Concave Perron–Frobenius Theoryand Applications

Ulrich KrauseFachbereich Mathematik und Informatik, Universitat Bremen, Germany

e–mail: krause@math.uni–bremen.de

Abstract

Many important results of classical Perron–Frobenius Theory can be extendedfrom linear selfmappings of the standard cone in finite dimensional real space toconcave selfmappings of this cone. This is in particular true for minima of linearmappings, albeit the spectrum of these special concave mappings is more intri-cate than that for linear mappings. As classical Perron–Frobenius Theory hasnumerous applications there are many new applications for its concave extension.

1991 Mathematical Subject Classification: Primary 39A11, 15A48; sec-ondary 47H07, 47H12.

1 Introduction

The classical Perron–Frobenius Theory for nonnegative matrices is a rich and beautifultheory with numerous applications to Markov Chains, population dynamics, electricalengineering, and economics, to mention only a few fields. (Cf., e..g., [1, 8, 10].) Itis the aim of the present article to show how some of the most important resultsof classical Perron–Frobenius Theory can be extended from linear selfmappings of thestandard cone induced by nonnegative matrices to concave mappings of this cone. Thisextension yields new applications in the above mentioned fields as well as in others.Though some results of Perron–Frobenius Theory can be extended to more generalnonlinear mappings, it is only the concave type of mappings which allows an extensionof a large part of the body of classical theory. (For particular questions concerningperiodic points see, e.g., the recent extension in [9] to nonexpansive mappings.) Evenfor concave mappings one faces challenging questions, because, in contrast to the linearcase, minima of linear mappings may possess a number of real eigenvalues greater thanthe dimension of the space, or as in other cases, the spectrum may be even continuous.Since concave mappings include linear ones, there are much more concave than linearmappings leaving the standard cone invariant. This statement can be made moreprecise: For example, the convex cone of all linear selfmappings of R+ has just onegenerator whereas, as a simple application of the Stone–Weierstraß Theorem shows,the convex cone of all concave selfmappings of R+ is not even finitely generated. Thus,

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in extending results from the linear to the concave framework additional assumptions onthe concave mappings will come into play which depend on the strength of conclusionsone is interested in.Section 2 treats a conditional eigenvalue problem for concave mappings and presents inparticular a convergence result for the iterates. In Section 3, the well–known notions ofirreducibility and primitivity for nonnegative matrices as well as corresponding resultsare extended to concave mappings. Section 4 treats concave mappings that are ho-mogeneous and presents for concave mappings which are positively homogeneous andprimitive results almost as strong as for primitive matrices. Section 5 demonstrates thatthe eigenvalue corresponding in the concave setting to the Perron–Frobenius eigenvaluehas indeed many dominance properties, provided the concave mapping is irreducibleand positively homogeneous. Section 6 sketches two applications to difference equa-tions and population dynamics, respectively. The paper contains no proofs but theresults are illustrated by means of examples.

2 A conditional eigenvalue problem for concave map-

pings

In what follows let Rn be the n–dimensional Euclidean space with standard cone

K = Rn+ = {x ∈ Rn | x = (x1, . . . , xn), xi ≥ 0 for 1 ≤ i ≤ n}.

For x, y ∈ Rn let x ≤ y if xi ≤ yi for 1 ≤ i ≤ n and x < y if xi < yi for 1 ≤ i ≤ n;x � y means that x ≤ y and x 6= y.Let ‖ · ‖ be a norm on Rn that is monotone, i.e., 0 ≤ x ≤ y implies ‖ x ‖≤‖ y ‖. Let◦K be the interior of K with respect to ‖ · ‖.A mapping T : K −→ K is concave if

T (αx + (1− α)y) ≥ αTx + (1− α)Ty for all x, y ∈ K and all α ∈ [0, 1],

and monotone if 0 ≤ x ≤ y implies 0 ≤ Tx ≤ Ty.Concave selfmappings of K possess the useful properties of being monotone on K and

continuous on◦K.

Examples of concave selfmappings of K

• T affine linear, i.e., Tx = A · x + a, where A is a nonnegative n× n–matrix anda ∈ K

• T the pointwise infimum of affine linear selfmappings of K

• The mapping of Pielou–type in biology, given by

Tix =n∑

j=1

rijxj

xj + sij

with rij ≥ 0 and sij > 0

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where Ti is the i–th component mapping of T

• The Cobb–Douglas mapping in economics, given by

Tix =n∏

j=1

βijxαij

j with αij ≥ 0,n∑

j=1

αij = 1 and βij > 0

Theorem 1. For a concave mapping T : K −→ K with Tx > 0 for x 0 the followingstatements hold.

(i) The conditional eigenvalue problemTx = λx with λ ∈ R, x ∈ K, ‖ x ‖= 1has a unique solution x = x∗, λ = λ∗, and x∗ > 0, λ∗ > 0.

(ii) For the normalized mapping T x = Tx‖Tx‖ , x 0

limk→∞

T kx = x∗ for all x 0.

Idea of the proof for Theorem 1 (cf. [3, 4, 5])

For Hilbert’s projective metric on◦K given by

d(x, y) = − log

[(min

1≤i≤n

xi

yi

)(min

1≤i≤n

yi

xi

)]

one shows that X = {x ∈ ◦K| ‖ x ‖= 1} is a complete metric space and that T : X −→ X

is a contraction for d. Apply Banach’s contraction mapping principle to (X, d) and T .

Corollary 1. For any concave mapping T : K −→ K there exists λ ≥ 0 and x 0such that Tx = λx.

Remarks

1. In general, T k is different from T k.

2. Using metric d one can prove that the convergence in Theorem 1 (ii) isgeometrical.

3. Corollary 1 can be obtained also using Brouwer’s Fixed Point Theorem, there is,however, an advantage to get it via metric d.

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3 Irreducible and primitive concave mappings

An arbitrary mapping T : K −→ K is called

irreducible if for any non–empty subset I $ {1, . . . , n}, n ≥ 2, there exist i ∈ I andj 6∈ I such that Tie

j > 0 (ej the j–th unit vector in Rn),

primitive if there exists some p ≥ 1 such that Tmx > 0 for all m ≥ p and all x 0,

ray preserving if for any x 0 and any λ > 0 there exists λ′ = λ′(x, λ) > 0 such thatT (λx) = λ′Tx and T0 = 0; geometrically, this means that T maps a ray into a ray.

Lemma 1. Let T : K −→ K be monotone, irreducible and ray preserving. Then

(i) For any two indices i, j with 1 ≤ i, j ≤ n there exists p = p(i, j) such that(T p)ie

j > 0.

(ii) If in addition Theh > 0 for some 1 ≤ h ≤ n then T is primitive.

Remarks Consider the special case where T is given by multiplication with a non-negative matrix A. Since Tie

j = aij the notions of irreducibility and primitivity for Tcoincide with the well–known notions for matrices. T is automatically ray preserving.In the case of matrices, Lemma 1 (i) means that for an irreducible matrix A and indicesi, j some power of A has a strictly positive (i, j)–entry. This does not imply primitivityof A but if the diagonal of A has a strictly positive entry then it does, according toLemma 1 (ii).

The following Theorem sharpens under additional assumptions the conclusions of The-orem 1.

Theorem 2. For a concave mapping T : K −→ K which is irreducible, ray preservingand with The

h > 0 for some 1 ≤ h ≤ n the following statements hold.

(i) The equation Tx = λx has a solution x = x∗ > 0, ‖ x∗ ‖= 1, λ = λ∗ > 0.Moreover, Tx = λx for x 0 and λ ∈ R implies that x = rx∗ for some r > 0,and λ > 0.

(ii) limk→∞

T kx‖T kx‖ = x∗ for all x 0

and, equivalently,

limk→∞

(T kx)i

(T kx)j=

x∗ix∗j

for all x 0 and all 1 ≤ i, j ≤ n.

The following two examples illustrate Theorem 2, in particular the kind of uniqueness asstated in part (i). In both examples, K is the standard cone in R2 and ‖ x ‖= |x1|+|x2|.

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Example 1. Let T (x1, x2) = (√

x1+√

x2,√

x1+√

x2). Obviously, T is concave and ray

preserving because of T (λx) =√

λTx for λ ≥ 0. Furthermore, Te1 = Te2 = e1+e2 > 0and, hence, T is irreducible and, by Lemma 1 (ii), primitive. The solution accordingto Theorem 2 (i) is x∗ = (1

2, 1

2) and λ∗ = 2

√2. The eigenvalue problem Tx = λx,

however, has not a unique solution x 0, λ ≥ 0. On the contrary, the solutions aregiven by x(t) = 8

t2x∗ and λ = t for any strict positive value of the parameter t. Thus,

in this example, T has a continuous spectrum.

Example 2. Let T (x1, x2) = (4x1 + 2x2 +√

x1x2, min{x1 + x2, 2x1}). T is concaveand ray preserving. Actually, T is even positively homogeneous, i.e., T (λx) = λTx forλ ≥ 0. Furthermore, T1e

2 = 2 > 0 and T2e1 = 1 > 0 and, hence, T is irreducible. Since

T1e1 = 4 > 0, T is primitive by Lemma 1 (ii).

A direct, though lengthy, calculation yields λ∗ = 5 and x∗ = (15, 4

5). Alternatively,

Theorem 2 (ii) could be used to calculate x∗ approximately by choosing any x 0 as afirst approximation. From Tx = λx for x 0 it follows by Theorem 2 (i) that x = rx∗

and λrx∗ = λx = Tx = T (rx∗) = rTx∗ = rλ∗x∗

and, hence, λ = λ∗. That is, contrary to Example 1, in Example 2 equation Tx = λxhas a unique solution λ = λ∗, x = x∗ (up to a positive scalar). This phenomenon willbe investigated further in the next section.

4 Homogeneous concave mappings

An arbitrary mapping T : K −→ K is called positively homogeneous of degree d ∈ R ifT (λx) = λdTx for all x ∈ K and all λ > 0, and T0 = 0,and positively homogeneous if d = 1.Obviously, a mapping which is positively homogeneous of degree d is ray preserving,but not vice versa. Geometrically, a mapping which is positively homogeneous of degreed > 0 maps a ray onto a ray.The following Theorem shows that concave and primitive mappings which are positivelyhomogeneous share many properties with primitive matrices.

Theorem 3. For a concave mapping T : K −→ K which is primitive and positivelyhomogeneous the following statements hold.

(i) The equation Tx = λx has a solution x = x∗ > 0, λ = λ∗ > 0 and for anysolution x 0, λ ∈ R it must hold that x = rx∗ for some r > 0, and λ = λ∗.

(ii) Sx = limk→∞

T kx(λ∗)k exists on K and S: K −→ K+x∗ is a concave and positively

homogenous mapping with Sx > 0 for x 0.

(iii) limk→∞

T kx‖T kx‖ = x∗ for all x 0.

(iv) limk→∞

‖T k+1x‖‖T kx‖ = λ∗ = lim

k→∞‖ T kx ‖ 1

k for all x 0.

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Though positive homogeneity is a strong assumption there are still many applicationsas, e.g., the following one. (Another example of a concave mapping which is positivelyhomogeneous is the Cobb–Douglas mapping in economics.)

Example Concave Markov Chains

Let A = (aij) be a strictly positive m × n–matrix withn∑

j=1

aij = 1 for all 1 ≤ i ≤ m.

For a given collection of non–empty subsets Mk ⊆ {1, . . . , m}, 1 ≤ k ≤ n consider themapping T : K −→ K defined by

(Tx)k = mini∈Mk

(ai1x1 + ai2x2 + . . . + ainxn}.

Obviously, T is concave and positively homogeneous and, because of Tx > 0 for x 0,T is primitive. Thus, Theorem 3 applies and, because of T (1, . . . , 1) = (1, . . . , 1), The-

orem 3 (i) yields for the norm ‖ x ‖=n∑

i=1

|xi| that λ∗ = 1 and x∗ = ( 1n, 1

n, . . . , 1

n). By

Theorem 3 (ii), Sx = limk→∞

T kx exists on K. For the special case where m = n and

Mk = {k} for 1 ≤ k ≤ n this is the famous Basic Limit Theorem for regular MarkovChains. Thus, by Theorem 3, this Theorem is extended to concave Markov Chains.

Within the realm of concave mappings one may consider also mappings that are posi-tively homogeneous of degree d 6= 1, which is impossible for linear mappings. This caseis interesting in applications because a unique positive equilibrium cannot be modeledby linear systems.

Theorem 4. For a concave mapping T : K −→ K which is primitive and positivelyhomogeneous of degree 0 ≤ d < 1 the following statements hold.

(i) The equation Tx = x has a unique solution x∗ 0, and x∗ > 0.

(ii) limk→∞

T kx = x∗ for all x 0.

Example Let T :R2+ −→ R2

+ be given byT (x1, x2) = (min{1

5

√x1 +

√x2,

12

√x1 + 1

4

√x2}, 2

√x1 +

√x2).

Obviously, T is concave and T (λx) =√

λTx, i.e., T is positively homogeneous of degreed = 1

2. T is primitive because of Tx > 0 for x 0. Theorem 4 applies and yields

limk→∞

T kx = x∗ for all x 0 where x∗ = (1, 4) is the unique non–zero fixed point of T

in K.

5 Dominance properties

In classical Perron–Frobenius Theory the Perron–Frobenius eigenvalue λ∗ exhibits spe-cific dominance properties for what reason λ∗ is also called the dominant eigenvalue.

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Theorem 5. For a concave mapping T : K −→ K which is irreducible and positivelyhomogeneous the following statements hold:

(i) The equation Tx = λx has a solution x = x∗ > 0, ‖ x∗ ‖= 1, λ = λ∗ > 0 and forany solution x 0, λ ∈ R it must hold that x = rx∗ for some r > 0, and λ = λ∗.

(ii) If λx ≤ Tx for x 0 then λ ≤ λ∗.If λ∗x ≤ Tx for x 0 then x = rx∗ with r > 0.

(iii) Suppose, T can be extended to T :Rn −→ Rn and consider the eigenvalue problem

Tx = λx for x ∈ Rn, x 6= 0 and λ ∈ R. Let |x| ∈ K for x ∈ Rn be thecomponentwise taken absolute value.

(a) If |Ty| ≤ T |y| for all y ∈ Rn then |λ| ≤ λ∗.

(b) If |Ty| ≤ T |y| for all y ∈ Rn and if for some k ≥ 1 it holds that |T ky| < T k|y|for all y ∈ Rn with y 6= |y| then |λ| < λ∗ for λ 6= λ∗.

In case, the mapping T is induced by a nonnegative matrix Theorem 5 yields the mostrelevant dominance properties in the matrix case. The assumption in part (iii) (a) issatisfied for any irreducible matrix and the assumptions in part (iii) (b) are satisfiedfor any primitive matrix. As the following example shows, however, not all dominanceproperties extend from linear to concave mappings.

Example Let T :R2 −→ R2 be given byT (x1, x2) = (min{2x1 + x2, x1 + 2x2}, x1 + x2).Obviously, T :R2

+ −→ R2+ is concave, positively homogeneous and primitive. Thus,

Theorem 5 applies and in particular part (iii). The eigenvalue problem Tx = λx on R2

has four (real) solutions, namelyλ1 = 1

2(3 +

√5) with eigenvector x1 = −(1

2(1 +

√5), 1)

λ2 = 12(3−√5) with eigenvector x2 = (1

2(1−√5), 1)

λ3 = 1 +√

2 with eigenvector x3 = (√

2, 1)

λ4 = 1−√2 with eigenvector x4 = (√

2, −1).Since λ3 is the only eigenvalue with an eigenvector in K = R2

+ it follows that λ∗ =

λ3 = 1 +√

2.One has |λ2| ≤ λ∗ and |λ4| ≤ λ∗ but λ1 > λ∗. This is in line with Theorem 5(iii) because in this example one does not have |Ty| ≤ T |y| for all y ∈ R2; e.g., fory = (−1, 0) one has Ty = (−2,−1) but T |y| = (1, 1). Thus, in contrast to the matrixcase even for primitive concave mappings it is not necessarily true that |λ| ≤ λ∗ for all(real) eigenvalues.

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6 Applications

Concave Perron–Frobenius Theory has many applications in various areas. In whatfollows two applications are sketched, to difference equations and Leslie models, re-spectively.

6.1 Concave difference equations

Consider the difference equation of order n ≥ 1

u(t + n) = f(u(t), u(t + 1), . . . , u(t + n− 1)) for t ∈ N = {0, 1, 2, . . .}with f :Rn

+ −→ R+ and initial conditions u = (u(0), u(1), . . . , u(n− 1)) ∈ Rn+.

For T (x1, x2, . . . , xn) = (x2, . . . , xn, f(x1, x2, . . . , xn))

one obtains from Theorem 3 the following result. (Cf. also [7].)

Theorem 6. Let f be concave and positively homogeneous and suppose there existn1, . . . , nr ∈ {1, . . . , n} with 2 ≤ r, n1 = 1 and gcd{n − ni + 1 | 1 ≤ i ≤ r} = 1 suchthat

0 ≤ x and 0 < xnifor some i imply 0 < f(x).

The the following statements hold.

(i) The characteristic equation λn = f(1, λ, λ2, . . . , λn−1) has a unique root λ∗ > 0.

(ii) For every solution u(·) of the difference equation with initial conditions u ∈ Rn+

limt→∞

u(t)(λ∗)t = s(u), where

s:Rn+ −→ R+ is concave, positively homogeneous and s(x) > 0 for x 0.

(iii) For every nonnegative solution u(·) that is not identically zero

limt→∞

u(t+1)u(t)

= λ∗ = limt→∞

u(t)1t .

Example Let A = (aij) be a nonnegative m × n–matrix for which the first columnand at least one other column are strictly positive. Consider a difference equation givenby

f(x) = min1≤i≤m

(ai1x1 + . . . + ainxn).

The characteristic equation

λn = min1≤i≤m

(ai1 + ai2λ + . . . + ainλn−1)

has a unique solution λ∗ > 0 and limt→∞

u(t)(λ∗)t = s(u).

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6.2 Stability in a concave Leslie model

Consider a population consisting of n age classes, let xi(t) be the number of individualsin age class i in period t ∈ N and let x(t) be the column vector with components xi(t).

The density dependent Leslie matrix is given by

L(x) =

b1(x) b2(x) . . . bn−1(x)bn(x)s1(x)

s2(x) 0

0. . . sn−1(x)sn(x)

where bi(x) and si(x) denote for class i density dependent birth rate and survival rate,respectively.

The dynamics of the population is modeled by the nonlinear Leslie model

x(t + 1) = L(x(t))x(t) for t ∈ N. (Cf. also [2].)

To discuss an example suppose that for some 0 ≤ d ≤ 1 one has

bi(x) = bixd−1i and si(x) = six

d−1i if xi > 0, and

bi(x) = bi, si(x) = si if xi = 0.

Assume si > 0 for 1 ≤ i ≤ n − 1 and sn ≥ 0; assume further bj ≥ 0 for all j and fork1, . . . , kr ∈ {1, . . . , n} relatively prime with 2 ≤ r, kr = n that bki

> 0 for 1 ≤ i ≤ r.For this model and Tx = L(x)x one obtains from Theorem 3 and Theorem 4 that thereexists a unique (up to a factor) equilibrium path x(t) = x(0)(1+ g)t and that any path

with x(0) 6= 0 grows finally with g, that is limt→∞

xi(t)xi(t)

= c(x(0)) > 0 for all 1 ≤ i ≤ n. For

d = 1 this is the well–known behavior of the classical Leslie model with a Leslie matrixL = L(x) which does not depend on x. In this case growth rate g and x(0) (up to afactor) are given as the solution of the linear eigenvalue problem Lx(0) = (1 + g)x(0).

For d < 1, T is concave, the growth rate is 0 and x(0) is the unique non–zero fixedpoint of T . For the special case of n = 2 and putting p(t) = x1(t) the latter situationreduces to the concave Fibonacci model p(t + 2) = p(t + 1)d + p(t)d2

. The results ofconcave Perron–Frobenius Theory allow the analysis of further concave versions of theLeslie model.

References

[1] Berman, A., Plemmons, R.J., Nonnegative Matrices in the Mathematical Sciences,Academic Press, New York, 1979.

[2] Fujimoto, T., Krause, U., Asymptotic properties for inhomogeneous iteration ofnonlinear operators, SIAM J. Math. Analy., 19 (1988), 841–853.

[3] Krause, U., Perron’s stability theorem for non–linear mappings, J. Math. Econ.15 (1986), 275–282.

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[4] Krause, U., A nonlinear extension of the Birkhoff–Jentzsch Theorem, J. Math.Anal. Appl. 114 (1986), 552–568.

[5] Krause, U., Relative stability for ascending and positively homogeneous operatorson Banach spaces, J. Math. Anal. Appl. 188 (1994), 182–202.

[6] Krause, U., Positive nonlinear systems: some results and applications, in Pro-ceedings of the First World Congress of Nonlinear Analysts 1992 (Edited by V.Lakshmikantham), 1529–1539, W. de Gruyter, Berlin, 1996.

[7] Krause, U., Positive nonlinear difference equations, Nonlinear Analysis (TMA) 30(1997), 301–308.

[8] Minc, H., Nonnegative Matrices, Wiley, New York, 1988.

[9] Nussbaum, R.D., Verduyn Lunel, S.M., Generalizations of the Perron–FrobeniusTheorem to nonlinear maps, Memoirs Am. Math. Soc 138 (659), 1999.

[10] Seneta, E., Non–negative Matrices and Markov Chains, 2nd edition, Springer,Berlin, 1980.

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