Demographic Models for Subdivided Populations ... - Science Direct

Theoretical Population Biology TP1261 theoretical population biology 49, 291313 (1996) article no. 0015

Demographic Models for Subdivided Populations: The Renewal Equation Approach J. D. Lebreton Centre d'Ecologie Fonctionnelle et Evolutive, CNRS, BP 5051, 34033 Montpellier Cedex 1, France Received May 5, 1994

Le Bras (Theor. Popul. Biol. 2, 100121, 1971) and Rogers (Demography 11, 473481, 1974), in two neglected papers, have generalized to the multisite case the EulerLotka renewal equation and demographic characteristics such as age structure and reproductive value. The purpose of this paper is twofold: first, to restate the multisite renewal equation in the matrix context; second, to derive results on age structure, net reproduction rate, generation time, and sensitivities, as generalizations of the one site case. The potential of this approach for population biology is illustrated using a model of a black-headed gull Larus ridibundus population. 1996 Academic Press, Inc.

1. Introduction The balance between births and deaths in natural populations is strongly influenced by the movement of individuals (e.g., Greenwood and Harvey, 1982). Demographic models that account for migration are thus needed. In human demography, ``multiregional models'' have been developed (Rogers, 1966, 1974) to link age-structured models over different spatial population units. In population biology, the role of extinctions and recolonizations in population dynamics and questions regarding the evolution of dispersal have led to consideration of the dynamics of subdivided populations, or metapopulations in the broad sense (e.g., Gilpin 6 Hanski, 1991). Models of subdivided populations have been developed on lines similar to those used for human populations, with an emphasis on the relationship between dispersal and regulation (e.g., Lebreton and Gonzalez-Davila, 1993). Discrete time multiregional or multisite models can be built in a fairly straightforward way: the numbers of individuals aged i in site j at time t are grouped in a column matrix N t . Assuming constant parameters, N t+1 is obtained from N t by a matrix recurrence equation which generalizes the well-known Leslie matrix model. The transition matrix can be presented in two equivalent ways (Caswell, 1989, p. 52). In the first way (Table IA), the 291 0040-580996 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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J. D. LEBRETON TABLE I Two Ways of Presenting the Same Multisite Matrix Model (A)

Site

Age

1 1 2 2

1 2 1 2

Site: Age:

1 1

1 2

2 1

2 2

_

0 s1 0 0

f 11 s1 f 12 0

0 0 0 s2

f 21 0 f 22 s2

(B)

Age

Site

1 1 2 2

1 2 1 2

Age varies within sites

&

Sites vary within age

Age: Site:

1 1

1 2

2 1

2 2

_

0 0 s1 0

0 0 0 s2

f 11 f 12 s1 0

f 21 f 22 0 s2

&

Note. (A) the age index varies within the site index; (B) the site index varies within the age index. There are two age classes, with reproduction at age 2 and dispersal. In (B) the overall matrix is a Leslie matrix with scalar parameters replaced by matrices.

age index, i, varies within the site index, j. The emphasis is then on the coupling of Leslie matrix models (e.g., Rogers, 1966). In the second way (Table IB), the site index, j, varies within the age index, i. Natural matrix generalizations of the fecundity and survival parameters appear (e.g., Caswell, 1989, pp. 52, 53), and, in block matrix notation, the overall transition matrix is similar to the Leslie matrix. Under mild conditions of connectivity, the matrix has a positive dominant eigenvalue which is equal to the asymptotic multiplication rate of the population over all sites (Rogers, 1966). Such a model has been used for animal populations by, e.g., Fahrig and Merriam (1985). In the one-site Leslie matrix model, the dominant eigenvalue is also the root of the EulerLotka equation (Lotka, 1907; Keyfitz, 1968), which is the renewal equation associated to the model. By generalizing the EulerLotka equation to the multisite case, Le Bras (1971) and Rogers (1974), in two neglected papers, generalize several quantities derived from the elementary demographic parameters to the multisite case. These quantities are the age


DEMOGRAPHIC MODELS

293

structure, the reproductive values, and the net reproduction rate (see also Rogers, 1973; Rogers and Willekens, 1978). The present paper has two objectives: First, we restate the multisite LotkaLe Bras equation in the matrix context to make it easily usable. The aim here is, in particular, to make it suitable for animal populations in a seasonal environment with an annual birth pulse. Second, we derive explicit results on age structure, net reproduction rate, generation time, and sensitivities, which generalize similar results for the one-site case (Houllier and Lebreton, 1986). The potential of explicit results for multisite models in animal population biology is illustrated using a model of a black-headed gull Larus ridibundus population.

2. Notation, Model, and Basic Results Let us consider p sites and n age classes, plus age class 0 for newborn individuals. For the sake of simplicity, only ``birth pulse'' populations are considered, i.e., populations in which reproduction is supposed to take place instantaneously. The time step and the length of age classes are assumed to coincide. Parameters are assumed to be constant over time. The following notation is used throughout: j and k (=1, ..., p) vary over sites; i and l (=0, ..., n) vary over age classes. f ijk is the number of newborn females in site j per female aged i in site k; F i =( f ijk ) is the p_p fecundity matrix per female aged i (i1); F i can in general be considered as diagonal, migration occurring early after birth being incorporated in the survival process; s ijk is the proportion of survivors in site j among females aged i&1 in site k; S i =(s ijk ) is the p_p migration-survival matrix of females from age i&1 to i (i1). In particular S 1 is the migration-survival matrix from birth to age 1. When there is no migration between i&1 and i, S i is diagonal. S n+1 is 0 if n is the maximum age. When there is no upper limit on age, F n and S n+1 are the fecundity and survival matrices of individuals aged n or more (F i =F n and S i =S n+1 , i>n), respectively. L i =S i S i&1 ... S 1 is the migration-survival matrix from birth to age i.


294

J. D. LEBRETON

N ki (t) is the number of females aged i in site k; N i (t)=(N ki (t)) is the p_1 matrix (vector) of numbers of individuals aged i. N 0(t) is the vector of newborn individuals; N *(t) is the pn_1 matrix (vector) of numbers of individuals at time t in each age class and site, the site index varying within the age index, which starts at age 1. N(t) is the similar p(n+1)_1 matrix (vector) including age 0. 0 is the zero matrix, I is the identity matrix, and 1 is a column matrix of ones, with dimensions obvious from the context. Under this notation and these assumptions, one may write a fecundity matrix F for the birth pulse and a survival matrix S from immediately after the birth pulse to immediately before the next pulse (Table II). If one considers that the birth pulse takes place immediately after each date, i.e., individuals enter the population at age 1, the recurrence equation for population numbers is, in block matrix notation, N 1(t+1) N 2(t+1) }}} N i (t+1) }}} N n(t+1)

_ &_

S1 F1 S1 F2 S2 0

=

0

0

}}} }}} }}}

}}}

S1 Fi

}}}

S i&1

0 Sn

S 1 Fn

N 1(t) N 2(t)

&_ & N i (t)

S n+1

N n(t)

or N*(t+1)=SFN*(t)=PN*(t).

(1)

If the birth pulse takes places immediately before each date, then individuals enter the population at age 0, and the recurrence equation for population numbers becomes N 0(t+1) N 1(t+1) N 2(t+1) }}} N i (t+1) }}} N n(t+1)

_ &_ =

F 1 S 1 F 2 S 2 } } } F 1 S i } } } F n S n F n S n+1 S1 0 }}} S2 } } } }}} Si 0 }}} 0

0

}}}

Sn

S n+1

N 0(t) N 1(t) N 2(t)

&_ & N i (t)

N n(t)

or N(t+1)=FSN(t)=MN(t).


(2)

295

DEMOGRAPHIC MODELS TABLE II

The Fecundity (F) and Survival (S) Matrices Associated with the Multisite Population Model F

_

S

F1 F 2 } } } F i } } } F n I

0 }}} I }}} I }}} }}}

0

0 }}}

I

& _

S1 S2

0 S3

0 }}} Si

}}} 0 }}} S n S n+1

&

Note. F maps age classes 1, ..., n, onto age classes 0, 1, ..., n and is thus p(n+1)_pn. S maps age classes 0, 1, ..., n onto age classes 1, ..., n, and is thus pn_p(n+1).

P and M generalize the well-known Leslie (1945) matrix, or the Usher (1972) matrix when S n+1 >0, with scalar fecundity and survival parameters replaced by p_p matrices. If p=1 (one site), P and M reduce to the Leslie or Usher matrix. In the following, M, a squared matrix of order p(n+1) including age class 0, is used. The same results can be obtained from matrix P. Caswell (1989, p. 13) compares the two approaches (before or after birth pulse) in the one site case. Mild conditions of connectivity between sites and age classes ensure that M is irreducible and aperiodic (e.g., Feeney, 1971). M is thus primitive and the PerronFrobenius theorem (e.g., Gantmatcher, 1959; Seneta, 1981) applies as in the one-site case: M has a real positive dominant eigenvalue *, which is simple. * is the largest positive root of det(M&*I)=0.

(3)

The corresponding left and right eigenvectors U and V are positive and verify MV=*V,

U$M=*U$.

(4)

They are usually scaled such as U$V=1. Then, lim N(t)=(U$N(0)) * tV.

(5)

t

The multisite model thus obeys the well-known asymptotic exponential growth regime (Rogers, 1966). V represents a stable age by site structure. U represents age by site reproductive values, which weight the influence of initial values in the asymptotic growth curve. V 0 and U 0 denote the column matrices (vectors) of the first p components of, respectively, V and U, corresponding to newborn individuals.


296

J. D. LEBRETON

3. The Multisite Renewal Equation The matrix approach considers all age classes over one time step. The renewal approach uses a single age class, that of newborn individuals, over many time steps. Backward calculations, similar to those used in the onesite case, allow for a shift from the matrix representation to the renewal equation approach (Table III). From Table III, one deduces N 0(t)=: F i S i } } } S 2 S 1 N 0(t&i),

or

N 0(t)=: F i L i N 0(t&i).

(6)

Once an asymptotic exponential growth with multiplication rate * has been reached, N 0(t)=* iN 0(t&i)

N 0(t&i)=* &iN 0(t),

or

which, after having dropped the index t, gives N 0 =: F i L i * &iN 0

(7)

or, noting A(*)= A i (*)= F i L i * &i, i1

\

+

(A(*)&I) N 0 = : F i L i * &i &I N 0 =0.

(8)

This is the basic equation of Le Bras (1971) and Rogers (1974). It allows for non-null solutions for N 0 if and only if (Rogers, 1974, Eq. 13, p. 475)

\

+

,(*)=det(A(*)&I)=det : F i L i * &i &I =0.

(9)

,(*) is the determinant of a p_p matrix. Its largest root is the asymptotic multiplication rate *, the largest eigenvalue of M obtained from (3), as a consequence of det(M&*I)=(&*) np det(S n+1 &*I) det(A(*)&I).

(10)

This equation is obtained by repeated application of a formula for the determinant of a patterned matrix (Graybill, 1983, p. 184). F n should be non-null to make M irreducible; then the largest eigenvalue of M is larger than the largest eigenvalue of S n+1 . Using (10), this makes Eqs. (3) and (9) equivalent. The price to pay for the reduction in determinant size from (3) to (9) is that * appears in polynomial form in the matrix in (9).


297

DEMOGRAPHIC MODELS TABLE III Backward Calculation Relating the Number of Newborn Individuals in the p Sites at Time t, N 0(t), to Past Numbers of Newborn Individuals N 0(t&i) (i=1, ...) Time: Age 0 1

t&i

t&i+1

}}}

t&1

t

N 0(t)

N 0(t&i) S

1 w N 1(t&i)

S

2 w N 2(t&i+1)

2 } } } } i }

}}} w Fi S

i N i (t&1) w

The element ( j, k) of the p_p matrix A i (*) is made up of the numbers of newborn individuals in site j per female aged i and born in site k, discounted for the overall geometric growth at rate *. Various demographic characteristics can be deduced from the matrices A i (*), as generalizations of quantities deduced from the scalars f i l i * &i in the one-site case. 4. Stable Site by Age Structure and Reproductive Values One may check, from MV=*V, that the stable age by site structure V is proportional to

V=

V0 V1 V2 }}} Vi }}}

I &1

_ &_ & Vn

=

* L1 * &2L 2 }}} * &iL i }}}

V0 .

(11)

: * &iL i

in

This result is classical for the one-site model, when V 0 and the L i are scalars. From (11), multiplying V by the first row of M shows that solutions for N 0 in Eq. (8) are proportional to V 0 , i.e. (Rogers, 1974, Eq. 6), (A(*)&I) V 0 =0.


(12)

298

J. D. LEBRETON

Let us partition U$, the left eigenvector of M, according to age classes into n+1 vectors with p components each, as U$=(U$0 , U$1 , ..., U$n ). Let us denote S l } } } S i+1 as l L i (l>i). Then, after some algebra, for i0, U$i =U$0

\* : F i

l l

+

L i * &l .

l>i

(13)

These quantities have been introduced by Rogers and Willekens (1978, Eq. 9, p. 508) as the exact generalizations of the reproductive values in the one-site case. Equation (13) indicates that, as in the one-site case (see, e.g., Caswell, 1989, p. 107), the reproductive values are obtained from the left dominant eigenvector of the Leslie matrix. For i=0, (13) reduces to (Rogers and Willekens, 1978, p. 507) U$0(A(*)&I)=0.

(14)

* can be obtained as the dominant eigenvalue of M or by solving (9). Then A=A(*) can be calculated. U 0 and V 0 are then respectively the left and right eigenvector of A associated to its dominant eigenvalue 1 ((14) and (12)). Then, Eqs. (12) and (14) may be solved to provide the stable distribution of newborn individuals among sites V 0 and the reproductive value at birth over sites U$0 , respectively. One may then explicitly proceed to further age classes from (11) and (13). At first sight, A resembles a stochastic matrix describing the change in the distribution of newborn individuals among sites from t&1 to t. However, the elements in each column of A do not in general sum to 1, and, as a consequence, the components of U 0 are not equal. The differences between components of U 0 reflect differences among sites in the future contribution to growth of newborn individuals. This is why they are called reproductive values at birth over sites (Rogers and Willekens, 1978). Provided the scaling U$0 V 0 =1 is used, the above results imply that A t converges to a rank one matrix lim A t =V 0 U$0 .

(15)

t

Indeed, the limit in (5), if the initial population consists exclusively of newborn individuals, becomes, under the original normalization U$V=1, lim N 0(t)=* tV 0 U$0 N 0(0).

(16)

t

N(t) can then be deduced from N 0(t) following (11). Equation (16) implies that, apart from certain scaling problems (see below), V 0 U$0 summarizes in the long term the transition between sites in the proportion of newborn individuals.


299

DEMOGRAPHIC MODELS

5. Sensitivities, Generation Time, and Scaling The implicit function of * in (6) leads to explicit results for the sensitivity of * to changes in parameters. Following the approach used by Keyfitz (1971) in the one-site case, an infinitesimal change d3 in a parameter 3 induces an infinitesimal change d * in *, linked by 0=d(,(*))=,3 d3+, * d *.

(17)

Hence, as for the one-site Lotka equation, *3= &(,3)(, *),

(18)

which gives the relative sensitivities ln * ln 3=

&3 ,3 . , * *

(19)

From results for the derivative of a determinant (Graybill, 1983), it is convenient to write, under U$0 V 0 =1 (Appendix), &*

, =:T=:U$0 : iF i L i * &i V 0 , *

\

+

(20)

where the coefficient : depends on demographic parameters. The quantity

\

+

T=U$0 : iF i L i * &i V 0

(21)

has a straightforward interpretation. First, notice that U$0 F i L i * &iV 0 = U$0 AV 0 =U$0 V 0 =1. Hence the quantities p ijk =U 0( j)(F i L i * &i )( j, k) V 0(k)=U 0( j) A i ( j, k) V 0(k) sum up to 1. These quantities can be interpreted, in a population having reached the asymptotic exponential regime, as defining the distribution of a random variable. This random variable, X, is the age (i) of mothers (born in k) at childbirth (in j), accounting for multiple births, because of the presence of F i , and for the future potential contribution of the newborn individuals, as measured by their reproductive value U 0( j). Hence T=E(X), the mean age of mothers at birth in the stable structure population, weighted by the distribution of mothers over sites, by multiple births, and by the reproductive value of newborns, i.e., by their future contribution to the population. As such T is a compound generation time over sites. In the one site case, T reduces to if i l i * &i, the generation time sensu Leslie (1966) also equal to &*(, *) (Houllier and Lebreton, 1986).


300

J. D. LEBRETON

Here, T is also the unconditional mean, weighted by i p ijk , of the means t jk conditional on j and k, i.e., on the site of childbirth and the site of mother birth

\

+