4.4 HigherOrder Linear Differential Equations
149
solution is "stable," that is, whether a solution to the initial value problem
will tend to zero. Stability of the zero solution depends on the eigenvalues,
the roots Al of the characteristic equation P(A
4.8 Phase Plane Analysis
159
proper when the eigenvalues arc equal and there are two linearly independent eigenvectors; otherwise it is called improper. A proper node is also
referred to as a star point or star solution (Gushing, 2004; Gulick,1992). The
r
60
Chapter 4 Linear Differential Equations: Theory and Examples
if the real part a = 0, then solutions are closed curves, encircling the origin.
The origin is classified as a spiral (or focus) if a 0 and a center if a = 0.
3.
4.
Spiral or locus: Eigcnvalu
7.9 Exercises for Chapter 7
3N
t
335
ZN
= f(N) +
x
with zero flux boundary conditions and a positive equilibrium at N.7he system
linearized about N is
av
0t
a2v
= cr11v + D.
Assume all = f'(N) < 0. Let v = eU'cos(kx) and show that it is impossible for
the
7 7 Pattern Formation
325
7.T Pattern Formation
Turing's work has had a tremendous impact on theoretical biology, EdelsteinKeshct (1988) stated that7uring's contribution is "one of the most important contributions mathematics has made to the realm of deve
150
Chapter 4 Linear Differential Equations. Theory and Examples
4.5 RouthHurwitz Criteria
Important criteria that give necessary and sufficient conditions for all of the
roots of the characteristic polynomial (with real coefficients) to lie in the left
332
Chapter 7 Partial Differential Equations: Theory, Examples, and Applications
(a) If 0 < b(a) < p(a) for all ages a > 0 (birth rate is less than death rate),
show that lime
n(t, a) = 0.
(h) If 0 < (a) < b(a) for all ages a > 0 (death late is less than
4.8 Phase Plane Analysis
161
Recall that the eigenvalues, the roots of the characteristic polynomial
A2  Tr(A)A + det(A) = A2  TA + h, satisfy
T+
The discriminant is denoted as y and defined as follows;
y=T243.
Figure 4,2 Solutions to
Example 4.12 are
4 8 Phase Plane Analysis
157
eigenvalue of A and V is an cigenvector corresponding to A. We summarize
the form taken by the general solution in the case of a 2 X 2 matrix A.The cigen
values are the solutions to the characteristic polynomial, dct(A  Al)
162
Chapter 4 Linear Differential Equations, Theory and Examples
Matrix A =
"1
a
. For the zero equilibrium to be a stable spiral, the discrim
inant and trace T must be negative. Tr A = T = a < o, det A = b = 1, so that
y = T2  46 = a2  4 < o. The two
154
Chapter 4 Linear Differential Equations: Theory and Examples
Consider the case of a firstorder system with constant coefficients,
dx
(111X + a 12y,
clt
cly
a21x
clt
c122Y
In matrix notation, dX/dt = AX, where A = (a1)andX = (x, y)1, Differentiating d
158
Chapter 4 Linear Differential Equations: Theory and Examples
Solutions to the linear system (4.16) are characterized by the eigenvalues of
the matrix A, which in turn depend on the trace and determinant of A. `I'he
origin will he classified as a node,
4.7 firstOrder Linear Systems
i 55
interval of existence. (Compare with the Wronskian.) Hence, the inverse (I (t)
exists for all t on the interval of existence. The unique solution to the IVI' for the
linear nonhomogeneous system can he expressed in the
156
Chapter 4 Linear Differential Equations: Theory and Examples
Example 4.9
Suppose A is diagonal,
A=
Then A =
and
0
cr1c
zz
C '4t
U
ki
ko
Ca
J
(a22t)k
U
'
U 1
U
Ic .
<o
The solution to the system ctX/dt = AX is
X(t) = eiX0 =_x0e' hI
(0)
J
rrh1r
+ Y(1
326
Chapter 7 Partial Differential Equations:`Theory, Examples, and Applications
First, we derive conditions for local asymptotic stability of (N1, N2) in the
reaction system. Linearizing the system about the equilibrium (Nl, N2) leads to
die
1
a11u1 + a1
4.6 Converting HigherOrder equations to FirstOrder Systems
i 53
can he expressed as an equivalent firstorder system. Define n new variables,
x1, . , xf7, as follows:
_
X2
_dx
dt'
Then
dx1
dx
dt ` tit ^ x2'
dx2
d2x
dt ` dt2
dx d"x
dt ^ dt"
x3
aix
_ a2x
148
Chapter 4 Linear 1)ifterential Equations: Theory and Examples
Hence, the roots or eigenvalucs are 0, 0, and 4 and the three linearly independent solutions are 1, t, and e4', respectively. The general solution is
x(t) = c1 c2t + c3e4r.
To verify that t
324
Chapter 7 Partial Differential Equations: Theory, Examples, and Applications
Figure Ti' A traveling wave
solution to Fisher's equation.
p(t,4 = N(z)
x
Note that there is one solution, a heteroclinic trajectory, that satisfies the following
conditions:
An Introduction to
kl`
EJiLTI CAL
7)
Linda J. S. Allan
Some Comments from Reviewers
"I have already recommended this book for use as a basic text in the upcoming upper
undergraduate class in mathematical biology. The main reasons are that it presents
a si
328
Chapter 7 Partial Differential Equations Theory, Examples, and Applications
Corollary 7. I
Suppose the reaction system (7.23) linearized about the equilibrium (N1, N2) has a
Jacobian matrix J = (a11). If the reactiondiffusion system (7.22) exhibits d
338
Chapter 7 Partial Differential Equations: Theory, Examples, and Applications
Sharps, F. R. and A. J. Lotka,1911. A problem in age distribution. Philos. Mag.
21:435438.
Shigesada, N. and K. Kawasaki. 1997. Biological Invasions, 'T'heory and Practice.
334
Chapter 7 Partial Differential Equations:`l'heory, Examples, and Applications
Figure 7,10 Solutions to the
NP system graphed in the
phase plane, v = 5, i= 4,
D = '1
a,1
2, and
v>2 rDa(a).
0.4
03
02
4,1
1rL
,Ll
'
'rLL,. 'k '1l
Gw w
tJ.
J' "
t
1
w
7.7 Pattern Formation 327
Substituting the solutions v1
leads to
av1
at
c1e(Ttw1(x), i = L, 2 into the linearized system
cjueU`w,(x) = ovj
and
(92V1
x2  c1 k e wl(x)  k v1.
z
2
SFr
The latter identity follows from the eigenvaluc problem (7.26). Expre
336
Chapter 7 Partial Differential Equations: Theory, Examples, and Applications
7.10 References for Chapter 7
Abramson, G., V. M. Kenkre,T. L. Yates, and R. R. Parmenter. 2003. Traveling waves
of infection in the hantavirus epidemics. Bull. Math. Rio!. 6
152
Chapter 4 Linear Differential Equations: Theory and Examples
positive, which implies that if the characteristic equation is expanded and
simplified,
A11
A"
a1
1
r a2A"
2 _F
. + a, = 0,
then all of the coefficients must satisfy a; > 0, i
1,., n
.
I
As
322
Chapter 7 Partial Differential Equations: Theory, Examples, and Applications
from outside the domain, and no selection, we can apply the HardyWeinberg laws
of genetics, SO that
The terms p2, 2 pq, and q2 represent the proportion of gene pairs AA, aA,
for Chapter 7 333
7,9
9. Consider the diffusive exponential growth model with an initial point
source,
DN
c?2N
1)  +i'N, xER, tE(0,oo),
dx
i)l
N (0, x) = N0b(x), x E R.
(a) Show that the solution to this IVP is given by
N (t, x) = N0

21/DTr t
exp
rt 
7.7 Pattern Formation
329
and Mimura and Murray (1975) give examples of predatorprey models that
exhibit diffusive instabilities.
The conditions in Theorem 7.2 give restrictions on the parameters that must
be satisfied for the onset of diffusive instabil