Midterm Exam - Math 4/5480 - Mathematical Modeling
Spring 2015 - Dr. Radu Cascaval - DUE Wednesday, April 1st, 1:40pm
Problem 1. Continuous Dynamical System
Consider the logistic model for (single) population dynamics P = P (t), which is governed by the following
differential equation:
dP
= 2P (1 − P )
(1)
dt
[where a = 2 stands for the intrinsic growth rate and K = 1 for the maximum sustainable population.]
(a) Find the steady states (equilibria). What is the eigenvalue method telling us about the stability of
these equilibria? [IN PART (a) DO NOT USE Matlab or any other computer algebra system]
(b) Find the general solution P = P (t) (explicit formulas please, NO computers here either!).
(c) Plot the direction field in the (t, P ) plane; then, using the explicit formulas found in (a), sketch a
few solution curves.
Problem 2. Discrete Dynamical System
(a) For a discrete-time dynamical system
x(n+1) = g(x(n) )
state the criterion for the stability of an equilibrium state x∗ [= g(x∗ )].
(b) Newton’s method for approximating solutions of a scalar (1D) equation f (x) = 0 is described by the
discrete-time dynamical system (x(0) = x0 ):
f (x(n) )
.
f 0 (x(n) )
√
which approximates x∗ = 3, the solution of f (x) = x2 − 3 = 0.
x(n+1) = x(n) −
Derive the dynamical system for x(n)
Problem 3. Investment and Expenditure Problem
Joanna just won the jackpot and she is considering her lifetime plan of investment and expenditures.
Presently she has an initial amount of money x0 = 107 [= $10, 000, 000] to be invested and plans no future
income other than the one she obtains from investment at a fixed interest rate r = 4%. She plans to
withdraw and spends money at a rate u(t). Her total capital x = x(t) at time t is therefore governed by
the equation
x0 (t) = rx(t) − u(t)
where r > 0 is the interest rate and u(t) denotes her rate of expenditure at time t. Her immediate enjoyment
due to expenditure is given by the (so-called utility) function E(u). Future enjoyment, at time t > 0, is
counted less today, at time 0, by incorporation of a discount term e−βt . Thus, she wishes to minimize the
amount of money at time T while maximizing her total enjoinment over a given period of time [0, T ]. The
objective is to
Z
T
minimize
J = x(T ) −
e−βt E(u(t))dt
0
Find the optimal strategy, assuming her utility function is E(u) = u and she has a constraint on the
rate of spending 0 ≤ u(t) ≤ M .
[Take T = 30 years and M =$500,000/year , β = 0.01]