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Stable population analysis in periodic environments
THEORETICAL
POPULATION
BIOLOGY
Stable Population
11,
49-59 (1977)
Analysis in Periodic Environments ROBERT S.
N.Y.
GOURLEY
*
Cooperative Wildlife Research Unit and Section of Ecology and Systematics, Fernow Hall, Cornell University, Ithaca, New York 14850 AND
CHARLES E. LAWRENCE? Division
of Systems Engineering, Sage Laboratory, Rensselaer Polytechnic Institute, Troy, New York 12181 Received February
19, 1975
A discrete time model is developed for periodic survivorship and maternal frequency rates. The Leslie matrix is subdivided by an additional variable representing time of birth (season of birth in the example presented) to accommodate both age-specific and time-specific variations in vital rates. Thus, in contrast to the standard time-invariant model, significant periodic alterations in age-specific birth and death rates are explicitly accounted for and may realistically include observed recurrent changes, such as zero or reduced birth rates during unfavorable seasons, etc. Conditions for stability of the extended projection matrix are developed and are shown to be analogous to those of the Leslie model. The periodic model is applicable to populations with overlapping generations in seasonal environments.
1. INTRODUCTION Mathematical demography is largely based on the stable population theory developed by Lotka (1924) and Leslie (1945). The stable model depends on the basic assumption that the vital rates (I, and m,) remain constant over time. This assumption is reasonably appropriate in certain cases, for example, many human populations, but it cannot be applied to most natural populations of other species. There are few natural environments that lack annual rhythms (of temperature, precipitation, etc.), and it is obvious that for most species the vital rates change in concert with the periodic changes in season. Our * Present Address: Department of Biology, Amherst College, Amherst, Mass. 01002. ‘Present Address: Cancer Control Bureau, New York State Department of Health, Albany, N.Y. 12237. Copyright All
rights
49 0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
004C-5809
50
GOURLEY AND LAWRENCE
focus here is on the type of demographic behavior (with overlapping generations) in which survivorship rate depends not only on age but also on season of birth, and in which reproduction varies with both age and time of year as well. Thus, for example, in addition to the rate changes normally associated with age, fall-born and spring-born individuals may survive at different rates in the winter, winter reproduction may be different than in other seasons, etc. Skellam (1967) provided lucid numerical and graphical representations of such processes, and emphasized both the general significance of periodic models in population biology and the importance of firmly establishing the mathematical properties of seasonal demographic behavior. Temperate zone populations of microtine rodents provide a well-known example of “seasonal generations” subject to different vital rates (reviewed by Schwarz et al., 1964).
2. BASIC MODEL AND EXISTING THEORY The basic form of the discrete time Leslie model of population &+,
= Axt
,
growth
is
(1)
where X is an (rz x 1) column vector (of population age distribution) and A is an (rr x rz) matrix (with vital rates as appropriate elements) describing the transition characteristics from time t to t + 1. The diverse biological applications of this model were reviewed by Usher (1972). 0 ur central interest is the model property of ultimate stability. A proof of stability in the discrete case depends on nonnegative, irreducible, and primitive conditions in the matrix A (see Sykes, 1969; Parlett, 1970; Demetrius, 1971a). Under these conditions the matrix has a single dominant eigenvalue h (2 1n - 1 others I) with corresponding latent vector (of nonnegative real elements); for particular vital rates these matrix parameters prescribe the finite rate of increase, stable age distribution, age-specific reproductive values, and other important population statistics. Population cycles induced by density-dependent perturbations have been studied by Keyfitz (1972), Beddington (1974) and Lee (1974). In this report we consider cycles arising from seasonal periodicities in the life table parameters themselves. Caughley (1967) adapted stable theory to life histories with a single restricted season of births. Bernardelli (1941) first addressed the issue of gradual and seasonal changes in birth and death rates and pointed up occurrences of age structures and population sizes with periodic form. Coale (1972, p. 214) made an heuristic extension of the weak ergodic theorem (Lopez, 1961) to periodic vital rates. Skellam (1967) and Namboodiri (1969) examined periodic models by representing cycle stages with separate matrices of vital rates and then repeating the multiplicative process of Eq. (1) in cyclical order.
DEMOGRAPHY
51
IN PERIODIC ENVIRONMENTS
Skellam linked explicitly seasonal birth and death patterns in this way and showed that if all seasons have a single dominant root, then all have the same root. That is, the ultimate rate of increase associated with a single season also holds for all seasons and for the population as a whole; thus, a periodic population examined at corresponding points in successive years will be seen to grow at the ultimate rate. It remains to be shown that a periodic model has stable properties; we wish to establish the conditions under which an arbitrary population with periodic vital rates will achieve global stability.
3. THE EXTENDED MODEL The basic discrete time Leslie model subdivides a population by the single variable age. A set of n equations describes the impact of births and deaths on the single sex age distribution in the following way:
Xo,t+l= 2 miXi,t
t = l,...,
i=cs
Xi+l,t+l
i = o,..., N,
= Qxi.t
t = I,...,
(3)
where Xi,, = number of individuals
of age i at time t,
m, = age specific birth rate, si = fraction of individuals
of age i to survive to age i + 1,
(Y = age at first reproduction, w = age at last reproduction. We extend this model by introducing a second subdividing variable for time of birth. Birth periods may be variously defined for different applications, but to clearly distinguish the variables in this paper we use the four temperate seasons throughout. Let Yj,,,, be the number of individuals born at season j, age i, and time t. The new model for births to seasonal group j at time t + 1 becomes
and for survivorship Yj,i+l.t+l
within
a seasonal group
= Sj,iYj.i.t >
j=
I,..., p,
i=O
,..., N,
t=
I,...,
(5)
52
GOURLEY
AND
LAWRENCE
where
mjie . . = number of offspring born in season j to adults of seasonal group k of age i divided by the number of individuals of seasonal group k of age i, s~,~= fraction of individuals of seasonal group j of age i to survive to age i + 1, p = number
of seasonal groups (i.e., seasons) in a year.
Defining the model parameters in this way allows for both age- and seasonspecific variations in birth and death rates; for individuals born in a particular season the age and season specific changes are coupled. Thus, individuals born in the spring will be two seasons old as they enter their first winter season, four seasons old at the outset of their second summer, six seasons old at the second winter, etc., and the model accommodates unique vital rates associated with these age-time states. Since the season of birth is specified for ail individuals, example, the obvious certain of the matrix elements (mi,i,,) must be zero-for fact that spring-born individuals cannot be born in the summer, fall or winter is reflected in the model by this requirement. Thus,
mj,i,k = 0
for
i=O,andi#(Z.p)-(k-j)-lwherel=1,2,....
(6)
Note that this constraint does not preclude reproduction by any seasonal group in any season. The set of Eqs. (4) and (5) may now be written in matrix form. Let Ff be the age-season distribution vector:
FY 1.1.t Y 1,z.t +mt Y 2s.t yt=
:
Y 2,n.t
*
Y 9.1.t kLw.t
J
Then
yt+l = BFt ,
(7)
ii
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
----------------_--_-------
0000
0
0
0
0
0
0
0
0
P r 01
0
0
0
0
0
0
0
0:
ao
In 400 01
ZCY c4l
0
0
0
0
0
0
0
a
0
0
0
0
0
0
0
0
0
0
i I I I
54
GOURLEY AND LAWRENCE
where B is the extended population growth matrix incorporating seasonal rates. To present the entire generalized matrix would require unseemly repetition, but Fig. 1 shows a matrix partition of the first two seasonal groups with labeled row and column indices for the temperate microtine populations where 01= 1, w = 2p, and p = 4. We wish to establish the asymptotic behavior of stable population theory for the nonnegative matrix B.
4. STABLE ATTRIBUTES
OF THE PERIODIC
MATRIX
Demetrius (1971a) examined the irreducibility and primitivity conditions for growth matrices of the Leslie model by utilizing the well-known relationship between matrices and directed graphs. In a similar vein we will show that our extended model is irreducible and imprimitive with index p. We take some initial definitions and notational conventions from Varga (1962): Let A = /I aij j/ be any (n x n) matrix, and consider n distinct points (nodes) PI, Pz ,..., P, in the plane. We will say that node Pi is connected to node Pj by means of a path z directed from i to j if and only if a,, + 0. Thus every (n x n) matrix A can be associated with a finite directed graph G(A). Figure 2 represents the directed graph associated with our example matrix B. A directed graph is strongly connected if, for any ordered pair of nodes Pi and Pj , there exists a directed path
connecting
Pi to Pi . Such a path has length r. Rosenblatt
(1957) has shown
m2,7,2
FIG. 2. The directed graph associated with the example matrix spond to the labeled rows and columns in B.
B. The nodes corre-
DEMOGRAPHY
IN
PERIODIC
ENVIRONMENTS
55
that an (n x fl) matrix A is irreducible if and only if its directed graph G(A) is strongly connected. Consider 71= pw distinct points in the plane with a single node associated with each age-season category. The directed paths in the extended matrix are easily established. The Q’S assure that there are directed paths from the nodes associated with age (i) to nodes associated with age (i + 1) for any seasonal group. Two additional conditions must be met to have directed paths between all nodes: (1) There exist directed paths between nodes associated with groups of individuals in all seasonal cohorts, either directly in which case 33 mjei,k > 0 or indirectly example
'6 k
(i.e., through some intermediate group(s)) in which case, for Vj, k.
% 2*3 (mt,i.i)(fllk~,t) > 0
Note that this condition simply requires that all seasonal groups interbreed, i.e., are members of the same population. (2) There exists a path from nodes associated with all the oldest age classes to at least one youngest age class:
3j3 mj,w,k >
0
Vk.
This condition is analogous to the requirement for the Leslie matrix that m, > 0, and it could also be analogously relaxed where a knowledge of postreproductives is needed (Lopez, 1961). Thus, the modified growth matrix B > 0 satisfying conditions (1) and (2) is irreducible. To establish the asymptotic behavior of Bt and the associated multiplicative process y((t + 1) = By(t) we next examine the primitivity of B. The connection between primitivity conditions of the matrix and characteristics of the asymptotic growth is a result of the Perron-Frobenius theorem (e.g., Cox and Miller, 1965, p. 120). Let A >, 0 be an irreducible (n x n) matrix, and let h be the number of eigenvalues of modulus A. If h = 1, then A is primitive, If h > 1, then A is cyclic (imprimitive) with index (period) h. Varga (1962) showed that the primitivity conditions of an irreducible matrix depend on the greatest common divisor of closed path lengths: “Let A = (1uij 11> 0 be an irreducible (n x n) matrix, with G(A) as its directed graph. For each node Pi of G(A), consider all closed paths connecting Pi to itself. If Sj is the set of all the lengths mi
56
GOURLEY
AND
LAWRENCE
of these closed paths, let Ki be the greatest common divisor of their lengths, i.e. ki = g.c.d. (m,}, rniE.7, Then k, = k, = ... A, = R, where when A is imprimitive (periodic) of Thus to complete our analysis of the lengths of the closed paths in associate a node with each age group
1
k = 1 when A is primitive and k > 1 index (period) K.” the properties of B we need to examine the corresponding graph G(B). We can for each seasonal category, i.e.,
P,,, is a node associated with age group i for seasonal group j. It is clear that there exists a path of length one from Pj,, to Pj,i+l since s?,~ > 0, and that the path length from Pi,, to Pi,,+, is 4. Also there exists a path of length 1 from Pk,i to Pi,O if and only if mj,i,k > 0. Now we examine the path lengths from Pf,O to itself. First consider paths only within seasonal group j. Restating Eq. (6) for the case of k = j, for
mjsi*j = 0
i = 0, and i # (lop)
-
1, I = 1, 2 ,... .
Thus when births occur within a seasonal group mj,i,j > 0. But from Eq. (6) this can only be so for age i = (I ‘9) - 1 > 0 (where I = 1, 2,...). Then the l
path length Pj,OPi,Ois (I * p) since the length from PjSOto Pj,(z.p)--l is (1 . p) - 1, and the length from Pj,(l.D)--l to Pj,O is 1. For example in Fig. 2, node Pl,O can be connected to itself via three paths of length 1 from Pl,O to Pl,3 and one path of length 1 from Pl,3 to PI,, making a total of four. In like fashion to go through some other seasonal group (K) would require
Process
Path
Path Length
Pj.OPi,(Z,.9)-(i-lc)-,
(Z1* P) - (j - k) - 1
Begin in seasonal group j, survive to age (4 . P> - ( j - 4 - 1 Reproduce
into K from j,
(4 *P> - (k -8
-
1
Pi.(~~.~)-(j--k)-~Pk.~
-
1
P~,,P,,(~~.~)-(k-j)-1
1
Survive to age (4 *PI - @ -A
(4 .P) - (k -j)
-
1
Return to j via a birth froWb
*PI-(j-+-l
I
P~.(zl.9)-(Ic--i)--1Pi,0 Total Path =
(Zr + I,)(p) ZI = 1, 2,...
I, = 1, 2,... .
57
DEMOGRAPHY IN PERIODIC ENVIRONMENTS
For example (Fig. 2), node Pr,a may be connected to itself via four paths of length 1 from Pi,a to P1,4 , a path of length 1 from P1,4 to Pa,a , two paths of length 1 from Pa,a to P2,z , and a path of length one from P2,2 to Pl,o, for a total path length of 8. Other intervening groups give the same result. The path lengths are multiples of p; their greatest common divisor is p or a higher multiple of p. (The latter case-biennial, etc., populations-is not developed in this report.) We have shown then that the matrix B is nonnegative, irreducible, and cyclic (periodic) with index (period) p. Thus the associated multiplicative process is irreducible, and the process has a stationary state (Demetrius, 1971 b).
5. CONCLUSIONS Several consequences of this result stem directly from Frobenius’ Theorems. In our case the modified growth matrix B can be put into the cyclic form
r 0 B,, 0 ... 0 I B0PI
0
Bz3 0. . . ... .*. Bv-,a 0 1
with each of the Bj-l,j submatrices corresponding to a “season.” This property reflects seasonal growth-for example, individuals surviving out of spring must survive into summer. As in the case of classical stable population theory the asymptotic behavior of the population can be determined by examining the eigenvalues of modulus X. In this case population growth within a year is periodic with period (p). Population growth from an annual perspective can be assessed from another of Frobenius’ results: Let A be an (n x n) nonnegative, irreducible, and cyclic matrix of index (period) R. Then A jk is completely reducible for every j > 1, i.e., Ajk can be permuted into the form
where each diagonal submatrix Ci is square, irreducible, maximal characteristic roots I; of
p”(c,) = ;(c,)
= *-- ;(c,)
= p’“(A).
and primitive
with
58 Thus if we examine intervals,
GOURLEY AND LAWRENCE
our modified
X(t +p)
population
growth
process at annual
= P-T(t),
and the process associated with each seasonal group will be globally stable (Demetrius, 1971 b). Consequently any initial age distribution will approach a unique stationary state and on an annual basis will grow by some real positive amount p” (=A”):
Jqt + P) = p(t), where 6 is the annual finite rate of increase of the population. Thus classic stable population theory can be extended to periodic environments. This extension should be most useful for systematic comparisons of population rates, the evaluation of life history features, and other contexts where realistic rates of increase, reproductive values, etc., are needed. In certain applications a periodic model is uniquely appropriate. Darwin and Williams (1964) applied the cyclical multiplicative process (cf. Skellam, 1967) to an important problem in population management. The age structure peaks and nadirs generated by periodic vital rates should permit more effective interpretation of previously intractable data on seasonal species’ age distributions 1974). Periodic models should be useful and population cycles (Williamson, for understanding life history rates as solutions to the problem of optimal demographic “tracking” in seasonal environments (e.g., MacArthur, 1968), and for pursuing the questions Fretwell (1972) has raised on seasonal population regulation. Among life histories of the type we have modeled (viz., overlapping-generation, iteroparous), Caughley (1967) discussed the important special case of sharply seasonal breeding and developed the treatment of births from a restricted annual reproductive period as if they occurred at one point in time each year. In the sense that annual environmental rhythms are pervasive and natural populations exhibit a continuum of adaptive life history responses to these rhythms, Caughley’s annual-pulse model and the classical time-invariant model are opposite limiting cases of the periodic model.
ACKNOWLEDGMENTS We are grateful for support from U.S. Dept. of Interior Award Contract 14-16-00081037 to R.S.G. and Milo E. Richmond and NSF Grant GK-37399 to C.E.L. This work was conceived at the Center for Environmental Quality Management, Cornell University, where we benefited from initial discussions of mouse populations with K. Hall and R. Swain.
DEMOGRAPHY
IN PERIODIC ENVIRONMENTS
59
REFERENCES BEDDINGTON, J. R. 1974. Age distribution and the stability of simple discrete time population models, J. Theor. Biol. 47, 65-74. BERNARDELLI, N. 1941. Population waves, J. Burma Res. Sot. 31, I-18. CAUCHLEY, G. 1967. Parameters for seasonally breeding populations, Ecology 48, 834-839. COALE, A. J. 1972. “The Growth and Structure of Human Populations, a Mathematical Investigation,” Princeton Univ. Press, Princeton, N. J. Cox, H. R. AND MILLER, H. D. 1965. “The Theory of Stochastic Processes,” Spottiswoode and Ballantyne, London. DARWIN, J. H. AND WILLIAMS, R. M. 1964. The effect of time of hunting on the size of a rabbit population, N.Z. J. Sci. I, 341-352. DEMETRIUS, L. 1971a. Primitivity conditions for growth matrices, Math. Biosci. 12, 53-58. DEMETRIUS, L. 1971b. Multiplicative processes, Math. Biosci. 12, 261-272. FRETWELL, S. D. 1972. “Populations in a Seasonal Environment,” Princeton Univ. Press, Princeton, N. J. KEYFITZ, N. 1974. Population waves, in “Population Dynamics” (T. N. E. Greville, Ed.), pp. l-38, Academic Press, New York. LEE, R. 1974. The formal dynamics of controlled populations and the echo, the boom and the bust, Demography 11, 563-585. LESLIE, P. H. 1945. On the use of matrices in certain population mathematics, Biometrika 35, 183-212. LOPEZ, A. 1961. “Problems in Stable Population Theory,” Office of Population Research, Princeton, N. J. LOTKA, A. J. 1924. “Elements of Physical Biology,” Williams and Wilkins, Baltimore, Md. MACARTHUR, R. 1968. Selection for life tables in periodic environments, Amer. Natur. 102, 381-383. NAMBOODIRI, N. K. 1969. On the dependence of age structure on a sequence of mortality and fertility schedules: an exposition of a cyclical model of population change, Demography 6, 287-299. PARLETT, B. 1970. Ergodic properties of populations I: the one sex model, Theor. Popd. Biol. I, 191-207. ROSFNBLATT, D. 1957. On the graphs and asymptotic forms of finite Boolean relation matrices and stochastic matrices, Naval Res. Logist. Quart. 4, 151-167. SCHWARZ, S. S., POKROVSKI, A. V., ISTSCHEAKO,V. G., OLENJEV, V. G., ORTSCHINNIKOVA, N. A., AND PJASTOLOVA, 0. A. 1964. Biological peculiarities of seasonal generations of rodents, with special reference to the problem of senescence in mammals, Acta Theriol. 8, I l-43. SKELLA~I, J. G. 1967. Seasonal periodicity in theoretical population ecology, in “Proceedings of the 5th Berkeley Symposium, Mathematical Statistics and Probability” (L. M. LeCam and J. Neyman, Eds.), Vol. 4, pp. 179-205, University of California Press, Berkeley. SYKES, Z. M. 1969. On discrete stable population theory, Biometrics 25, 285-293. USHER, M. B. 1972. Developments in the Leslie matrix model, in “Mathematical Models in Ecology” (J. N. R. Jeffers, Ed.), pp. 29-60, Blackwell, London. VARGA, R. S. 1962. Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, N.J. WILLIAMSON, M. 1974. The analysis of discrete time cycles, in “Ecological Stability” (M. B. Usher and M. H. Williamson, Eds.), pp. 17-33, Chapman and Hall, London.
POPULATION
BIOLOGY
Stable Population
11,
49-59 (1977)
Analysis in Periodic Environments ROBERT S.
N.Y.
GOURLEY
*
Cooperative Wildlife Research Unit and Section of Ecology and Systematics, Fernow Hall, Cornell University, Ithaca, New York 14850 AND
CHARLES E. LAWRENCE? Division
of Systems Engineering, Sage Laboratory, Rensselaer Polytechnic Institute, Troy, New York 12181 Received February
19, 1975
A discrete time model is developed for periodic survivorship and maternal frequency rates. The Leslie matrix is subdivided by an additional variable representing time of birth (season of birth in the example presented) to accommodate both age-specific and time-specific variations in vital rates. Thus, in contrast to the standard time-invariant model, significant periodic alterations in age-specific birth and death rates are explicitly accounted for and may realistically include observed recurrent changes, such as zero or reduced birth rates during unfavorable seasons, etc. Conditions for stability of the extended projection matrix are developed and are shown to be analogous to those of the Leslie model. The periodic model is applicable to populations with overlapping generations in seasonal environments.
1. INTRODUCTION Mathematical demography is largely based on the stable population theory developed by Lotka (1924) and Leslie (1945). The stable model depends on the basic assumption that the vital rates (I, and m,) remain constant over time. This assumption is reasonably appropriate in certain cases, for example, many human populations, but it cannot be applied to most natural populations of other species. There are few natural environments that lack annual rhythms (of temperature, precipitation, etc.), and it is obvious that for most species the vital rates change in concert with the periodic changes in season. Our * Present Address: Department of Biology, Amherst College, Amherst, Mass. 01002. ‘Present Address: Cancer Control Bureau, New York State Department of Health, Albany, N.Y. 12237. Copyright All
rights
49 0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
004C-5809
50
GOURLEY AND LAWRENCE
focus here is on the type of demographic behavior (with overlapping generations) in which survivorship rate depends not only on age but also on season of birth, and in which reproduction varies with both age and time of year as well. Thus, for example, in addition to the rate changes normally associated with age, fall-born and spring-born individuals may survive at different rates in the winter, winter reproduction may be different than in other seasons, etc. Skellam (1967) provided lucid numerical and graphical representations of such processes, and emphasized both the general significance of periodic models in population biology and the importance of firmly establishing the mathematical properties of seasonal demographic behavior. Temperate zone populations of microtine rodents provide a well-known example of “seasonal generations” subject to different vital rates (reviewed by Schwarz et al., 1964).
2. BASIC MODEL AND EXISTING THEORY The basic form of the discrete time Leslie model of population &+,
= Axt
,
growth
is
(1)
where X is an (rz x 1) column vector (of population age distribution) and A is an (rr x rz) matrix (with vital rates as appropriate elements) describing the transition characteristics from time t to t + 1. The diverse biological applications of this model were reviewed by Usher (1972). 0 ur central interest is the model property of ultimate stability. A proof of stability in the discrete case depends on nonnegative, irreducible, and primitive conditions in the matrix A (see Sykes, 1969; Parlett, 1970; Demetrius, 1971a). Under these conditions the matrix has a single dominant eigenvalue h (2 1n - 1 others I) with corresponding latent vector (of nonnegative real elements); for particular vital rates these matrix parameters prescribe the finite rate of increase, stable age distribution, age-specific reproductive values, and other important population statistics. Population cycles induced by density-dependent perturbations have been studied by Keyfitz (1972), Beddington (1974) and Lee (1974). In this report we consider cycles arising from seasonal periodicities in the life table parameters themselves. Caughley (1967) adapted stable theory to life histories with a single restricted season of births. Bernardelli (1941) first addressed the issue of gradual and seasonal changes in birth and death rates and pointed up occurrences of age structures and population sizes with periodic form. Coale (1972, p. 214) made an heuristic extension of the weak ergodic theorem (Lopez, 1961) to periodic vital rates. Skellam (1967) and Namboodiri (1969) examined periodic models by representing cycle stages with separate matrices of vital rates and then repeating the multiplicative process of Eq. (1) in cyclical order.
DEMOGRAPHY
51
IN PERIODIC ENVIRONMENTS
Skellam linked explicitly seasonal birth and death patterns in this way and showed that if all seasons have a single dominant root, then all have the same root. That is, the ultimate rate of increase associated with a single season also holds for all seasons and for the population as a whole; thus, a periodic population examined at corresponding points in successive years will be seen to grow at the ultimate rate. It remains to be shown that a periodic model has stable properties; we wish to establish the conditions under which an arbitrary population with periodic vital rates will achieve global stability.
3. THE EXTENDED MODEL The basic discrete time Leslie model subdivides a population by the single variable age. A set of n equations describes the impact of births and deaths on the single sex age distribution in the following way:
Xo,t+l= 2 miXi,t
t = l,...,
i=cs
Xi+l,t+l
i = o,..., N,
= Qxi.t
t = I,...,
(3)
where Xi,, = number of individuals
of age i at time t,
m, = age specific birth rate, si = fraction of individuals
of age i to survive to age i + 1,
(Y = age at first reproduction, w = age at last reproduction. We extend this model by introducing a second subdividing variable for time of birth. Birth periods may be variously defined for different applications, but to clearly distinguish the variables in this paper we use the four temperate seasons throughout. Let Yj,,,, be the number of individuals born at season j, age i, and time t. The new model for births to seasonal group j at time t + 1 becomes
and for survivorship Yj,i+l.t+l
within
a seasonal group
= Sj,iYj.i.t >
j=
I,..., p,
i=O
,..., N,
t=
I,...,
(5)
52
GOURLEY
AND
LAWRENCE
where
mjie . . = number of offspring born in season j to adults of seasonal group k of age i divided by the number of individuals of seasonal group k of age i, s~,~= fraction of individuals of seasonal group j of age i to survive to age i + 1, p = number
of seasonal groups (i.e., seasons) in a year.
Defining the model parameters in this way allows for both age- and seasonspecific variations in birth and death rates; for individuals born in a particular season the age and season specific changes are coupled. Thus, individuals born in the spring will be two seasons old as they enter their first winter season, four seasons old at the outset of their second summer, six seasons old at the second winter, etc., and the model accommodates unique vital rates associated with these age-time states. Since the season of birth is specified for ail individuals, example, the obvious certain of the matrix elements (mi,i,,) must be zero-for fact that spring-born individuals cannot be born in the summer, fall or winter is reflected in the model by this requirement. Thus,
mj,i,k = 0
for
i=O,andi#(Z.p)-(k-j)-lwherel=1,2,....
(6)
Note that this constraint does not preclude reproduction by any seasonal group in any season. The set of Eqs. (4) and (5) may now be written in matrix form. Let Ff be the age-season distribution vector:
FY 1.1.t Y 1,z.t +mt Y 2s.t yt=
:
Y 2,n.t
*
Y 9.1.t kLw.t
J
Then
yt+l = BFt ,
(7)
ii
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
----------------_--_-------
0000
0
0
0
0
0
0
0
0
P r 01
0
0
0
0
0
0
0
0:
ao
In 400 01
ZCY c4l
0
0
0
0
0
0
0
a
0
0
0
0
0
0
0
0
0
0
i I I I
54
GOURLEY AND LAWRENCE
where B is the extended population growth matrix incorporating seasonal rates. To present the entire generalized matrix would require unseemly repetition, but Fig. 1 shows a matrix partition of the first two seasonal groups with labeled row and column indices for the temperate microtine populations where 01= 1, w = 2p, and p = 4. We wish to establish the asymptotic behavior of stable population theory for the nonnegative matrix B.
4. STABLE ATTRIBUTES
OF THE PERIODIC
MATRIX
Demetrius (1971a) examined the irreducibility and primitivity conditions for growth matrices of the Leslie model by utilizing the well-known relationship between matrices and directed graphs. In a similar vein we will show that our extended model is irreducible and imprimitive with index p. We take some initial definitions and notational conventions from Varga (1962): Let A = /I aij j/ be any (n x n) matrix, and consider n distinct points (nodes) PI, Pz ,..., P, in the plane. We will say that node Pi is connected to node Pj by means of a path z directed from i to j if and only if a,, + 0. Thus every (n x n) matrix A can be associated with a finite directed graph G(A). Figure 2 represents the directed graph associated with our example matrix B. A directed graph is strongly connected if, for any ordered pair of nodes Pi and Pj , there exists a directed path
connecting
Pi to Pi . Such a path has length r. Rosenblatt
(1957) has shown
m2,7,2
FIG. 2. The directed graph associated with the example matrix spond to the labeled rows and columns in B.
B. The nodes corre-
DEMOGRAPHY
IN
PERIODIC
ENVIRONMENTS
55
that an (n x fl) matrix A is irreducible if and only if its directed graph G(A) is strongly connected. Consider 71= pw distinct points in the plane with a single node associated with each age-season category. The directed paths in the extended matrix are easily established. The Q’S assure that there are directed paths from the nodes associated with age (i) to nodes associated with age (i + 1) for any seasonal group. Two additional conditions must be met to have directed paths between all nodes: (1) There exist directed paths between nodes associated with groups of individuals in all seasonal cohorts, either directly in which case 33 mjei,k > 0 or indirectly example
'6 k
(i.e., through some intermediate group(s)) in which case, for Vj, k.
% 2*3 (mt,i.i)(fllk~,t) > 0
Note that this condition simply requires that all seasonal groups interbreed, i.e., are members of the same population. (2) There exists a path from nodes associated with all the oldest age classes to at least one youngest age class:
3j3 mj,w,k >
0
Vk.
This condition is analogous to the requirement for the Leslie matrix that m, > 0, and it could also be analogously relaxed where a knowledge of postreproductives is needed (Lopez, 1961). Thus, the modified growth matrix B > 0 satisfying conditions (1) and (2) is irreducible. To establish the asymptotic behavior of Bt and the associated multiplicative process y((t + 1) = By(t) we next examine the primitivity of B. The connection between primitivity conditions of the matrix and characteristics of the asymptotic growth is a result of the Perron-Frobenius theorem (e.g., Cox and Miller, 1965, p. 120). Let A >, 0 be an irreducible (n x n) matrix, and let h be the number of eigenvalues of modulus A. If h = 1, then A is primitive, If h > 1, then A is cyclic (imprimitive) with index (period) h. Varga (1962) showed that the primitivity conditions of an irreducible matrix depend on the greatest common divisor of closed path lengths: “Let A = (1uij 11> 0 be an irreducible (n x n) matrix, with G(A) as its directed graph. For each node Pi of G(A), consider all closed paths connecting Pi to itself. If Sj is the set of all the lengths mi
56
GOURLEY
AND
LAWRENCE
of these closed paths, let Ki be the greatest common divisor of their lengths, i.e. ki = g.c.d. (m,}, rniE.7, Then k, = k, = ... A, = R, where when A is imprimitive (periodic) of Thus to complete our analysis of the lengths of the closed paths in associate a node with each age group
1
k = 1 when A is primitive and k > 1 index (period) K.” the properties of B we need to examine the corresponding graph G(B). We can for each seasonal category, i.e.,
P,,, is a node associated with age group i for seasonal group j. It is clear that there exists a path of length one from Pj,, to Pj,i+l since s?,~ > 0, and that the path length from Pi,, to Pi,,+, is 4. Also there exists a path of length 1 from Pk,i to Pi,O if and only if mj,i,k > 0. Now we examine the path lengths from Pf,O to itself. First consider paths only within seasonal group j. Restating Eq. (6) for the case of k = j, for
mjsi*j = 0
i = 0, and i # (lop)
-
1, I = 1, 2 ,... .
Thus when births occur within a seasonal group mj,i,j > 0. But from Eq. (6) this can only be so for age i = (I ‘9) - 1 > 0 (where I = 1, 2,...). Then the l
path length Pj,OPi,Ois (I * p) since the length from PjSOto Pj,(z.p)--l is (1 . p) - 1, and the length from Pj,(l.D)--l to Pj,O is 1. For example in Fig. 2, node Pl,O can be connected to itself via three paths of length 1 from Pl,O to Pl,3 and one path of length 1 from Pl,3 to PI,, making a total of four. In like fashion to go through some other seasonal group (K) would require
Process
Path
Path Length
Pj.OPi,(Z,.9)-(i-lc)-,
(Z1* P) - (j - k) - 1
Begin in seasonal group j, survive to age (4 . P> - ( j - 4 - 1 Reproduce
into K from j,
(4 *P> - (k -8
-
1
Pi.(~~.~)-(j--k)-~Pk.~
-
1
P~,,P,,(~~.~)-(k-j)-1
1
Survive to age (4 *PI - @ -A
(4 .P) - (k -j)
-
1
Return to j via a birth froWb
*PI-(j-+-l
I
P~.(zl.9)-(Ic--i)--1Pi,0 Total Path =
(Zr + I,)(p) ZI = 1, 2,...
I, = 1, 2,... .
57
DEMOGRAPHY IN PERIODIC ENVIRONMENTS
For example (Fig. 2), node Pr,a may be connected to itself via four paths of length 1 from Pi,a to P1,4 , a path of length 1 from P1,4 to Pa,a , two paths of length 1 from Pa,a to P2,z , and a path of length one from P2,2 to Pl,o, for a total path length of 8. Other intervening groups give the same result. The path lengths are multiples of p; their greatest common divisor is p or a higher multiple of p. (The latter case-biennial, etc., populations-is not developed in this report.) We have shown then that the matrix B is nonnegative, irreducible, and cyclic (periodic) with index (period) p. Thus the associated multiplicative process is irreducible, and the process has a stationary state (Demetrius, 1971 b).
5. CONCLUSIONS Several consequences of this result stem directly from Frobenius’ Theorems. In our case the modified growth matrix B can be put into the cyclic form
r 0 B,, 0 ... 0 I B0PI
0
Bz3 0. . . ... .*. Bv-,a 0 1
with each of the Bj-l,j submatrices corresponding to a “season.” This property reflects seasonal growth-for example, individuals surviving out of spring must survive into summer. As in the case of classical stable population theory the asymptotic behavior of the population can be determined by examining the eigenvalues of modulus X. In this case population growth within a year is periodic with period (p). Population growth from an annual perspective can be assessed from another of Frobenius’ results: Let A be an (n x n) nonnegative, irreducible, and cyclic matrix of index (period) R. Then A jk is completely reducible for every j > 1, i.e., Ajk can be permuted into the form
where each diagonal submatrix Ci is square, irreducible, maximal characteristic roots I; of
p”(c,) = ;(c,)
= *-- ;(c,)
= p’“(A).
and primitive
with
58 Thus if we examine intervals,
GOURLEY AND LAWRENCE
our modified
X(t +p)
population
growth
process at annual
= P-T(t),
and the process associated with each seasonal group will be globally stable (Demetrius, 1971 b). Consequently any initial age distribution will approach a unique stationary state and on an annual basis will grow by some real positive amount p” (=A”):
Jqt + P) = p(t), where 6 is the annual finite rate of increase of the population. Thus classic stable population theory can be extended to periodic environments. This extension should be most useful for systematic comparisons of population rates, the evaluation of life history features, and other contexts where realistic rates of increase, reproductive values, etc., are needed. In certain applications a periodic model is uniquely appropriate. Darwin and Williams (1964) applied the cyclical multiplicative process (cf. Skellam, 1967) to an important problem in population management. The age structure peaks and nadirs generated by periodic vital rates should permit more effective interpretation of previously intractable data on seasonal species’ age distributions 1974). Periodic models should be useful and population cycles (Williamson, for understanding life history rates as solutions to the problem of optimal demographic “tracking” in seasonal environments (e.g., MacArthur, 1968), and for pursuing the questions Fretwell (1972) has raised on seasonal population regulation. Among life histories of the type we have modeled (viz., overlapping-generation, iteroparous), Caughley (1967) discussed the important special case of sharply seasonal breeding and developed the treatment of births from a restricted annual reproductive period as if they occurred at one point in time each year. In the sense that annual environmental rhythms are pervasive and natural populations exhibit a continuum of adaptive life history responses to these rhythms, Caughley’s annual-pulse model and the classical time-invariant model are opposite limiting cases of the periodic model.
ACKNOWLEDGMENTS We are grateful for support from U.S. Dept. of Interior Award Contract 14-16-00081037 to R.S.G. and Milo E. Richmond and NSF Grant GK-37399 to C.E.L. This work was conceived at the Center for Environmental Quality Management, Cornell University, where we benefited from initial discussions of mouse populations with K. Hall and R. Swain.
DEMOGRAPHY
IN PERIODIC ENVIRONMENTS
59
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