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1.5 One-Dimensional Dynamical Systems 65
growth rate in volume equals the assimilation rate minus the rate food is
used. Food is assimilated at a rate proportional to its surface area because
food must ultimately pass across the cell walls; food is used at a rate
proportional to its volume because ultimately cells are three-dimensional.
Show that the differential equation governing its size L(t) can be written
L ′ (t) =a − bL,
where a and b are positive parameters. What is the maximum length the
organism can reach? Use separation of variables to show that if the length
of the organism at time t =0isL(0) = 0 (it is very small), then the length
is given by L(t) =(a/b)(1−e −bt ). Does this function seem like a reasonable
model for growth?
9. (Pest outbreaks) In a classical ecological study of budworm outbreaks in
Canadian fir forests, researchers proposed that the budworm population N
was governed by the law
(
N ′ = rN 1 − N )
− P (N),
K
where the first term on the right represents logistics growth, and where
P (N) is a bird-predation rate given by
P (N) = aN 2
N 2 + b 2 .
a) Sketch a generic plot of the bird-predation rate P (N) versusN.
b) What are the dimensions of all the constants and variables in the
model?
c) Define new dimensionless independent and dependent variables by
τ =
t
b/a , n = N b ,
and reformulate the differential equation in terms of those variables
and certain dimensionless constants. (For this problem, a dimensionless
form is extremely tractable compared to the dimensioned model.)
d) Working with the dimensionless equation, show that there is at least
one and at most three positive equilibrium populations. What can
be said about their stability? Hint: Write the equation in the form
dn/dτ = nF (n) and plot the two terms of F (n) to find their intersection
points.