminal segments will
there be after 1 day? Now find general solution for an with these initial
conditions. How many terminal segments will there be after 10 days?
c) Under the same conditions as in b) what will be the total number of
segments after 10 days?
2. The Peter Pan model. Suppose in a population adults can produce offsprings (juveniles) that in a year can vanish, become adults or stay juveniles forever. Define m as the number of offsprings each adult produces
per year, σ - fraction of juveniles reaching adulthood in a year, p - fraction of adults surviving to next year; and γ - fraction of juveniles staying
juvenile.
a) Write a model for the dynamics of number of adults and juveniles over
the years.
b) Suppose there are two different populations. In the first m = 0.5, p =
0.5 and σ = 0.9. In the second m = 2., p = 0.5 and σ = 0.2. Find the
leading eigenvalue for each population. What would happen to each in
the long run?
c) Extra credit Suppose instead that the two types of years alternate.
Find the model describing what the population would look like after two
years.
3. Consider the red blood cells model from class
Rn+1 = (1 − f )Rn + Mn
Mn+1 = γf Rn
a) Reduce thesystem to one equation for Rn
b) Find general solution of the equation.
c) Show that the number of red blood cells can only remain approximately
constant if γ = 1. Why does this result make sense?
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