Math 56
September 13, 2007 [corrected Sept. 18]
Problem Set 1 (due September 20)
These problems relate to the interplay between discrete and continuous population models.
1. (“Hard” exponential model)
a. Consider the discrete model
xk+1 = xk (1+R) for k = 0, 1, 2, … (1)
where x0 and R are parameters of the model (i.e., treat them
as if they are known constants).
Show that this model has the analytic solution
xk = x0 (1+R)k for every integer k ≥ 0. (2)
b. Consider the continuous model
x’(t) = r x(t) for t ≥ t0 (3)
x(t0) = x0
where t0 is given and x0 and r are parameters of the model.
Show that this model has the analytic solution
x(t) = x0 exp ( r(t-t0) ) for all real t ≥ t0. (4)
[corrected 9/18 to insert “-t0”]
[ Note: “exp(z)” means ez but is easier to type. ]
c. Reconcile these two models. Show that these models agree exactly if
t0, t1, t2, … are evenly spaced times with
increment Dt = tk+1 - tk
xk means x(tk)
and if r and R are related by
exp ( r Dt ) = 1 + R. (5)
2. (“Soft” exponential model)
Note that the discrete model
xk+1 = xk (1 + Rk) (6)
(where x0 is a parameter but Rk can vary with k)
has the (useless) analytic solution
xk = x0 ( 1+R0 ) ( 1+R1 ) … ( 1 + Rk-1 ). (7)
a. Consider the continuous model
x’(t) = r(t) x(t) (8)
x(t0) = x0
(where r(t) can vary with time).
Show that this model has the analytic solution
x(t) = x0
exp ( 
) (9)
for all real t ≥ t0.
b. Show that these models (equations 6-7 and equations 8-9) agree if
1 + Rk
= exp ( 
)
for each k. (10)
[corrected 9/18; l.h.s. was just “Rk”]
3. (Rate changing linearly)
a. Consider the continuous model
x’(t) = r(t) x(t)
x(t0) = x0
(as before)
with the additional equation
r(t) = r0 – at (11)
where r0 and a (and x0) are parameters of the model. That is,
the growth rate decreases linearly.
Show that this model has the analytic solution
x(t) = x0 exp ( r0 t – a (t2 – t02) /2 ). (12)
b. Suppose that x0, r0, and a are all positive. Show that x(t) is an increasing
function of t when t is slightly larger than t0, but that then it decreases.
What is the limit (as t -> infinity) of x(t) ?
c. Show that if we turn this model into a discrete model by observing x(t)
only when t = t0, t1, …, and if we define Rk as in the previous problem
then Rk is NOT LINEAR as a function of k. (Computing three values of
Rk in an example might be enough to show this.)
4. (Discrete logistic model)
a. Consider the discrete logistic model
xk+1 = xk (1 + Rk) (13)
with
Rk = R* ( (M – xk ) / M ). (14)
Here R* and M are parameters of the model.
Show that this implies that
xk+1 = (1+R*) xk (1 – Z xk ) (15)
where Z is some combination of R* and M.
b. Suppose that 0 < x0 < M and that 1 + R* = 1.02. Show that xk increases
as a function of k, from x0, and that it approaches a limit of M as
t approaches infinity.
c. Suppose that x0 = 1 million, M = 2 million, and 1 + R* = 3.5. Show
that xk DOES NOT increase as a function of k (at least, not for all k)
and that xk DOES NOT approach a limit as t -> infinity.
(A computational illustration is enough, if it is convincing.)
5. (Continuous logistic model)
a. Consider the continuous model
x’(t) = r(t) x(t)
x(t0) = x0
with
r(t) = r* ( (M – x(t) ) / M ). (16)
(Now r* and M are parameters of the model.)
Show that this model is equivalent to the equation
x’(t) = r* x(t) – (r*/M) x(t)2 . (17)
b. Show that this model has the analytic solution
x(t) = 
(18)
for all real t ≥ t0.
c. Show that if 0 < x(t) < M and r* is positive, then x(t) increases
as a function of t and has limit M as t approaches infinity,
regardless of the exact value of r*.