A Genealogical Overlapping Generations Model of The Rendille of Northern Kenya*

Transcription

1 Comments welcome A Genealogical Overlapping Generations Model of The Rendille of Northern Kenya* by Merwan Engineer and Ming Kang University of Victoria Revised May 2015 ABSTRACT The Rendille tribe of Northern Kenya is an age-group society in which cohorts of similarly aged males known as age sets are organized along periodic lines of descent. This paper develops a matrix population model that reveals the genealogical-generation dynamics inherent in the Rendille social organization. The rules governing marriage reduce the size and the growth rate of the population and dramatically favor one of the lineages. More generally, the analysis suggests that matrix population modeling is a powerful tool for analyzing age-group societies.

2 1. Introduction Ethnographers have classified societies that are formally organized according to age and/or lineage as age-group societies. Such societies are found around the world and have been classified in set-theoretic terms by Stewart (1977). Age-set societies are those that recruit similar aged males into cohorts age sets whose members transit the lifecycle together. Paternal-linking societies are societies that link each father s age-set with his sons in lineages. This paper examines a particular age-group society highlighted by Stewart (1977), the Rendille tribe of northern Kenya. Rendille society incorporates both forms of age-group organization, age sets and paternal linking. The operation of the Rendille age-group rules is well described by a lineage-based genealogical-generation matrix population model. The model is solved analytically using Perron-Frobenius matrix methods (e.g. Tuljapurkar and Caswell (1997)), and the solution to the model reveals the population dynamics inherent in the age-group marriage rules and dramatic differences in the evolution of the various lineages. Using stable population theory, we prove that the population dynamics of the fahan-based model yield convergence to a constant growth path. The rules governing marriage according to lineage: substantially reduce the level and the growth rate of the population, and dramatically favors one of the lineages. In the steady state, the imbalances in the population among the lineages yield a population cycle within fahan-time intervals but not across fahan-time intervals. We show these results apply generally over the range of parameters that are demographically realistic. The analytical system is interesting in itself because it reveals that the dynamic flows can be explained in terms of subsystems for each lineage. We explain when subsystems can exist on their own and when they are dependent on other systems. These exercises are motivated by an important age-group rule, called Sepaade, that delays the marriage of the daughters from one of the lineages. The paper proceeds as follows. Section 2 describes the Rendille age-group system and previous models of this system upon which this paper builds. Section 3 formally lays out the dynamic model. The steady state is characterized analytically in Section 4. Section 5 provides further analysis with the aid of simulations. Section 6 concludes. 2. The Age-Group System and Models of the System A historical description of the Rendille age-group system is presented in Table 1 (reproduced from Engineer, Roth and Welling, 2003). The system s history is remarkably regular: every 14 years a new age-set of males between the ages of is initiated. The initiation ceremony 1

3 involves circumcision and hence the circumcision year is used to identify the age-set. Age-sets are distinguished by fahan and lineage. Three adjacent age-sets make up what the Rendille call a fahan. A fahan is an epoch, which is organized along three lines of descent: each fahan n contains age-sets ordered X n, Y n and Z n, one from each of the three lineages denoted X, Y and Z. Lineage X, starts each fahan; the Rendille name for it is Teeria, meaning first-born lineage. (Table 1 here) In Stewart s (1977) categorization of age-group societies, Rendille society is described as a negative paternal-linking system that requires a minimum 3-age-set (42-year) interval between the initiation of father and sons. Thus, sons belong to the fahan(s) subsequent to that of their fathers, and each fahan can be thought of as a genealogical generation. Further, paternallinking systems aim to maintain fathers and early-born sons in the same lineage, as these are the sons that inherit property and carry on the family name. 1 Thus, each fahan n can be described as a 3-age-set genealogical generation equal to 3 x 14 = 42 years with the generation cycling through the lineages X n, Y n, and Z n ; their early-born son s are initiated into age-sets in the next fahan that have the same lineages as their father, i.e. respectively to X n+1, Y n+1, and Z n+1. The cyclical features of the Rendille age-group system makes it possible to construct a genealogical-generation model based on fahan-time intervals that is congruent with the age-set formulation. This paper builds on previous work, in particular, Engineer, Kang, Roth and Welling (2005). 2 They model the Rendille age-group rules and show the rules imply overlapping generations models where the population dynamics can be studied from either the age-cohort (age-set) or the genealogical (fahan) perspective. 3 They formally model the age-group rules and develop an age-set model, which uses 14-year periods that correspond to the interval between age-set initiations. This model is not solved analytically but rather calibrated to fit Roth s (1993, 1999) data. The paper examines political economy issues generated by conflicting interests of the various lineages (also see Engineer, Roth and Welling (2003)). The current paper in contrast concentrates on modeling approaches and analytical solution techniques, which provide generality to the results. 1 Amongst the Rendille most sons are early-born. Early-born sons are born within the first 17 years of marriage. These sons meet the minimum enrolment age of 14 years at the time of initiation (see Table 1). Sons that are less than age 14 at the initiation of their older brothers are initiated with the next age set. Some may climb back into the age-set of their older brothers and reclaim the lineage of their fathers. 2 The history and structure of the Rendille age-group rules is discussed in Beaman (1981), Roth (1993, 1999, 2001). The reader is referred to Engineer, Roth and Welling (2006) for a detailed description of the Rendille age-group system and for evidence that the Rendille in fact closely adhere to the age-group rules. 3 The overlapping generations model is a widely used theoretical model used in economics. It has for the most part not been used as an applied demographic model and genealogy is ignored. See Engineer and Welling (2004) for a further description and reference to the literature. 2

4 The age-set model in Engineer et. al (2005) uses two identifiers (dimensions), the age-set and birth order. This approach is simple and works surprisingly well in capturing most demographic transitions. However, an awkward auxiliary assumption is needed to shift some people into other groups into order to correctly model transitions that relate to lineage. Engineer et. al. (2005) show that a genealogical-generation model which uses three identifiers, fahan, lineage and birth order, can correctly map the demographic transitions. This model gets the timing correct because initiation, marriage and child-rearing activities are conceived in the age-group rules as relations among the ordered lineage age-groups in a fahan. The use of three identifiers results in the genealogical-generation model being written more compactly with 2 nd -order difference equations in fahan time rather than 5 th -order difference equations in age-set time. The 2 nd -order system is not only more accurate and intuitive, it also can be solved analytically. 3. The genealogical-generation model To study the population dynamics of the Rendille age-group system, we first identify groups of daughters in relation to their father s fahan and lineage. The lineage-based marriage customs and age-group rules are then used to identify marriages between daughter cohorts and men of various age-sets. The model is maternal in that reproductive rates only depend on when women marry not whom. The maternal reproductive assumption permits a matrix analysis with fixed coefficients. 4 In our maternal model, the process of mothers begetting daughters fully determines the population dynamics. In particular, the model is described by a system of difference equations that relate how daughters beget daughters by lineage, fahan, and birth order. We proceed by stating the assumptions upon which the lineage matrix population model is based, illustrating the model dynamics with the aid of a flow diagram, and then deriving the system of equations governing the dynamics. Assumptions of the model Our model postulates the existence of generation groups distinguished by lineage and fahan, the basic units of genealogical and generational distance used by the Rendille. Recall from Table 1 that a fahan is to a three age-set epoch equal to 3 x 14 = 42 years with each epoch cycling 4 The maternal reproductive assumption is particularly applicable to the Rendille. Amongst the Rendille, women are not allowed to bear children until they marry. Married women space bearing children roughly every three years; the Rendille live a nomadic existence in dessert environment making it difficult to carrying more than one infant at a time. All women marry; there is no shortage of husbands as polygyny is allowed. Women may continue to bear children and raise them as their husband s even after their husband has died. See Engineer, Roth and Welling (2006) for a further discussion. 3

5 through the lineages X, Y, and Z. This cyclical feature of the Rendille age-group system makes it possible to construct a genealogical model that is congruent with the age-set formulation. Because marriage and child-rearing activities are conceived as relations among the ordered generation groups, additional assumptions that address the coordination of these processes in real time are unnecessary. The following assumptions are based on the age-group structure described above and the marriage and fertility assumptions used in Engineer et. al (2005). Assumption 1 (Age-Sets, Fahans and Lineages). A fahan is a 3-age-set epoch equal to 3 x 14 = 42 years with each epoch cycling through the lineages X, Y, and Z. The Teeria lineage starts each fahan. Without loss of generality, the Teeria are associated with lineage X and non-teeria are associated with lineages Y and Z. Generation [Fahan] n consists of male age-sets (X n, Y n, Z n ). Lineage assignment is patrilineal: early-born sons of fahan n males belong to fahan n+1 and belong to the same lineage as his father, (X n+1, Y n+1, Z n+1 ). Each male generation group corresponds uniquely to an age set. A generation group, male or female, is identified by its lineage and fahan. For the female generation groups, we also need to distinguish between the early-born and late-born in each female generation group in order to pin down the population dynamics. This is because differential timing of birth leads to differences in the age at which daughters marry, even if they belong to the same lineage. Assumption 2. (Female groups). Female groups are identified by the fahan, lineage and birth order (early- vs. late-born): (X n,x' n,y n,y' n,z n,z n '), where late-born female groups are denoted by the symbol ('). Daughters are identified by the lineage of their fathers and are assigned to the fahan subsequent to their fathers. The following table relates father s group and daughters. For example, female group X n+1 represents late-born daughters from father s group X n. For simplicity similar notation is used for female and male groups. Assumption 3 (Marriage and lines of descent). All women marry. All men marry (if possible). Men and women of particular generation groups marry and rear children according to the following table: Men of generation group Are eligible to marry women-marrying-old (womenmarrying-young) from groups: Rear daughters belonging to groups: X n Z' n-1, X n ( X' n,y n ) X n+1, X' n+1 4

6 Y n X' n,y n ( Y' n, Z n ) Y n+1, Y' n+1 Z n Y' n, Z n ( Z' n, X n+1 ) Z n+1, Z' n+1 In Rendille society most daughters for line Y and Z men marry early. Daughters of line X (Teeria) men on the other hand tend to marry late. Because age-at-marriage for women is heterogeneous with respect to lineage and birth order, we assign each female group a parameter governing the proportion of women-marrying-young. The parameter restrictions and ranges are consistent with the findings in Engineer et. al. (2005). Assumption 4 (Marriage timing by lineage). Of each female group k=x,x',y,y',z,z', a fixed proportion p k are women-marrying-young and (1 - p k ) are women-marrying-old. The p k s satisfy the following: - Sepaade restricts the majority of early-born daughters of Teeria men to be womenmarrying-old: p X 0.4. Complete Sepaade is the special case p X = 0 - The vast majority of early-born daughters in the non-teeria lines are women-marryingyoung: p Y = p Z = p Virtually all late-born daughters are women-marrying-young: p X '= p Y ' = p Z ' = p', and p p'. For generality, we do not impose these marriage timing parameter restrictions until we discuss the conditions needed for growth. y Assumption 5 (Net reproductive rates). Women-marrying-young each rear n 1 early-born daughters and y n 2 late-born daughters, where n n y y 1 2 > 0. Women-marrying-old each rear o n early-born daughters. These net reproductive rates are fixed and satisfy the restrictions y y y o n n1 n2 n and o n n y 2 > 0. The net reproductive rates from mother to daughter in Assumption 5 are independent of lineage, consistent with our maternal model assumption. Women-marrying-old are assumed to rear no o y more children than women-marrying-young, n n, but at least as many children as womenmarrying-young rear from children born in the second 14-period of child bearing, n n2. This o y assumption is consistent with the evidence. 5 5 The net reproductive rates for Women-marrying-old try to have at least one son as it is sons that look after moms in old age. Women-marrying-young are likely to have already of borne at least one son and may not be interested in more children in her middle age. 5

7 In summary, assumptions 1 and 2 define the model primitives to be generation groups. Assumption 3 states that participation in marriage is universal and restricts the possible marital relations between generation groups, conditional on the woman s age at marriage. Assumption 4 assigns group-specific marriage timing parameters that vary by lineage but not by fahan. Assumptions 4 and 5 form the core of the maternal reproductive specification. Life-cycle flows To develop intuition for the population dynamics, we constructed a life-cycle flow diagram that represents the maternal model as a Markov chain with growth. In Figure 1, the Markov states correspond to the six female generation groups extant at any time: X, X', Y, Y', Z, and Z'. Transitions between states are understood to be reproductive flows with mothers of one lineage rearing daughters of another lineage by virtue of marrying fathers who belong to that lineage. In the flow diagram, the nodes represent the states and the edges represent the state transitions. The arrows are labeled with the conditional rearing rates and the reproductive flows are read from mother s group to daughter s group. For instance, there are four flows into the X state from the Z, X, X, and Y states. That is, daughters of group X are reared by mothers of groups Z, X, X, and Y. This is a direct translation of the maternal equation for group X. The other states are treated similarly. (Figure 1 here ) The maternal equations Assumptions 1-5 yield the following system of maternal equations: X n+1 = no (1-p Z' )Z' n-1 + no (1-p X )X n + ny1 p X' X' n + ny1 p Y Y n X' n+1 = ny2 p X' X' n + ny2 p Y Y n Y n+1 = no (1-p X' )X' n + no (1-p Y )Y n + ny1 p Y' Y' n + ny1 p Z Z n (1) Y' n+1 = ny2 p Y' Y' n + ny2 p Z Z n Z n+1 = no (1-p Y' )Y' n + no (1-p Z )Z n + ny1 p Z' Z' n + ny1 p X X n+1 Z' n+1 = ny2 p Z' Z' n + ny2 p X X n+1 The system is linear, second-order, and homogeneous. Observe that, conditional on birth order, the differences only show up in the proportion that marry young in each generation group (p k ). In the next section, we characterize the explicit solution to the maternal system. 6

8 To understand how these equations are derived, consider the intergenerational mapping from mother to daughter group given in Table 2. The mapping is based on the age-at-marriage rules specific to each lineage (Assumption 3). Specifically, the table identifies the daughter groups that marry men of fahan n and become mothers of daughter groups of fahan n+1. Notice that the daughter groups marrying men of fahan n are spread over fahans n-1, n, and n+1. This spread arises due to the variation within a group in terms of the timing of marriage (young or old) and of child bearing (early- or late-born). DAUGHTER GROUPS N+1 Table 2: Mother generation groups identified with daughter generation groups n+1 IDENTIFIED MOTHER GROUPS (DAUGHTER GROUPS MARRYING MEN OF FAHAN n) Z n-1 X n X n Y n Y n Z n Z n X n+1 X n+1 WMO WMO WMY WMY X n+1 WMY WMY Y n+1 WMO WMO WMY WMY Y n+1 WMY WMY Z n+1 WMO WMO WMY WMY Z n+1 WMY WMY WMY women (mothers) marrying young, WMO women (mothers) marrying old Each lineage marries its men once per fahan. Men of lineage Y from fahan n sire daughters Y n+1 and Y n+1. Consider early-born daughters Y n+1 in the far left column. Reading across the row, we see that four groups of fahan n daughters marry into lineage Y and become mothers bearing children Y n+1. These mothers comprise X n and Y n daughters marrying old, and Y n and Z n daughters marrying young. Now consider late-born daughters Y n+1. These daughters come from only those groups able to marry young: Y n and Z n. Men of lineage Y never marry Z or X daughters because these daughters are born in the same period as when the men are supposed marry, thus they are too young to marry. Under assumptions 4 and 5, the size of each daughter group is simply the number of mothers rearing daughters into that group multiplied by the associated net reproductive rate. Then it is straightforward to derive the maternal equations for lineage Y: Y n+1 = no (1-p X' )X' n + no (1-p Y )Y n + ny1 p Y' Y' n + ny1 p Z Z n Y' n+1 = ny2 p Y' Y' n + ny2 p Z Z n The maternal equations for lineages X and Z are derived similarly. This establishes the system of maternal equations asserted above. 7

9 4. The steady state In the model, mothers beget daughters within and across cohorts at different rates associated with different vital coefficients, which suggests that population growth and the age distribution will change over time. When the reproductive flows are balanced so that all extant maternal cohorts grow at constant rates, we say that the dynamical system has reached a steady state. Below, we first characterize steady state assuming one exists; then we prove that there is a unique and stable steady state, and finally, we develop conditions for steady-state growth. Characterizing the steady state Consider a steady state where all groups grow at the same rate. Let r* be the gross rate of steady-state growth. If r* is known, the steady-state population dynamics are simply: X n+1 = r*x n Y n+1 = r*y n Z n+1 = r*z' n X' n+1 = r*x' n Y' n+1 = r*y' n Z' n+1 = r*z' n Substituting into the maternal equations, r*x n = no (1-p Z' )(r*) -1 Z' n + no (1-p X )X n + ny1 p X' X' n + ny1 p Y Y n r*x' n = ny2 p X' X' n + ny2 p Y Y n r*y n = no (1-p X' )X' n + no (1-p Y )Y n + ny1 p Y' Y' n + ny1 p Z Z n r*y' n = ny2 p Y' Y' n + ny2 p Z Z n r*z n = no (1-p Y' )Y' n + no (1-p Z )Z n + ny1 p Z' Z' n + ny1 p X r*x n r*z' n = ny2 p Z' Z' n + ny2 p X r*x n we obtain a system of six linear homogeneous equations. In matrix form: x n+1 = r*x n = (r*) x n, or [ (r*) - r* I]x n = 0 where x n ( X n, X' n, Y n, Y' n, Z n, Z' n ), I is the identity matrix, and no(1-p X ) ny1 p X' ny1 p Y no(1-p Z' )(r*) -1 ny2 p X' ny2 p Y (r*) no(1-p X' ) no(1-p Y ) ny1 p Y' ny1 p Z ny2 p Y' ny2 p Z 8

10 ny1 p X r* ny2 p X r* no(1-p Y' ) no(1-p Z ) ny1 p Z' ny2 p Z' Characterizing the steady state is then equivalent to solving an ordinary eigenvalue problem. The solution is obtained by finding the roots of the characteristic polynomial f(r) = det [ (r) - r I] taking r* to be the largest root. After simplifying, the characteristic polynomial reduces to: f(r) = (r 2 - b X r + c X )(r 2 - b Y r + c Y )(r 2 - b Z r + c Z ) - r 4 (ny1) 3 p X p Y p Z where b X = no(1-p X ) + ny2 p Z' c X = no ny2(p Z' - p X ) b Y = no(1-p Y ) + ny2 p X' c Y = no ny2(p X' - p Y ) b Z = no(1-p Z ) + ny2 p Y' c Z = no ny2(p Y' - p Z ) Interestingly, there is a correspondence between the three factors that comprise the first term in f(r) and three lineage subsystems constituting a partition of the life-cycle flow diagram. For example, the second term, r 2 - b Y r + c y, can be derived as the characteristic polynomial f Y (r) that solves the maternal equations: r*x' n = ny2 p X' X' n + ny2 p Y Y n r*y n = no (1-p X' )X' n + no (1-p Y )Y n. These are the same equations for X' n and Y n above minus the interaction terms ny1 p Y' Y' n and ny1 p Z Z n. In the flow diagram, one can visualize an isolated subsystem comprised of X' and Y with no flows coming in or out of the subsystem. Similarly, the other terms of f(r) can be derived as characteristic polynomials: f X (r) = r 2 - b X r + c X from the closed maternal system of Z' n and X n, and f Z (r) = r 2 - b Z r + c Z from Y' n and Z n. The characteristic polynomial can thus be written as: f(r) = f X (r) f Y (r) f Z (r) - r 4 (ny1) 3 p X p Y p Z (2) Viewing the characteristic polynomial from the subsystems perspective allows us to intuit the propositions and their implications. This is particularly true of the case of Complete Sepaade where p X = 0 eliminates the last term which captures interactions. 9

11 Note that f(r) is of degree six. Below we show that the largest root r* is the steady-state growth rate. We then derive qualitative conditions for steady-state growth using inequalities based on the boundary condition f(r*) = f(1) = 0. Existence, uniqueness and stability of the steady state The above characterization assumes a steady state. Here we show not only does the steady state exist but also that regardless of the initial population the system converges to a steady state wherein all maternal groups grow at rate r*. To prove this result, we impose the following. regularity conditions. Restriction 1 (Regulatory conditions). p' j p k > 0 for j = Z,X,Y and k = X,Y,Z. These conditions are consistent with Assumption 4 (which requires p X < p Y, p Z p' j ) except for the singular case of Complete Sepaade, p X = 0. Proposition 1. The system of maternal equations (1) has a unique and stable steady state characterized by the largest root, r*, of the characteristic equation, f(r) = 0. In the steady state, all female groups (i.e. X, X', Y, Y', Z, Z') grow at rate r*. Proof. See the Appendix. Briefly, the proof proceeds as follows. First, we transform the maternal equations into a first-order system x n+1 = Ax n where x is a vector of state variables, one for each maternal group, and A is a square transition matrix. With the above regularity conditions, the transition matrix A has special properties that allow us to apply the Perron-Frobenius theorem for nonnegative matrices. A direct implication of the theorem is that the system has a unique and stable steady state which can be identified using r*, x 0 and the eigenvalues of A. The system has the feature that regardless of initial condition x 0 > 0, the system always converges to a steady state where all groups grow at rate r*. Our method of proof is unable to establish Proposition 1 for the case of Complete Sepaade. Technically, with p X = 0, the proof fails because the incidence digraph of A is no longer strongly connected. The reader is referred to Tuljapurkar & Caswell (1997) for a more comprehensive treatment of the relevant matrix methods. Growth in the steady state 10

12 To simplify the growth analysis and concentrate on the impact of Sepaade we restrict ourselves to the marriage rates in Assumption 4, i.e. p X < p Y =p Z =p p X' =p Y' =p Z' =p'. Substituting: f(r) = (r 2 - br + c) 2 (r 2 - b X r + c X ) - r 4 (ny1) 3 p 2 p X where: b = no(1-p) + ny2 p' c = no ny2(p' - p) b X = no(1-p X ) + ny2 p ' c X = no ny2(p' - p X ) To establish the necessary and sufficient condition for growth we impose the following restriction, which is consistent with Engineer et. al. (2006) evidence that ny2 < Restriction 2. ny2 p' < 1 Restriction 2 but not Restriction 1 is used to establish the following lemma. Lemma 1. f(1) (<,=,>) 0 iff r* (>,=,<) 1. Proof. See the Appendix. The lemma, in turn, yields the main growth proposition directly. Proposition 2. A necessary and sufficient condition for steady state growth r* ( ) 1 is given by: (ny1) 3 p 2 p X ( ) [(1-no)(1-ny2 p') + (1-ny2)no p] 2 [(1-no)(1-ny2 p') + (1-ny2)no p X ]. Proof: Rewriting the characteristic polynomial, we have: f(r) = (r 2 - br + c) 2 (r 2 - b X r + c X ) - r 4 (ny1) 3 p 2 p X Substituting, for b, c, b X and c X gives f(r) = [(r - no)(r - ny2 p') + (r - ny2)no p] 2 [(r - no)(r - ny2 p') + (r - ny2)no p S ] - r 4 (ny1) 3 p 2 p X (3) Then: f(1) = [(1 - no)(1 - ny2 p') + (1 - ny2)no p] 2 [(1 - no)(1 - ny2 p') + (1 - ny2)no p X ] - (ny1) 3 p 2 p X By Lemma 1, f(1) (<,=,>) 0 iff r* (>,=,<) 1 which yields the desired condition and proves the Proposition. We can now apply Proposition 2 to study some special parametric cases: 1. Full symmetry: p X = p = p' 11

13 r* ( ) 1 (ny)p + no(1-p) ( ) 1 where ny = ny1 +ny2. As ny no by assumption, a sufficient condition for growth, r* > 1, is no >1. Another sufficient condition for r* > 1 is ny > 1/p, which yields growth even if no = Partial symmetry: p X = p p' r* ( ) 1 (ny)p + no(1-p) + (1-no)ny2(p'-p) ( ) 1 With partial symmetry, p'-p > 0, the growth rate is greater than with full symmetry. 3. Complete Sepaade: p X = 0 r* ( ) 1 (1-no)(1-ny2 p')[(1-no)(1-ny2 p') + (1-ny2)no p] 2 ( ) 0 Inspection of this last condition reveals that the following corollary. Corollary. With Complete Sepaade, no ( ) 1 is both necessary and sufficient for r ( ) 1. With Complete Sepaade, whether the population increases or declines depend entirely on the rearing rate of women-marrying-old. Unlike the generic case (p X > 0), Complete Sepaade does not satisfy the regularity conditions in Restriction 1 which guarantee that all lineages will grow at r* in the steady state. We now show that Complete Sepaade leads to groups growing at different rates. Complete Sepaade and the Growth Rates of Individual Groups The case of Complete Sepaade is conceptually distinct because it shuts off all flows from lineage X into lineage Z. This has the effect of creating subsystems in which there are no inflows of females (as can be seen in the flow diagram Figure 2) and different group growth rates. Proposition 3. With Complete Sepaade (p X = 0), there are three nested subsystems that have distinct steady states from the Full System. 12

14 (a) Subsystem (Z ). This group is always in steady state; the population declines, r* a = ny2(p ) < 1. (b) Subsystem (Z, Z, Y ). The steady-state growth rate is bounded ny2(p ) r* b no. (c) Subsystem (Z, Z, Y, Y, X ). The steady-state growth is r* c = r* b. (d) Full System. The steady-state growth is r* = no, which is at least a large as any of the subsystems. The Teeria lineage X grows at rate r* = no in the steady state and completely dominates the population when r* > r* c. Proof: First consider the full system. We have p X = 0 and p X ' = p' which implies that the characteristic polynomial (3) reduces to f(r) = [(r - no)(r - ny2 p') + (r - ny2)no p] 2 [(r - no)(r - ny2 p')] = [f N (r)] 2 [f X (r)]; where f N (r) = f Z (r) = f Y (r) Clearly, the roots of f X (r) include no and ny2 p'; no is the largest of these two roots since ny2 p' ny2 no by Assumption 5. Solving f N (r) = 0 yields the other two roots, each with multiplicity two. We are interested in r*, the largest root r* = max(no,r). But f N (no) = (no - ny2)p 0 which means that r no since f N (r) is strictly increasing near its largest root. Thus, r* = no. We associate the roots of f X (r) and f N (r) with the growth rates of their corresponding lineages. With r* = no, f X (no) = 0 so the Teeria growth rate is no. Since f N (no) = (no - ny2)no p 0, the non- Teeria grow at a slower rate when ny2 < no. Then the Teeria population completely dominates in the sense that the ratio of non-teeria to Teeria converges to zero. Subsystem (Z ) is self-contained Z n+1 = (ny2 p') Z n which implies r* a = ny2 p'. Subsystem (Z, Z, Y ) gives characteristic polynomial f(r) = f N (r) f Z (r), where f Z (r)=(r - ny2 p'). From above f N (no) 0 which gave r no. f N (ny2 p') = ny2(p' - 1)no p 0 for p' 1 which implies r ny2 p'. Thus, for this subsystem ny2(p ) r* b no. Subsystem (Z, Z, Y, Y, X ) gives characteristic polynomial f(r) = [f N (r)] 2 f Z (r). Therefore r* c =r* b. From above r* c < no when ny2 < no. This completes the proof. Examination of the flow diagram Figure 1 provides an intuitive explanation of the proof. In the diagram setting p X = 0 results in no external flows into lineage Z. This group forms a subsystem where flows are to itself at rate ny2 p Z. Restriction 2 requires ny2 p Z < 1 so group Z declines geometrically. With lineage Z dying out there are effectively no flows into the subsystem (Z, Y ). Recall from equation (2) and Assumption 4 that the growth rate of this subsystem can be described by f Z (r) = r 2 - br + c. The proof shows that the greatest root is bounded ny2(p ) r* b no. Now flows from subsystem (Z, Z, Y ) at rate r* b flow into subsystem (Y, X ). Similarly, this latter system is also given by f Y (r) = r 2 - br + c implying r* c = r* b. Lastly there are flows from subsystem (Z, Z, Y, Y, X ) into X at rate r* c. But X replenishes itself at rate no. As no r* c, the Teeria line growth rate is never exceeded and may exceed that of the other lines in which case it effectively comprises the whole population. The above proposition only describes the steady state. Though we have not analytically proven that the system converges to a stable unique steady state, simulation analysis suggests that the 13

15 system does always converges to a steady state satisfying Proposition 3. The steady state with p X = 0 as described in Proposition 3 permits lines to grow at different rates unlike the steady state for p X > 0 as described in Proposition 1. In this sense, the Complete Sepaade does not constitute a limit point of the Incomplete Sepaade case as p X 0. However, we have found in simulations that as p X 0 the steady state does display features that correspond to points (i) and (iii) of Proposition 3. Further, even though the growth rates of the lineages in the steady state are the same, the Teeria completely dominates the non-teeria as p X 0 in the sense that the ratio of the non-teeria to Teeria population goes to zero. There can be dramatic differences in the lineage population levels in the steady state. 5. Simulations and Transition Dynamics Here we examine some of the features of the transition dynamics. Simulation 1 below is from Engineer et al (2005). In that paper, the existing data about the Rendille is used to calculate the population and marrying parameters within ranges. A simple calibration exercise is done to pick the most plausible parameters consistent with the 1990 cross sectional dominance of the Teeria lineage (age-set line X) is in the data. Simulation 1: Incomplete Sepaade Shock starting in period 5 y y p X =.3125, p =.88, p =.945, n 1 =1.286, n 2 = n o =.246 Populations of Women by Line Population Index Line X Line Y Line Z Period 14

16 The simulation attempts to roughly fit the known history. The institution of Sepaade is introduced in age-set period 5. Each age set period corresponds to 14 years and the beginning of period 5 corresponds to 1825, see Table 1. Sepaade is modelled as a shock where the parameter for women-marrying-young in age-set-line X drops from the average for the other agents which is p =.88 to p X = In actuality, compliance with the Sepaade rule was incomplete, the most plausible calibration finds that 31.25% of early-born daughters of line-set X men did not follow the tradition. The model in period 1 is started in the steady state under the assumption that all lines are symmetric. Over the first 5 periods the each line and the total population grow at 16.36% per 14- year period. This is because most almost all women marry young and have rear more than one daughter. After the Sepaade shock, most early born daughters of age-set line are forced to marry when old to age-set line X men. Thus, age set X increases and other lines decrease. Overall, the population continues to growth and converges to 4.2% growth. Now consider the implications of complete adherence to the Sepaade rule. With Complete Sepaade we know that in the steady state r*=no. As no =.246 < 1, all lineages will eventually die out. However, this might take a long time and there might be dramatic growth before this happens. This is exactly what happens in the following simulation. Simulation 2: Complete Sepaade Shock starting in period 5 y y p X = 0, p =.88, p =.945, n 1 =1.286, n 2 = n o =.246 Populations of Women by Line 7 Population Index Line X Line Y Line Z Period 15

17 In the example, the rate of convergence is particularly slow because complete Sepaade has birthing taking place every two periods. Perturbation analysis allows us to measure the contribution of each vital coefficient to population growth. Growth elasticities with respect to the vital coefficients are calculated and compared to simulation results. 6. Conclusion The paper examines a particular age-group society, the Rendille tribe of Northern Kenya, and develops and solves a lineage matrix population model representing this society. The Rendille are a particularly interesting case study because their lineage system is integrated with their ageset and generation-group organization. Age-sets of similarly aged males are organized into ageset lines along three lines of decent (which we denote X, Y, and Z). The lineage system relates fathers and their children according to genealogical generations. A rotation through the three ageset lines distinguishes the generation group of fathers from the next generation group of their children. The matrix model maps from one genealogical generation to the next to the next and reveals the genealogical dynamics inherent in the Rendille social organization. We are able to analytically solve the model and prove the existence of a unique globally stable dynamic path that converges to a (periodic) steady-state growth path. The steady state growth path is characterized with necessary and sufficient conditions for growth of the age-set lines. The rules governing marriage as they relate to lineage considerations substantially reduce the size and the growth rate of the population; they also induce periodicity in the demographics, and dramatically favour one of the age-set lines. Specifically, the institution of Sepaade lead to the Teeria (age-set line X) dominating. In the extreme case where Sepaade held back all early-born daughters from marrying young, the other age-set lines quickly disappear leaving only the Teeria line with negative population growth. More generally, the analysis suggests that matrix population modeling is a powerful tool for analyzing the social dynamics of age-group societies. As these societies incorporate similarly aged individuals into age-groups that transit the lifecycle together, they display more homogeneity than other human societies. In our analysis of the Rendille, heterogeneity was also limited by clear and simple rules on marriage and lineage. Other societies with straightforward marriage rules should also be amenable to relatively simple matrix population modeling. 16

18 We hope our analysis is also of interest to theoretical population science because of the way in which we apply the classical Perron-Frobenius methods. In biology, for instance, structuredpopulation models typically introduce population dynamics along a single dimension, usually age or size; similarly, mathematical models have been used in cultural anthropology to study the dynamics of kinship relations or age-group rules but few models tractably accommodate both kinds of dynamics simultaneously. In contrast, our model tracks reproductive transitions across lineage groups (inter-genealogical transitions) as well as across age cohorts (inter-generational transitions). The work here advances the goal of integrating the systems of socioeconomic, demographic and lineage organization within a coherent dynamic model. References Beaman, A.W. (1981), "The Rendille Age-Set System in Ethnographic Context: adaptation and integration in a nomadic society", unpublished Ph.D. dissertation, Department of Anthropology, Boston University. Engineer, M., Kang, M., Roth, E. and L. Welling (2005), Overlapping Generations Models of An Age-Group Society: The Rendille of Northern Kenya, mimeo, University of Victoria. Engineer, M., Roth, E. and L. Welling (2003), Modeling Social Rules and Experimental History: The Impact of the Sepaade Tradition on the Rendille of Northern Kenya, mimeo, University of Victoria. Engineer, M. and L. Welling (2004), "Overlapping Generations Models of Graded Age-Set Societies", Journal of Institutional and Theoretical Economics, 160.3: Roth, E. (1993), "A Re-examination of Rendille Population Regulation", American Anthropologist, 95: (1999), Proximate and Distal Variables in the Demography of Rendille Pastoralists, Human Ecology, 27.4: (2001), "Demise of the Sepaade Tradition: Cultural and Biological Explanations", American Anthropologist, 103.4:

19 Stewart, F.H (1977), Fundamentals of Age-Group Systems, Academic Press. Tuljapurkar, S. and H. Caswell (1997), eds., Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Chapman & Hall. 18

20 Table 1: Historical Timeline (incomplete) Fahan (n) Line Age-set (t) Name Circumcision Year Marriage Year 0 X 0 0 Y Z X Y Z 5 Irbaandf X 6 Ilkibugu Y 7 Libaale Z 8 D bgudo 1867a 1878a 3 X 9 Dismaala 1881a 1892a 3 Y 10 Irbaangudo 1895a 1908* 3 Z 11 D'fgudo * X 12 Irbaalis Y 13 Libaale * 4 Z 14 Irband if X 15 D'fgudo Y 16 Irbaangudo Z X Y Z 20 0 Each fahan opens with the circumcision of line X (Teeria) men and close with the marriage of line Z men 39 years later. The start of fahan n corresponds to the initiation of line X men of age-set t =3n. Sepaade begins in t =5 and end in t=17. There are 11 intervening periods. Sources: Roth 1991, 2004,? 19

21 Figure 1: Life-Cycle Flow Diagram no1(1-p x ) ny2 p x' X ny1 p x' X' no1(1-p z' ) ny2 p x ny1 p y ny2 p y no1(1-p x' ) Z' ny1 p x Y ny2 p z' ny1 p z no1(1-p y ) ny1 p z' ny1 p y' Z ny2 p z Y' no1(1-p y' ) no1(1-p z ) ny2 p y' 20

22 APPENDIX A1. The steady state: existence, uniqueness, stability Proposition 1. The system of maternal equations (1) has a unique and stable steady state characterized by the largest root, r*, of the characteristic equation, f(r) = 0. In the steady state, all female groups (i.e. X, X', Y, Y', Z, Z') grow at rate r*. Proof. The proof proceeds in three steps: a transformation of the maternal equations into a firstorder system and a characterization of the associated transition matrix, a direct application of the Perron-Frobenius theorem, and identification of the steady state. a. The first-order transformation. Recall that the maternal equations are given by: X n+1 = no(1-p Z' )Z' n-1 + no(1-p X )X n + ny1 p X' X' n + ny1 p Y Y n X' n+1 = ny2 p X' X' n + ny2 p Y Y n Y n+1 = no(1-p X' )X' n + no(1-p Y )Y n + ny1 p Y' Y' n + ny1 p Z Z n Y' n+1 = ny2 p Y' Y' n + ny2 p Z Z n Z n+1 = no(1-p Y' )Y' n + no(1-p Z )Z n + ny1 p Z' Z' n + ny1 p X X n+1 Z' n+1 = ny2 p Z' Z' n + ny2 p X X n+1 Substituting Z n-1 = (ny2 p Z' ) -1 Z n - (p X /p Z )X n in the first equation and expanding X n+1 in the fifth and sixth equations, we obtain: X n+1 = [no(p Z' -p X )/p Z ]X n + ny1 p X' X' n + ny1 p Y Y n + [(no/ny2) (1-p Z )/p Z ]Z n X' n+1 = ny2 p X' X' n + ny2 p Y Y n Y n+1 = no(1-p X' )X' n + no(1-p Y )Y n + ny1 p Y' Y' n + ny1 p Z Z n Y' n+1 = ny2 p Y' Y' n + ny2 p Z Z n Z n+1 = [ny1 no(p Z' -p X )(p X /p Z )]X n + (ny1) 2 p X p X X n + (ny1) 2 p X p Y Y n + no(1-p Y' )Y' n + no(1-p Z )Z n + [ny1 p Z' +(ny1 no/ny2) (1-p Z )(p X /p Z' )]Z n Z' n+1 = [ny2 no(p Z' -p X )(p X /p Z )]X n +ny2 ny1 p X p X X n + ny2 ny1 p X p Y Y n + [ny2 p Z' +no(1-p Z )(p X /p Z' )]Z' n This yields the first-order linear system x n+1 = Ax n where x n ( X n, X' n, Y n, Y' n, Z n, Z' n ) and: no(p Z' -p X )/p Z ny1 p X' ny1 p Y (no/ny2)(1-p Z )/p Z ny2 p X' ny2 p Y A no(1-p X' ) no(1-p Y ) ny1 p Y' ny1 p Z ny2 p Y' ny2 p Z ny1 no(p Z' -p X )(p X /p Z ) (ny1) 2 p X p X (ny1) 2 p X p Y no(1-p Y' ) no (1-p Z ) ny1 p Z' +(ny1 no/ny2)(1-p Z )(p X /p Z' ) ny2 no(p Z' -p X )(p X /p Z ) ny2 ny1 p X p X ny2 ny1 p X p Y ny2 p Z' +no(1-p Z )(p X /p Z' ) b. Characterization of A. The crux of the proof is the following result: THEOREM (PERRON-FROBENIUS). Let A be an irreducible, aperiodic, nonnegative square matrix. Then there exists an eigenvalue of A, r*, satisfying the following: 21

23 i) r* is real and positive, ii) r* > r i for all other eigenvalues r i, iii) the left and right eigenvectors u and v associated with r are uniquely positive up to scaling, iv) r* is a simple root of the characteristic equation of A. (Seneta, 1981) DEFINITION. Let T be a square matrix and let t ij (n) denote the entry in the ith row and jth column of T n where n is a positive integer. i. T is nonnegative when t ij (1) 0 for all (i,j). ii. T is irreducible when for each (i,j) there exists b such that t ij (b) > 0. iii. T is aperiodic when the greatest common divisor of those b for which t ii (b) > 0 is 1 for all i. To apply the theorem, we need only verify that the transition matrix A from the first-order transformation is nonnegative, irreducible, and aperiodic. Nonnegativity. With the regularity conditions, it is immediate that all of the entries of A are nonnegative. Irreducibility and aperiodicity. Consider the incidence digraph of A illustrated in Figure 3. The vertices and arcs correspond to maternal groups and flows, respectively. Accordingly, a ij > 0 iff there is an arc from vertex i to vertex j and a ij (n) > 0 iff there exists a path from vertex i to vertex j of length n. It turns out that A is irreducible iff the incidence digraph of A is strongly connected (Minc, 1988); that is, a path exists between any two vertices in the graph. To verify that A is strongly connected under the regularity conditions is straightforward. [With p X =0, the proof fails because the incidence digraph of A is no longer strongly connected.] Moreover, the presence of loops (1-cycles) in the graph immediately implies that A is aperiodic. Thus we have shown that A satisfies the conditions of the Theorem. 22

24 Figure 2: Incidence digraph of the transition matrix A X X' Z' Y Z Y' 23

25 c. Identification of the steady state The Perron-Frobenius theorem says that A has a single dominant eigenvalue r* > 0 with positive left and right eigenvectors u and v. We know that r*, u, and v are jointly determined by A and its characteristic equation, f(r) = 0. When u and v are normalized so that u T v = 1, the singular value decomposition (SVD) of A yields: A n = (r*) n vu T + O(n m-1 r 2 n ) where r 2 is the second-largest eigenvalue of A and m is the multiplicity of r 2. [The term O(n m-1 r 2 n ) is bounded in the sense that there exist real numbers C, D such that C n m-1 r 2 n O(n m-1 r 2 n ) D n m-1 r 2 n for all n.] Since the theorem guarantees that r* > r 2, it follows that (r*) n >> r 2 n for large n: as n increases, the second matrix term in the SVD expression becomes increasingly dominated by the first term so that A n (r*) n vu T. We then have x n = Ax n-1 = = A n x 0 (r*) n (vu T x 0 ) or x n+1 (r*) x n. Regardless of the initial population x 0 > 0 the system always converges to a state wherein all maternal groups grow at a constant rate r*: it is in this sense that the steady state is globally stable. Furthermore, since u and v are uniquely positive among normalized eigenvectors, the steady-state group distribution is unique given x 0. We conclude that (r*,u,v,x 0 ) completely identifies the steady state. A2. A necessary and sufficient condition for growth Lemma 1. f(1) (<,=,>) 0 iff r* (>,=,<) 1. Proof: Our task is to show that f(r) (<,=,>) f(r*) = 0 when r (<,=,>) r* for all values of r under consideration. First, rewrite the characteristic polynomial as: f(r) = (r 2 - br + c) 2 (r 2 - b X r + c X ) - r 4 (ny1) 3 p 2 p X = [(r - k 1 N )(r - k 2 N )] 2 [(r - k 1 X )(r - k 2 X )] - r 4 (ny1) 3 p 2 p X = [f N (r)] 2 [f X (r)] - r 4 (ny1) 3 p 2 p X where the k i X 's and k i N 's are the roots of f X (r) and f Y (r) = f Z (r), resp.; assume k 1 j k 2 j without loss of generality. Then: k j j 1 + k 2 j k 1j k 2 = b j = no(1-p j ) + ny2 p' = c j = no ny2(p'-p j ) k 1 j = (b j - d j )/2, k 2 j = (b j + d j )/2 where d j = (b j 2-4c j ) 1/2. All of the k i j 's are real since: b j 2-4c j = [no(1-p j ) + ny2 p'] 2-4no ny2(p'-p j ) = no 2 (1-p i ) 2 + 2no(1-p)ny2 p' + (ny2 p') 2-4no ny2(p'-p i ) no 2 (1-p i ) 2 + 2no(1-p i )ny2 p' + (ny2 p') 2-4no ny2(1-p i )p' [no(1-p j ) - ny2 p'] 2 0 Moreover, the k i j 's must be non-negative since b j 0 and c j 0 under the assumption p' p j. Now consider the following factorization: f j (r) = (r - no)(r - ny2 p') + (r - ny2)no p j 24

26 Note that f N (r) (<,=,>) f X (r) when r (<,=,>) ny2, given p X p. By assumption, 0 ny2 p' ny2 no and p j p' which implies that f j (0) 0, f j (ny2 p') 0, f j (ny2) 0, and f j (no) 0. Also, f X (r) (, ) 0 implies [f N (r)] 2 [f X (r)] (, ) 0. Since f(r) is continuous, we can apply the intermediate value theorem repeatedly to obtain 0 k 1 N k 1 X ny2 p' and ny2 k 2 N k 2 X no. Figure 3: Illustration of Lemma proof f N (r) f X (r) r [f N (r)] 2 f X (r) r 4 (ny1) 3 p 2 p X k 1 N k 1 X w k 2 N k 2 X r * r wp' To locate the roots of f(r) proper, note that [f N (r)] 2 [f X (r)] (<,=,>) r 4 (ny1) 3 p 2 p X whenever f(r) (<,=,>) 0 by definition and [f N (r)] 2 [f X (r)] 0 r 4 (ny1) 3 p 2 p X for ny2 p' r k 2 X. The coefficient of the r 6 term in the expansion of f(r) is positive so f(r) is strictly increasing near its largest root r*; clearly r* k 2 X. It follows that f(r) 0 for ny2 p' r r* and f(r) > 0 for r* > r. Since 1 ny2 p' by assumption, we have f(1) (<,=,>) 0 iff r* (>,=,<) 1. This completes the proof of the Lemma. References Minc, H. (1988). Nonnegative matrices. Wiley-Interscience: New York. Seneta, E. (1981). Nonnegative matrices and Markov chains, 2 nd ed. Springer: New York. 25