Multiplicative processes-II

MATHEMATICAL

BIOSCIENCES

20, 345-357 (1974)

345

Multiplicative Processes-Ii* LLOYD DEMETRIUS Center for Dynamical Systems, Division of Applied Mathematics, Brown Vniversi@, Providence, R.I. 02912 Communicated

by Richard

Bellman

ABSTRACT This paper continues the study of the asymptotic properties of a unified class of biological models which are represented by positive homogeneous operators on ordered Banach spaces. Ergodic properties of these models are analyzed in terms of the spectral properties

of the operators.

In this paper, we study the asymptotic properties of certain biological models with an infinite number of states. In recent years there have been several studies of models on an infinite state space. Among these we note the work of Bailey [I] on birth and death processes for populations composed of an infinite number of colonies, also the results of Eshel [9] and Karlin [13] on evolution of a population with polygenic characters. Mathematically, these models are characterized by positivity and homogeneity properties. Our aim is to analyze a general class of models with these features and to apply our results to particular biological systems. This paper is a sequel to [5] where the term “multiplicative process” was used to describe models with the positivity and homogeneity properties. Our methods apply also to finite-dimensional models. In particular, we refer to the work of Usher and Williamson [23] on colonies in a spatially distributed population, and the work of Maruyama [ 181 on the rate of decrease of heterozygosity in a subdivided population. These models are both linear, positive, and act on finite state spaces. Among nonlinear positive models, we note the work of Stein and Ulam [22] where a class of nonnegative quadratic transformations that arise in genetics was analyzed; the results of *This research 15132.

was supported

by the National

0 American

Science

Elsevier

Foundation

Publishing

under

Company,

grant

GP

Inc., 1974

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Blakley [3] on nonnegative symmetric quadratic transformations, and the comprehensive treatment by Kesten [ 141. The paper is organized as follows. Section ldetails a class of models to which the general formulation applies. The mathematical aspects are given in Sec. 2, 3, and 4. Section 2 revolves around some rules on positive operators due to Schaefer [20], Vere-Jones [25,26], and the author [6]. In Sec. 3 we analyze three types of ergodicity and discuss relations between them. In both of these sections we treat applications to infinite population models and stepping-stone models. Nonlinear models are treated in Sec. 4. Here the projective metric concept due to Birkoff [2] is applied to models that are represented by homogeneous positive operators. This section tidies up some of the ideas originally sketched in [5]. The approach we give is primarily applicable to economic growth models and the main result is an infinite-dimensional extension of the Solow-Samuelson theorem. Few new mathematical results are given; the aim of this paper is to organize a body of results from a point of view which has an easy applicability to biological models and to indicate how certain geometric properties of the models determine their ergodic behavior.

(a) Consider a population consisting of an infinite array of colonies situated at integer points on the real line. Assume that each colony is subject to a birth and death process with birth and death rates X and p, respectively. Assume that migration occurs between neighboring colonies. The migration rate is 4~. If x,(t) denote the colony size at time t, then the change in colony size is given by the infinite set of simultaneous ordinary differential equations

In discrete time, the change is given by the equation x(t+ I)= M.x(t) where irreducible matrix. This model was introduced by Bailey [I] and the asymptotic properties were studied in [1] using differential equation methods. The finitedimensional version was analyzed by Usher and Williamson [24] by means of matrix techniques.

M = (a,,) 2 0 is a nonnegative

(b) Consider an infinite number of colonies with their positions represented by integers on a line. Assume that migration occurs between neighboring colonies. Consider a single locus with a pair of alleles A ,,A,. Assume that changes in gene frequency are due to mutation and migration. If x;(r)

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denote the relative frequency

of A, in the ith colony at time t, then

xi(t+ l)=(l-m)x,(t)+

:(-Xi-l(t)+Xi+I(t)),

where m stands for the rate of migration per generation to neighboring colonies. The change in gene frequency is given by the transformation x( t + 1) = Mx( t), where M = (aJ > 0 is a nonnegative irreducible matrix. The problem is to determine the ergodic properties of M. This model is adapted from the stepping-stone model of genetic correlation due to Kimura and Weiss [ 151. In [ 151 terms involving a change in x,(t) due to random sampling of gametes in reproduction are also considered. The stepping-stone model of Kimura and Weiss and the Bailey model are prototypic of a large class of models in population processes. The models are irreducible in the sense of matrix theory and this irreducibility reflects the geometric connectedness of the colonies. We shall observe in section 2 that the ergodic properties of these models are a consequence of this geometry. (c) Consider a population divided into a countable number of age classes. Let x,(t) denote the number of individuals in age class i at time (t) and let a, denote the number of individuals produced by individuals in the ith age class at the next instant of time and let b, denote the proportion of individuals surviving from the ith to the (i + 1)th age class. The dynamics of the population is given by x (t + 1) = MT (t), where M is a nonnegative matrix. This model, which is the infinite-dimensional version of the Leslie model, was mentioned in [5] and some of its properties were discussed. The theorems we shall state will allow us to obtain more information about this model and certain finite-dimensional variants due to Goodman [ 10,111.

Let Y be a Banach space over the real or complex numbers. A nonempty subset C of I/ is called a positive cone if it satisfies (1) C + C c C; (2) aC c C for (Y> 0; (3) C n - C= (0). A Banach space V with closed positive cone defines a partial order on Y. We write x < y if y - x E C. An operator T on V is called positive if T(C)C C. T is homogeneous of degree k if T(ax)= a“T(x). A multiplicative process is any operator T on V which satisfies the following properties: 1. 2.

T is positive and continuous T is homogeneous of degree k.

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348

We say that x is a stationary state if T(x) =xX for some x E C,h > 0. X is the growth parameter of the process. We shall be concerned with three types of ergodicity: I. II. III.

T is strongly ergodic if T”/X”~P; T is ergodic if (l/n)2~=,Tk/Xk-+P; T is weakly ergodic if (I+ T)“/(lZ+

TII”+P.

P is a projection operator and convergence in each case is understood to be in the uniform operator topology. Let V be a partially ordered Banach space with closed positive cone C. In the work of Karlin [12], Krein and Rutman [18], and others, it is assumed that the cone C has interior points. This is a severe restriction. The $ spaces 1 < p < CO,with the natural ordering, do not have interior points. In models (a),(b),(c), a suitable Banach space is 1, or I,. It is thus useful to have a theory which accommodates these examples. The notion of a quasi-interior point, introduced by Schaefer [20], achieves this. An element eE C is called a quasi-interior point if the order interval [O,e] is a total subset of V. The I, spaces equipped with the natural ordering have cones with quasi-interior points. Let T denote a positive linear operator on V and let r(T) = p denote-the spectral radius of T. T is said to be irreducible if for some X> p, T(x) = TR,(x) is a quasi-interior point of C for every x E C, x #O. R, denotes the resolvent of T. Let A = (aV) denote a non-negative matrix. A is said to be irreducible if given any (i,j) there exists an n such that a$“) > 0. It is clear that if A defines a bounded operator on some sequence space, the two notions of irreducibility coincide. We recall that irreducibility of the matrix implies that the state space S = { 1,2,. . } can be decomposed into d disjoint subclasses C,, C,, . . ,C, where C, n C, = Cp if i#j and u’,!! ,Cr= S. The integer d can be obtained as the g.c.d. of all integers n for which a,$“) is nonzero. If we associate a directed graph G(A) with the matrix A, then irreducibility implies that G(A) is strongly connected; that is, given any two nodes i and j, there is a directed path from i to j. The next two results are due to Schaefer [20]. Theorem (2.1) gives conditions under which stationary states of multiplicative processes exist. We shall let p denote the spectral radius of the operator T. THEOREM

2.1

Let V be (I partially ordered Banach space whose positice cone C has quasi-interior points. Let T denote a positive linear operator on V. Suppose p is a pole o_f the resohent R,. Then corresponding to p, there exist positioe eigencectors xO.yO of T and T* respectively.

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Proof Let p be a pole of order k. The Laurent expansion of R, near p is given by R,= EF= _kAn(A-p)“, where A, and B, are members of [VI, the space of bounded linear operators on V. A _ k = lim,,,(h - p)“RX. Let (C] denote the cone of positive operators. Since [C] is closed and R, > 0 for X > p, it follows that A _k > 0. Since C has quasi-interior points, there exists u E C such that A _k(~) > 0. Now, (pl- T)A _k = 0; therefore for some x,EC, namely x,=A_,(u), (pl-T)x,=O. Since r(T)=r(T*), a similar argument shows that T*( ye) = py,, for some ye E C*, ye # 0.

If we assume that the operator is irreducible, then we conclude that the pole p is simple and the eigenvector x,, is a quasi-interior point of C. We have the following theorem. THEOREM

2.2

Let T be an irreducible positive linear operator on the Banach space V. Suppose p is a pole of the resolvent R,. Then

(I)

p> 0 and is a simple pole,

(2)

corresponding to p there exist positive eigenvectors xO,yOof T and T*

such that x0 is a quasi-interior point of C and (yO, x) > 0 for any x E C,x # 0. (3) if V is a Banach lattice, then the geometric multiplicity of p = 1. Proof: (1) By Theorem (2.1) there exist eigenvectors x,E C, y,E C* such that T(x,)= px, and T*(y,) = pye. We I now show that x,, is a quasiinterior point of C. Since T is irreducible, T(x,,) = TR,(x,) is a quasi-interior point of C whenever h > p. Now

f(x,)=

$

T”(x,),X”=x,~

($. n=l

Hence x,, is a quasi-interior point of C. It also follows that p > 0. We now show that ye is a strictly positive linear form. Consider with x#O. For some h >p,

=

2 (&)(T*“(Yo~)= 2( II=1

n=l

f(x)).

= (YO>

~)(Yo.W)>

x E C,

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Since T(x) is a quasi-interior point of C, we conclude that (y,,,x)>O; that is, y0 is a strictly positive linear form. (2) Suppose p is a pole of order k > 1. We have already shown that A_, >O. Choose an xEC such that A-,(x)=x, with x,+=0. Lety,EC* be an eigenvector of T* for p. Now (y,,,x,) > 0. However,

(Yo~xcJ = (~~4 -,) = (yo,A _,(Pz-

T)~-‘(~)),

which is impossible. Hence k = 1. The proof of (3) involves ideas of a more technical nature. We refer to [20] for a proof and details on Banach lattice. We remark however, that the I, spaces with 1 < p < cc, equipped with the standard ordering are all Banach lattices. These theorems apply immediately to the matrix model for populations. Let A denote a population model. The element x is a stationary age distribution if there exists a h > 0 such that Ax =xX. The element y is a reproductive index if there exists h > 0 such that yA =hy. The vector y = (yi) is a measure of the relative contribution per head made to the stationary population in the future by the individual age groups. We now have Theorem 2.3. THEOREM

2.3

Let A denote a population model. Suppose I. Ca, 0 for infinitely many n 3. b,+O as n+oo. Then (1)’ A has an essentially unique stationary age distribution, and (2)’ A has an essentially unique reproductive index. (By “essentially unique” we mean unique up to a constant scalar factor.) ProojY Conditions (1) and (2) imply that A defines a compact operator A: 1,+/,. Condition (2) implies that A is irreducible. Hence there exist x = (xi), y = ( yi), xi > 0, y, > 0 such that Ax = px and yA = py. In the applications of Theorems (2.1) and (2.2) the matrix is considered as a bounded linear operator on a suitable Banach space. It is also possible to analyze the matrix itself using the concepts of positive recurrence and transience from the theory of Markov chains. This point of view has been developed in a series of papers by Vere-Jones [25,26] and Kingman [ 171. We shall assume that the iterates A” = { a$“)} are defined and finite. The

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irreducibility of A implies that lim[aUend) ] I/* = 1/R where index A = d and R is the convergence norm of the matrix. The notions of recurrence and transience can be defined in terms of the convergence of the series x;= ,uy)R”. These ideas are derived from analogous properties for Markov chains; we shall first describe the propertiies in tha! context. Let P=(pU) > 0 denote an irreducible aperiodic Markov chain. Let fi(“) denote the probability that the first return to state j occurs at time n > 1. If i is the probability that the system ever returns to state j, we have 4 = Cr= ,fi(“). The time required for a first return to state j is called the recurrence time of the state. If A = 1, the first moment p, = Zr= ,n$“) is the mean recurrence time for state j. The states of a Markov chain are classified as follows. (1) j is positive recurrent if f, = 1 and cl, < w, (2) j is null-recurrent if & = 1 and pj = cc, (3) j is transient if f, < 1. The irreducibility of A implies a “solidarity” result. If a single state is positive recurrent (null recurrent, transient) then all states are positive recurrent (null recurrent, transient). Properties (1), (2) (3) and the ErdosFeller-Pollard theorem prompt the following definition [25]: DEFINITION

1. A is positive recurrent if Za,$“)RN diverges and a,j”)R” does not tend to zero. 2. A is null recurrent if Eu$.“)RN diverges and a$“)RN tend to zero. 3. A is transient if Ea,l”)RN converges. We remark that R is the radius of convergence

of the series Ca,j”)z”.

Let T be a bounded linear operator on 1, (1 < p < w). The spectral radius p is given by p=lim]] T”II’/“. If T can be represented by an irreducible nonnegative matrix T=(u,), then lim[ay)]l/n= l/R, where index of T= d. This shows that l/R < p. The next result gives conditions under which 1/R = p. This result is implicit in Vere-Jones [25]. The proof given in [25] is based on a theorem concerning the positivity of the resolvent operator due to Schaefer. We give a direct and simpler proof. Refinements of this result will appear in [6]. THEOREM

2.4

Let T be a bounded linear operator on b(l < p < 00) with spectral radius p. Suppose T is represented by an irreducible nonnegative matrix with convergence parameter R . Then (a)

P>O

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(b) (c)

DEMETRIUS

If p is an isolated point of the spectrum of T then 1/R = p If p is a pole of the resoluent R,, then T is posit&e recurrent.

Proof. (a) T is an irreducible positive linear operator. Hence, by Theorem 2.2, there exists a quasi-interior point x0 such that 7(x,) = px,. The assumption p = 0 contradicts the irreducibility of T. Hence p > 0. (b) If p is an isolated point of the spectrum of T then p is a singularity of the function ~~S,oa(R)/z”+’ which can be identified with the (i,j) element in the matrix refiresentation of the resolvent operator of T. Since R is the radius of convergence of the series Z~SOa$~)zn, we conclude that 1/R > p. However, the reverse inequality holds. Hence 1/R = p. (c) If p is a pole of the resolvent, then R is a pole of the function C~=,,a,j”)z”. This implies that ZFx,,a,j”)R” diverges and lima,J”)R” does not tend to zero. Hence T is positive recurrent. In the case of the population matrices studied, the condition b,-0 implies compactness of the operator; hence l/R = r(T) and the matrix is positive recurrent. The matrix A arises in several stepping stone models [ 1,151. These matrices do not necessarily induce compact operators.

0

0

.......

c2

b,

0

.......

a3

c3

b,

b,

......... .......

where ai > 0, bi > 0, c, > 0. A is irreducible. If c, = 0 for all i and a, = a, b, = b with a + b = 1, then A describes a random walk on the nonabsorbing states S= { 1,2,. . }, and index A =2. The convergence norm R (see Seneta and Vere-Jones [21]) is given by R=

>l

-=1 ’ 26Z

ifa=b=1/2,

ifafb.

The process A is transient. More generally, if ci= c for all i and a+ b < 1 with a+ b+ c= 1, then A is transient for a > b [21]. This stepping stone model corresponds to a random walk with absorption.

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3. In this section we analyze three types of convergence of operator iterates and we discuss relations between them. The notions of ergodicity and strong ergodicity are strong ergodicity are well known. A suitable source is Karlin [ 121 where a more general statement of Theorem 3.1 is given. Theorem 3.2 is an extension of a result in [18]. The convergence of the iterates (I+ T)” which we call weak ergodicity seems to have originated with Bonsall [4]. In the next two results we shall assume p > 0 and p= r(T) = 1. The second assumption involves no loss in generality since if r(T)>O, any positive linear operator can be normalized so that r(T) = 1. THEOREM

3.1

Let T be a positive linear operator on V. Assume r(T) = 1. If 1 is a simple pole of R,, then 1/nC”,_ P.

, Tk converges in the uniform topology to a projection

Proof: Since 1 is a simple pole of R, then (h - 1) R,-+ P as h+ 1 + where P is a projection.

For ) pi< 1, cl-lR(l/y,T)=E

p”T” and

(~-cL)(~/~)R(~/cL,T)=(~-~)~~-P,~~~T”~P. An application

of a classical Tauberian

theorem gives

and T is ergodic. THEOREM

3.2

Let V be a Banach lattice with positive cone C. Let T be an irreducible positive linear operator on V. Suppose T has no spectral points on the circle IzI= 1 except a pole at I = 1. Then T” converges in norm to a projection P. Proof. Since 1 is the only eigenvalue of modulus 1, and 1 is a pole, it follows that 1 is a simple pole. Since T is irreducible, the eigenspace W associated with the eigenvalue 1 is one-dimensional. Let P denote the projection of V on W. Then T= P + Q where the spectral radius of Q, r(Q), < 1 and PQ = QP=O. Since P is a projection T” = P + Q”. However, r(Q)=lim]]Q”]]‘/n. Hence ]]Q”]] < [r(Q)r for n > N. Therefore ]IT” - PI < r(Q)‘+0 and we are done.

The following result was proved by Bonsall for compact operators. We shall only assume that p = r(T) is a pole of the resolvent R,. The proof is essentially the same.

LLOYD

354 THEOREM

DEMETRIUS

3.3

Let T be a positive linear operator acting on V. Suppose p the spectral radius, is a pole of order k of the resolvent R,. Then (Z+T)” A k-‘P I%& ))I+ TJJ” = \)A k-‘PI) ’ where A = T- pZ and P is a projection. Proof. Since p is a pole of the resolvent R,, there exist projection Q from V to the null space of A k and the range of A k so that

P and

V=R(Ak)CT3‘?X(Ak). Let U(T) denote the spectrum of T. Let 02, denote the disc IzI < p. Now u(T) c fiIp and p is an isolated point of a(T). 1+ p E a(Z + T) and 1 + p is an isolated point of u(Z+ T). This implies that for some CL,such that 1 < ZJ < 1 +p, u(Z+ T)c~, ~(1 +p). Hence u(Z+ T)Q c 9,,. We conclude that lim .,,II((Z+ T)Q)“ll”” G P. Since TQ= QT and Q2= Q, it follows that lim,,,ll(Z+ T)nQll’/” < p and since p < 1+ p we have -1

tICI+ T)“QII=O. Now A “P = 0 for m > k. Therefore can show, that

by expanding

(I + T)” in powers of T we

-I

(I+

T)“P=(l+p)‘-kAk-‘P.

Since (I + T)” = (I + T)“P + (I + T>“Q we conclude

that

-1

(I+ T)“=(l+p)l-kAk-‘P.

Since A k-‘P+

0 it follows that (I+

T)”

Ak-‘P &!% llZ+ Tll” = IIAk-‘PII

’

We remark that if p is a simple pole of R,, then the expression converges to the projection P. This will be the case, if for example, T is an irreducible

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operator. The biological models we have cited are all characterized by irreducible positive linear operators. In these cases ergodicity and weak ergodicity hold. However irreducibility is not a sufficient condition for strong ergodicity: Suppose A is an n x n nonnegative irreducible matrix with index A = d. Then (1) (2) (3)

A is ergodic, A is strongly ergodic if and only if index A = 1, A is weakly ergodic.

Thus if A =

’ ’ 1 0

, index A = 2 and A “/p” is periodic.

It is well-known

I I (see [ 121) that if l/nC’fTk-P where P is nonzero, then P is a projection and 1 is a simple pole of R,. Hence ergodicity implies weak ergodicity. Irreducible models that induce compact operators are all ergodic. This includes population models. Stepping stone models do not necessarily define compact operators nor is the spectral radius r(T) necessarily a pole of the resolvent R,. For these models an analysis in terms of recurrence and transience may be more useful. 4. This section summarizes facts about positive nonlinear operators. The results here seem to apply mainly to economic models. We refer to [7] for a detailed exposition. The argument and statement given here clarify the material presented in [5]. The crucial idea is that of a projective metric introduced by Birkhoff [2]. Let C be the positive cone in a partially ordered Banach space. Two elements x,y in C are said to be comparable if there exist positive real numbers h,~ such that AX < y < p. If x and y are comparable, we let a,=inf {A: x < Xy}, &,= inf {A: y Q hu} and we define d(x,y)=loga,&. The functional d is a pseudometric; that is, (i) (ii) (iii)

d(x,y) = 0 if and only if x = hy for X > 0, d(x,y) = d(y, x), d(x,y) < d(x,z)+

d(y,z).

A positive operator T is said to be uniformly positive if for and real number (Y,he < T(x) < ahe for all x > 0 and some said to be uniformly monotonic if for some fixed eE x,y E C,, x > y, xfy, there exists a positive real number that T(x)- T(y) > he. C,= 1/,n C where V, is the vector by e. We shall denote by I] \le the norm induced on V, by

some fixed e > 0 A = h(x) > 0. T is C, e >0 for all h=X(x,y) such space generated the element e.

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A mapping T of a metric space M into itself is said to be strictly nonexpansive if d( T(x), T(y)) < d(x,y)(x #y,x,y E M). The next result extends the Solow-Samuelson theorem [22]: THEOREM

4.1

Let V be a partially ordered Banach space with positive cone C. Assume C is closed and normal. Let T be a homogeneous operator of degree 1 on V. Suppose 1. 2. 3. Then (a) (b)

T is uniform& monotonic, T is uniform& positive, T is compact.

T has an essentialfy unique eigenvector u E C,, T(u) = hu. h > 0. Given any x,~ C,, d(T”(x,),u)+O as n+oO.

Proof. Since T is compact as an operator on V, T is also compact as an operator on the Banach space V,. Let Y?(X)= T(x)/11 T(x)[[~. Choose xOE C, and let X, = T(x,_ J. The sequence {x,} is bounded. Since T is compact, { T(x,)) has a convergent subsequence; that is, there exists a subsequence { T(x,)) and an element y E C, such that ]IT(x,,~)-y/1.-+0. T is uniformly positive; hence pe < 7(x,) where p = p(x,J and )I > 0. Since C, is closed, y > pe and llyll,#O. Hence ?‘(x,,,) converges toy/lly](e. Hence {xn} has a convergent subsequence. However, X, = ?‘(x,,). This means that there exists an lelement u such that I(?‘k(xO)- ull,-+O as k+co. This implies that d( T”*(x,,),u)+O as k+co. Since T is uniformly positive and uniformly monotonic, it follows that T is strictly nonexpansive [7]. Hence ? is strictly nonexpansive. By a theorem due to Edelstein [8] we conclude that d( T(u), u) = 0. Hence, for some X > 0, T(u) = Xu. In addition, d( T”(x,), u) converges to 0 as n+oo. REFERENCES 1 2 3 4 5 6 7

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