Murray J.D. Mathematical Biology: I. An Introduction
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Appendix B. Routh–Hurwitz Conditions,
Jury Conditions, Descartes’ Rule of Signs
and Exact Solutions of a Cubic
Appendix B.1 Characteristic Polynomials, Routh–Hurwitz
Conditions and Jury Conditions
Linear stability of the systems of ordinary differential equations such as arise in inter-
acting population models and reaction kinetics systems (cf. Chapters 3 and 6) is de-
termined by the roots of a polynomial. The stability analysis we are concerned with
involves linear systems of the vector form
dx
dt
= Ax, (B.1)
where A is the matrix of the linearised nonlinear interaction/reaction terms: it is the
Jacobian matrix about the steady state—the community matrix in ecological terms. So-
lutions are obtained by setting
x = x
0
e
λt
, (B.2)
in (B.1) where x
0
is a constant vector and the eigenvalues λ are the roots of the charac-
teristic polynomial
| A −λI |=0, (B.3)
where I is the identity matrix. The solution x = 0 is stable if all the roots λ of the
characteristic polynomial lie in the left-hand complex plane; that is, Re λ<0forall
roots λ. If this holds then x → 0 exponentially as t →∞and hence x = 0isstableto
small (linear) perturbations.
If the system is of nth order, the characteristic polynomial can be taken in the
general form
P(λ) = λ
n
+a
1
λ
n−1
+···+a
n
= 0, (B.4)
where the coefficients a
i
, i = 0, 1,... ,n are all real. We tacitly assume a
n
= 0since
otherwise λ = 0 is a solution, and the polynomial is then of order n − 1 with the
508 Appendix B. Conditions, Rule of Signs and Exact Solutions
equivalent a
n
= 0. We require conditions on the a
i
, i = 0, 1,... ,n such that the zeros
of P(λ) have Re λ<0. The necessary and sufficient conditions for this to hold are the
Routh–Hurwitz conditions. There are various equivalent forms of these, one of which
is, together with a
n
> 0,
D
1
= a
1
> 0, D
2
=
a
1
a
3
1 a
2
> 0, D
3
=
a
1
a
3
a
5
1 a
2
a
4
0 a
1
a
3
> 0,
D
k
=
a
1
a
3
····
1 a
2
a
4
·· ·
0 a
1
a
3
·· ·
01a
2
·· ·
······
00···a
k
> 0, k = 1, 2,... ,n.
(B.5)
These conditions are derived, using complex variable methods, in standard texts on the
theory of dynamical systems (see, for example, Willems 1970). As an example, for the
cubic equation
λ
3
+a
1
λ
2
+a
2
λ +a
3
= 0
the conditions for Re λ<0are
a
1
> 0, a
3
> 0; a
1
a
2
−a
3
> 0.
Frankly it is hard to imagine anyone actually using the conditions for polynomials of
order five or more.
Although (B.5) are the necessary and sufficient conditions we need, the usual al-
gebraic relations between the roots and the polynomial coefficients can often be very
useful. If λ
1
,... ,λ
n
are the distinct nonzero roots of (B.4) these are
n
i=1
λ
i
=−a
1
,
n
i, j
i=j
λ
i
λ
j
= a
2
,... λ
1
λ
2
...λ
n
= (−1)
n
a
n
. (B.6)
Lewis (1977) gives several useful ways of deriving qualitative results using the
Routh–Hurwitz conditions together with network concepts directly on the matrices.
In the case of systems of discrete models for interacting populations (cf. Chapter 3),
and with single population models with delay (cf. Chapter 2), stability is again deter-
mined by the roots of a characteristic polynomial (cf. Section 3.10 in Chapter 3). The
linearised systems again give rise to matrix forms like (B.3) and hence polynomials like
(B.4). With delay equations, with a delay of n time-steps say, we have to solve linear
difference equations typically of the form
u
t+1
= b
1
u
t
+···+b
n
u
t−n
Appendix B.2 Descartes’ Rule of Signs 509
((2.37) in Chapter 2 is an example). We solve this by setting u
t
∝ λ
t
which again results
in a polynomial in λ. Linear stability here however is determined by the magnitude of
λ: stability requires |λ | < 1 since in this case u
t
→ 0ast →∞. So, for the linear
stability analysis of discrete systems we require conditions on the coefficients of the
characteristic polynomial so that the solutions λ have magnitude less than 1. The Jury
conditions are the conditions for this to be the case.
The Jury conditions are given, for example, in the book by Lewis (1977), who
describes and illustrates several useful, analytical and numerical techniques concerning
the size and signs of the roots of polynomials. For the polynomial P(l) in (B.4), let
b
n
= 1 −a
2
n
, b
n−1
= a
1
−a
n
a
n−1
,... , b
n−j
= a
j
−a
n
a
n−j
,... ,
b
1
= a
n−1
−a
n
a
1
;
c
n
= b
2
n
−b
2
1
, c
n−1
= b
n
b
n−1
−b
1
b
2
,... ,
c
n−j
= b
n
b
n−j
−b
1
b
j+1
,... , c
2
= b
n
b
2
−b
1
b
n−1
;
d
n
= c
2
n
−c
2
2
,... , d
n−j
= c
n
c
n−j
−c
2
c
j+2
,... , d
3
= c
n
c
3
−c
2
c
n−1
;
and so on until we are left with only three elements of the type
s
n
= r
2
n
−r
2
n−3
, s
n−1
= r
n
r
n−1
−r
n−3
r
n−2
, s
n−2
= r
n
r
n−2
−r
n−3
r
n−1
.
The Jury conditions (necessary and sufficient) which ensure that the roots of the poly-
nomial P(λ) in (B.4) all have magnitudes less than 1 are:
P(1)>0,(−1)
n
P(−1)>0,
|a
n
| > 1, |b
n
| > |b
1
|,
|c
n
| > |c
2
|, |d
n
| > |d
3
|,... , |s
n
||s
n−2
|.
(B.7)
Appendix B.2 Descartes’ Rule of Signs
Consider the polynomial (B.4), and, as before, we take without loss of generality a
n
> 0.
Let N be the number of sign changes in the sequence of coefficients {a
n
, a
n−1
,... ,a
0
},
ignoring any which are zero. Descartes’ Rule of Signs says that there are at most N roots
of (B.4), which are real and positive, and further, that there are N, N −2orN −4,...
real positive roots. By setting ω =−λ and again applying the rule, information is
obtained about the possible real negative roots. Together these often give invaluable in-
formation on the sign of all the roots, which from a stability point of view is usually all
we require.
As an example consider
λ
3
+a
2
λ
2
−a
1
λ +a
0
= 0, a
i
> 0foralli = 0, 1, 2. (B.8)
There are two sign changes in the sequence of coefficients, and so there are either two
or zero real positive roots. If we now set λ =−ω, the equation becomes
ω
3
−a
2
ω
2
−a
1
ω − a
0
= 0,
510 Appendix B. Conditions, Rule of Signs and Exact Solutions
which has one change of sign in the sequence, and so there is at most one real positive
root ω. This means there is exactly one negative root λ of (B.8).
Appendix B.3 Roots of a General Cubic Polynomial
Sometimes it is helpful to have the actual roots of the characteristic polynomial, however
complicated they may be. Although it is possible to find these for polynomials higher
than order three, the complexity is usually not worth the effort. The roots of a cubic are
probably the most complicated that we would ever wish to have; simple derivations of
these have been given by Miura (1980) and Namias (1985). Table B.1 gives the explicit
forms of the roots of a cubic.
Table B.1. Explicit roots of the cubic polynomial p(λ) = λ
3
+ Aλ
2
+ Bλ +C, with A, B and C real.
λ
3
+ Aλ
2
+ Bλ +C = 0, A ≡ 3a, B ≡ 3b, α ≡ a
2
−b, β ≡ 2a
3
−3ab +C
α>0 β = 0 λ
1
=−a
λ
2
= (3α)
1/2
−a
λ
3
=−(3α)
1/2
−a
|β |≤2α
3/2
λ
1
= 2α
1/2
sin φ − a φ = (1/3) sin
−1
{β/[2α
3/2
]}
λ
2
=−2α
1/2
sin(π/3 +φ) − a −π/6 ≤ φ ≤ π/6
λ
3
= 2α
1/2
sin(π/3 −φ) − a
β>2α
3/2
λ
1
=−2α
1/2
cosh ψ −a ψ = (1/3) cosh
−1
{|β |/[2α
3/2
]}
λ
2
=−α
1/2
cosh ψ −a +i(3α)
1/2
sinh ψ
β<−2α
3/2
λ
1
=−2α
1/2
cosh ψ −a
λ
2
=−α
1/2
cosh ψ −a +i(3α)
1/2
sinh ψ
λ
3
=−α
1/2
cosh ψ −a −i(3α)
1/2
sinh ψ
α = 0 −∞ <β<∞ λ
1
=−β
1/3
−a
λ
2
= β
1/3
/2 − a +3iβ
2/3
/4
λ
3
= β
1/3
/2 − a −3iβ
2/3
/4
α<0 −∞ <β<∞ λ
1
=−2(−α)
1/2
sinh θ −a θ = (1/3) sinh
−1
{β/[2(−α)
3/2
]}
λ
2
= (−α)
1/2
sinh θ −a +i(−3α)
1/2
cosh θ
λ
3
= (−α)
1/2
sinh θ −a −i(−3α)
1/2
cosh θ
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