- Email: [email protected]
Polygamy in human and animal species
Polygamy in Human and Animal Species A. T. DASH Department of Mathematics and Statistics, University of Guelph, Guebh, Ontario, Canada NI G 2 WI
R. CRESSh4AN Department of Mathematics,
Wiifrid Laker
University,
Waterloo, Ontario, Canada N2L 3C5 Received
10 April, 1987; revised 9 September 1987
ABSTRACT An age-structured discrete-time population dent survival and fertility rates. Polygamous
model is developed that includes sex-depenas well as monogamous mating systems are
considered. If adult survival rates are sex independent, it is shown that optimum species growth is attained when the sex ratio at maturity balances the degree of polygamy of the species. Furthermore, if a positive equilibrium occurs when growth rates are density dependent, then stability criteria are established using either Perron-Frobenius theory for non-negative Leslie-like matrices or the Gershgorin Disc Theorem in more general settings.
1.
INTRODUCTION
Mathematical models are widely used to describe the population dynamics of human and animal species. The usual models in both ecology and demography are replete with many assumptions and restrictions. One of the common assumptions in the models of both these subjects (that describe the dynamics of a single species and of human populations, respectively) is that a population may be modelled by only considering one sex. That is, either the entire population is considered unisexual (asexual) or one of the sexes (usually the male) is completely ignored. However, as it became clear to several researchers in both disciplines that unisexual mathematical models in population dynamics are often unrealistic, various two-sex models started to be developed. In the last forty-five years, literature in demography has contained many two-sex models (see [7] for detailed discussions and further references). In all of these models, the mating system of human populations has been assumed to be monogamous. In fact, this is the customary mating system of most MATHEMATICAL OElsevier
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50
A. T. DASH AND R. CRESSMAN
present-day societies. Nevertheless, Murdock [4] reports that at least seventy percent of all societies exhibit some degree of polygamy. Smouse [8] has introduced population models for sexually-dimorphic animal species by considering intra and inter specific competitions (based on Lotka-Volterra equations) between the males and the females. He used these models to study the advantage of a dioecious over a monoecius mating system. However, Smouse assumed the mating system to be monogamous. This is clearly unreasonable for those animal species that exhibit a high degree of sexual dimorphism. Indeed, polygamous mating is quite common among animal species such as mammals and birds. The above discussions led Rosen [5] to develop mathematical models that describe the dynamics of both human and animal polygamous mating populations. He implicitly assumed that either the birth and death rates are independent of age or that population changes occur in such a way that the age distribution remains unaltered. However, many species contain various year classes, and recruitment to the reproductive population takes place several years after birth. From this point of view, Rosen’s polygamous mating model is somewhat unrealistic. In this paper, mathematical models will be developed to describe the dynamics of polygamous mating populations taking into account age structure and sex-dependent age of maturity. This type of model can be reduced to a non-linear delay system of difference equations. In Section 2, we introduce age-structured, sex-dependent population models that include such factors as survival and fertility rates. We establish the connection between adult sex-ratio, survival rates, and age of maturity for general reproductive functions. Section 3 examines particular reproductive functions that are tied closely to the rarer sex in the polygamous mating system. The next section analyzes the resulting dynamics. To attain the greatest species growth, the sex-ratio at maturity must be in balance with the degree of polygamy when adult males and females are equally likely to survive. If adult survival is sex-dependent, it is shown that the sex-ratio shifts to favour the “weaker” sex. Section 5 introduces density dependent factors into the age-structured polygamous-mating model. The conditions for local stability at a positive equilibrium of the age-structured dynamics are established and interpreted biologically. 2.
AGE-STRUCTURED,
SEX-DEPENDENT
POPULATION
MODEL
Consider a species where male and female offspring may be divided into distinct year classes until they reach reproductive age (i.e. maturity). Suppose the species breeds seasonally with male offspring maturing in a! years and females in /I years. Although these ,assumptions are not strictly met by human species, we will show the results qualitatively relate to human demography.
POLYGAMY IN HUMAN AND ANIMAL SPECIES
0
=:
z 20
. ..O@O
B
c
0
0
$j
2’
op
51
52
A. T. DASH AND It. CRESSh4AN
Let Mi (t)( 4 (t)) be the number of immature males (females) of age i in the population at year t. The index i varies over O,l, . . , a - 1 for males and O,l,..., j3 - 1 for females. Also, let M( t)(F( t)) be the number of mature males (females) in year t. The difference equation relating the population structure from one year to the next is given by the modified Leslie matrix of (2.1). Here Si” and Si” are the density independent survival rates of immature offspring of age i whereas SM and SF refer to adult survival rates. Notice that there is no class for adults that are no longer reproductive-they are not important in the dynamics. The functions f and g, the density dependent fertility functions, play an important role in our model. We rewrite them in the form
R&f(t)&)) =f(M(t)2’(t))M(t) R.(M(t),F(d) =dM(t),F(d)F(t)
(2.2)
where R,(R,) give the total number of male (female) offspring born in a particular year to a population with M mature males and F mature females. From Equation (2.2), the system (2.1) reduces to the following two delay-difference equations: M(t+l)
=S’+‘M(t)+
d’&(M(t-a),F(t-a)) (2.3)
F(t+l)
=SFF(t)+aFRF(M(t-P),F(t-P)).
The constants u~=S~~,.S~~. me. *SF and aF=S$?,.S$?,+ ... *SC are the respective proportions of male and female offspring that survive until adulthood, We will call these constants the pre-adult surviva.l rates. For the rest of the paper we will assume that the sex-ratio r of male versus female offspring is fixed at birth by the species. However, the total number of offspring is not fixed and indeed depends on both M(t) and F(t) in general. Thus &,(M(t),F(t))
=r&(M(t),F(t))
=rR(M(t),F(t))
where we define the number of female offspring R(M(t), reproductive function. The system (2.3) now becomes M(t+l)
=S”M(t)+
(2.4)
F(t)) as the
u%R(M(t-a),F(t-a))
F(t+l)=SFF(t)+uFR(M(t-@,F(t--8)).
(2.5)
POLYGAMY
IN HUMAN
AND ANIMAL
53
SPECIES
In the final analyses of this section, we make general observations on the limiting behaviour of adult sex ratios and interpret these results biologically. We divide the observations into two cases based on the relative magnitudes of adult survival fractions. Case A: SM=SF=S We can eliminate the reproductive
function
a”rF(t+/3+1)-aFM(t+cr+1)
=S(uMrFit+P)-uFM(t+a)).
from (2.5) to obtain
(2.6) This is very similar to the differential equation approach without time delay used in Rosen [5]. The solution to (2.6) is the geometric sequence u”rF(r+j3)-uFM(r+a)=AS’ for some constant A. Since O
F(r+P)
z----r
uM uF
(2.7)
That is, the sex-ratio r at birth when adjusted by the pre-adult survival rates and the maturity ages yields the adult sex-ratio in the long run (i.e. as r gets large). Case B: SM # SF. Assume SF > SM. Equation (2.6) changes to u”rF( r + j3 + 1) - u FM( r + CY+ 1) = SFuMrF( r + j3) - S”uFM( r + a). This new equation cannot be solved explicitly without knowing R but it can be rearranged to yield a qualitative result: u”rF(r+j3+1)-uFM(r+a+1)=SF(uMrF(r+/3)-uFM(r+a)) +(SF-S”)uFM(r+a). Since (SF - S”)aFM( r + a) is positive, solution of (2.8) to that of (2.6), that
b t-co
M(r+a) sup F(r+B)
one can show by comparing
uM
GTr.
(2.8) the
(2.9)
(The limit supremum of a sequence { u, } is lim, _ m(sup{ a,, a, + 1,. . . }).) In biological terms, formula (2.9) states that the adult sex-ratio of males to females when adult females have a higher survival rate is no larger than
A. T. DASH AND R. CRESSMAN
54 the same ratio when adults Clearly, a desirable result.
3.
of both
REPRODUCTIVE FUNCTIONS MATING SYSTEMS
sexes are equally
likely to survive.
OF POLYGAMOUS
To proceed further, we need some assumptions on the reproductive function R. If there are M(F) mature males (females) and the species has a monogamous mating system, then it is reasonable that R( M, F) should depend on the number of mature individuals of the rarer sex. With this in mind, Kendall [3] suggested R( M, F) = c min( M, F) where c is a constant for the species and min( M, F) is the minimum value of M and F. For polygamous mating where each male on the average has k mates each reproductive season, Rosen [5] has considered R(M,F)
=cmin(kM,F).
(3.1)
When k = 1 in this equation, we clearly have Kendall’s monogamous mating. The case of one male mating on average with more than one female (i.e. polygyny, k > 1) is extremely common in human societies while polyandry (k < 1) is quite rare in humans but does occur more often in animal species [2,4]. It should be emphasized that a reproductive function like (3.1) is density independent in the sense that if one doubles the number of males and females, then the total number of offspring also doubles. (In mathematical terms, such R are called homogeneous functions of two variables with order 1.) These types of reproductive functions cannot yield a positive limiting population size in the long run. As in a single-sex, density-independent population model, either the species becomes extinct or grows infinitely large in total number. The next section analyzes the exponential behaviour when R is given by (3.1). Section 5 on the other hand allows density dependence in the reproductive function and examines behaviour near resulting equilibria. 4.
DENSITY
Consider namely,
INDEPENDENT
POLYGAMOUS
the system that results
MODELS
from the Equations
M(t+a+l)
=S”M(f+a)+aMrcmin(kM(t),F(f))
F(t+B+l)
=SFF(t+/3)+aFcmin(kM(t),F(t)).
(2.5) and (3.1);
(4.1)
We will examine
the two cases in Section 2 but now with respect to (4.1).
POLYGAMY
Case A:
IN HUMAN
AND ANIMAL
SPECIES
55
SM = SF= s.
From (2.7), it is not difficult to solve (4.1) for large values of r. With this asymptotic behaviour, we will determine what sex ratio r maximizes the species growth rate. More explicitly, if the survival rates S, u M, aF; the degree of polygamy k; and the total number of offspring for a given M and F are all independent of r, we answer the question, for what r does the total adult population size grow the fastest? Although the discussion leading to (4.5) below assumes CY = /3, a similar conclusion holds in the general case except that adult population sire must be taken as M(t + a) + F(t + /I) to adjust for different maturity ages. We have taken maturity ages equal since the mathematical complications of the general case unnecessarily obscure the main point. The total number of offspring is (a”rc + aFc)min( M, F). If this is to depend only on M and F, then c must change with r according to b C= (u"r+uF)
for some constant b. Add the two equations of (4.1) to obtain M(t+cr+l)+F(t+~~+l)
=S(M(r+a)+F(t+a)) +bmin(kM(t),F(t)).
Asymptotically, M(t)
= ( u”r/uF) F( t), so that F(t+a)+bmin
That is, F(t+a+l)
=SF(t+a)+
uM;;uFmin
(4.2)
This is a linear, homogeneous difference equation of order Q + 1 whose solution must be a linear combination of solutions of the form t”x’ where x is a root of the polynomial equation P( x; r) = x4+l - Sx” -
buF
u”r + uF
(4.3)
and rr is a nonnegative integer less than the multiplicity of the root x. By Appendix A, there is exactly one simple (i.e. multiplicity l), positive root x0 larger than the absolute value of all other roots, real or complex. In
56
A. I-. DASH AND R. CRESSMAN
the long run, the term involving ~6 will dominate and F(t) will behave like Ax; asymptotically (F(t) - Ax;). In particular, if x0 is less than 1, F(t) is asymptotically zero and hence the species will become extinct. Although our analysis includes this possibility, we are more interested in situations where X,>l. We proceed to find r for which this positive root is as large as possible. Suppose
%
(4.4)
Then P(x; r) = xa+l - Sx” -(bku”r/aMr
dP(x;r) = Jr
bku”( -
+ uF) and
u”r + u”) - bka”ruM ( u”r + u”)’
1
bku”uF
=-
( u”r + IJ”)”
-=c0.
With x fixed, P(x; r) decreases with r. That is, P( x; ri) > P( x; r2) whenever r, < r,. If P(x,; rl) = 0, then P(x,; r2) < 0. From the shape of pendix A, the positive root of P(x; r2) must be larger than means is that r should be as large as possible while still have total population size growing fastest. The analysis for ( ku”r/uF) > 1 proceeds analogously.
the graph in Apx,,. What all this satisfying (4.4) to From (4.3),
baF
P(x;r)=x*+‘-Sx”-
u”r+uF’
Since the partial derivative is now positive, we conclude that r should be as small as possible. From these two factors, we now know that population size will increase at its fastest rate in the long run if
kaMr=l uF
.
(4.5)
The biological significance of (4.5) is that the sex ratio of birth should be in perfect balance with the degree of polygamy of those offspring that survive to adulthood. This balance is very clear if we let uM = uF for the moment. If one male mates with k females on average, from (4.5) the growth is maximized when k females are born for each male. Alternatively, when k = 1 as in most modem societies, result (4.5) says the sex-ratio should be the same as female to male pre-adult survival rates. This
POLYGAMY
IN HUMAN
AND ANIMAL
57
SPECIES
validates the argument that the higher male infant mortality rate of humans explains why r is slightly larger than 1 in modern western societies. The interesting possibility of significant shifts in r for human societies exhibiting a higher degree of polygamy will not be pursued here. Rather, we offer another possible explanation why r > 1 in humans. If adult females survive longer than adult males, a careful analysis of Case B below will show that the adult sex-ratio L of (4.6) is less than that in Case A. This implies that r must now be larger to maximize the growth of population size. Since both SF and uF are larger than SM and u”, respectively, for humans, these parameters of the model both act to make the offspring sex-ratio larger than 1. Case B: SF > SM. As before, take (Y= /3, but now assume that there exists an L such that
M(t)
h
r+m
EL
F(f)
(4.6)
.
That is, we assume that an adult sex-ratio eventually emerges. We will see that for some parameters such an L is impossible. In these cases, it will also be shown the species becomes extinct. With (4.6) the asymptotic behaviour of (4.1) reduces to M(t+a+l)=S”M(t+cy)+aMrcmin(kL,l)F(~) (4.7) F(t+ol+1)=S~F(t+cr)+aknin(kL,1)F(t). From the last equation
F(t) - Ax; where x,, is the positive root of
P,( x; r; L) = xa+l - SFxa - uFc min( kL , 1) . Substitution of
(4.8a)
of M(t) - A Lx; into (4.7) implies that x,, must also be a root
P,(x;r;
L) =xa+l
- S”xa - u”rcmin(
k,l/L).
(4.8b)
As an aside to help clarify the role of L, notice that Equations (4.8), when applied to Case A, can only have the same root if uFc min( kL,l) = u”rc min( k, l/L) or u’c = u”rc/L. That is, the only possible adult sex-ratio L is simply (u”/uF)r, a result we already knew from (2.7). Moreover, when SF> S”, then
S”x”0 + +min(
kL,l)
= SFx; + uFcmin( kL,l),
58
A. T. DASH AND R. CRESSMAN
and this implies
a”rcmin( kL,l) L=
(S~-S~)X~+&min(kL,1)
<%
which is consistent with (2.9). There is no guarantee that Equations (4.8) will have a common root for some adult sex-ratio L. Indeed we prove the following: There is an L for which Equations (4.8) have a common root if and only if
P,( SF; r;O) -e0. Furthermore,
(4.9)
when (4.9) holds, this L is unique.
Proof. For large I, the positive root of P,(x; r; Z) is approximately S”; and as I decreases, the positive root increases (see the shape of these functions in Appendix A), On the other hand, PF( x; r; 0) has root SF and the root of PF(x; r; I) decreases as 1 decreases. Thus the smallest root of PF( x; r; I) is SF and this occurs when I = 0. Since SM < SF, Equations (4.8) can both be satisfied if and only if the positive root of PM(x; r; 0) is larger than SF. This is equivalent to condition (4.9) by Appendix A. Furthermore, as I changes, the positive root of either (4.8a) or (4.8b) also changes. This implies there can be at most one I where (4.8a) and (4.8b) have the same root. The failure of condition (4.9) has an interesting biological consequence. If (4.9) is not true, then (SF)a” - S”(SF)o - a”rck > 0. Since SF< 1, we see from Appendix A that (4.10)
l-S”-u”rck>O.
Choose 6 < 1 such that SM + a”rck < S from (4.10). From (4.1), we have M(t+a+l)~S~M(t+a)+u~rckM(t) Q ( SM + u”rck)max(
M( t + a), M(t))
+8max(M(t+a),M(t)). It is not difficult to see from this inequality that lim,, ,M(t) = 0. This, in turn, implies lim,, mF(t) = 0. In summary, if there is no L for which (4.8a,b) are satisfied (i.e. (4.9) fails), then the species must tend to extinction in the long run.
59
POLYGAMY IN HUMAN AND ANIMAL SPECIES
5.
DENSITY
DEPENDENT
MODELS
In this section, we return to the system of difference equations in Section 2,
developed
M(t+ol+l)=S~M(t+a)+a%R(M(t),F(t)) (5.1) F(t+/3+1)
=sFF(t+p)+uFR(M(t),F(t)),
to study the local stability at a positive equilibrium point (M*, F*). That is, M* and F* are both positive and satisfy M* = S”M* + a”rR( M*, F*), F* = SFF* + aFR( M*, F*). As noted previously, such an equilibrium can exist only if the reproductive function displays some density dependence, in contrast to the situation in Sections 3 and 4. Stability is determined through the Jacobian matrix J corresponding to the matrix equation (2.1) but with the reproductive function (2.4) substituted appropriately. J is a square matrix of order (Y+ /I + 2 whose ij th entry is J._= aZ;(t+l) IJ aZJ(r)
where the partial
derivatives
are evaluated
=
,..., cx i=ar+l
M(t) Fr_ca+9(r)
i=cr+2,...,a+j?+l’ i=a+j?+2
F( t) The calculation
at (M*, F*) and
i=l
M,-,(t) ‘i(‘)
’
yields 0
...
0
C?R
r-gjq
0
.”
0
aR
‘37
GM
J= 0
g
0
L
ss”-1
SF
60
A. T. DASH AND R. CRESSMAN
By standard theory, (M*, F*) will be locally stable if all eigenvalues of J have modulus less than 1 and unstable if any eigenvalue has modulus greater than 1. From Appendix B, the characteristic equation is
det(xZ-J)
= [xa(x-~~)--~~~I~][x~(x-~f)--(rf~] _ &feF?!?
_?!? b’M dF’
(5.2)
Eigenvalues of J are the roots of (5.2). If we assume that the total number of offspring increases if either the number of mature males or females increases then all entries of J are nonnegative and, in fact, J is a positive matrix. Such matrices, called primitive in Perron-Frobenius theory [8], have a positive eigenvalue x0 that is larger than the modulus of any other root of (5.2). Stability will occur if x0 < 1 but not if x0 > 1. In other words, we want to know if det(xZ-J)>Oforallx>l. This is true if and only if all three of the following satisfied: (1)
l-S”-ru”--->O
(2)
1-SF-uF-->O
(3)
(l-S”-
conditions
are
8R dM dR aF
r+&)(
(5.3) I-SF-&+
>r#~~. 1
Proof. Suppose (5.3) is true. From Appendix A, both factors inside the square bracket of (5.2) are increasing for x > 1. By the last condition of (5.3), det(xZ- J) > 0 for x al. Now suppose one of the three conditions of (5.3) is not satisfied. We want to show x,, > 1. This is clear if (3) is not true. Consider the case where (1) is not true. Now there is some x1 > 1 with xF(xi - s”) - ra”( dR/ilM) = 0. For this value, det(x,Z - J) < 0. Since det( xZ - J) is eventually positive, x,, > xi > 1. A similar argument is used if (2) is not true. Condition (5.3) can be further reduced to a form that is quite easy to check for a given reproduction function. At equilibrium, (1- S”)M = a”rR(M, F) and (1- SF)F = uFR(M, F). With this substitution, the three
POLYGAMY
IN HUMAN
AND ANIMAL
61
SPECIES
parts of (5.3) simplify to R(M, F) > M( aR/aM), R(M, F) > F( aR/aF), and R( M, F) > M( aR/aM) + F( aR/aF), respectively. Since the partials are positive, the last of these inequalities implies the other two. We have proven Theorem: An equilibrium (M*, F*) of the system (5.1) is stable if and only if aR R(M,F)>Mm+FTF.
aR
(5.4)
Species that have, at equilibrium, either a large number of offspring or a reproduction function that remains relatively constant at nearby adult population sizes will exhibit the most stability by (5.4). A surprising aspect of this analysis is that the stability condition (5.4) does not involve any survival rate parameters or the sex-ratio at birth. Let us briefly consider the possibility that R (M, F) does not increase for both M and F increasing. For instance, suppose the number of males over a certain threshold has no effect on total number of offspring. Since the Jacobian is no longer primitive, we will use Gerschgorin’s Disc Theorem [9] to test for stability. Applied to the columns of matrix J, the theorem states that the eigenvalues of J are contained in the union of the following discs in the complex plane: a+p+2
Dj=
x: Ix-J;~I<
c i-l
i
i#j
Jlj
, j=1,2
,..., a+/3+2.
i
Unless j = LY+ 1 or OL+ /I + 2, it is clear Dj is a subset of the unit disc. Thus the equilibrium point will be stable if both Du+1= Da+/?+2
=
(x:
Ix-SMIC(I+1)$
{x:
Ix-S’,<(r-l)%)
are also subsets. We conclude all eigenvalues
of J are in the unit disc if
(5.5) We wish to emphasize that these new sufficient conditions (5.5) for stability are certainly not necessary. If, in fact, J is primitive, the inequalities of (5.3)
62
A. T. DASH AND R. CRESSMAN
are
satisfied in many cases when those of (5.5) are not. The problem is that the converse of Gerschgorin’s Theorem is not very useful here. Both Da +1 could be outside the unit disc and the system remain stable. We and L,uz will not pursue this possibility further except to refer the interested reader to a paper by Agnew [l] where delay-difference equations of the form (2.3) are examined using the approach of this paragraph. We argue instead that the natural approach to stability questions for our sex-dependent, age-structured populations is through nonnegative matrices that lead to condition (5.4). Thus the test for stability at a typical equilibrium does not depend on either the population’s survival or sex-ratio parameters. 6.
DISCUSSION
AND EXAMPLE
The techniques and conclusions of Sections 4 and 5 are quite different but there is a tenuous connection between the limiting adult sex-ratio of both sections that should be mentioned. At an equilibrium of (5.1)
M* F*
-=I-
CTM1-P uF l-SM.
(6.1)
With
unbounded growth in Section 4 we have L = r(u”/u’) or L G when S”= SF or SF> S”, respectively. When S”= SF, the limiting sex-ratios of both sections are the same. One might conjecture r(u”/uF)
uM l-SF L=r(l,
I_SM
for all cases of Section 4. A numerical example will show density independent growth is more subtle than this. Consider a species with equal sex-ratio at birth (r = l), no time delay to maturity (a = p = 0 and uM = uF= 1) and with SF> SM. From (4.1), the difference equations are
M(t+l)
=S”M(t)+cmin(M(t),F(t))
F(t+l)=SFF(t)+cmin(M(t),F(t)).
Eventually
(i.e. for all f > to) F( t) will be greater than M(t)
since SF > SM.
POLYGAMY
63
IN HUMAN AND ANIMAL SPECIES
Take t, = 0 and solve to obtain
M(r) =(S"+c)'M(O)
for some constant
Em I--trn
and
F(t)
=A(SF)‘+
CM(O) SM+c_SSF
(SM+c)t
A. Then
M(O)
M(t)= F(t)
+
CM(O) s”+c-SF
0
ifSF>SM+c
1-p SF-SM
= i
ifSF
C
It is only the last case SM + c > 1 > SF that we are interested the species become extinct. Comparison with (6.1) shows 1_
in. Otherwise,
SF-SM =- l-SF C
l-SM
if and only if SM + c =l. Since SM + c > 1, the adult sex ratio for unbounded growth is somewhere between 1 and the equilibrium ratio of (6.1). The more realistic time-delay models are much more difficult to solve completely for M(t) and F(t) but the analysis developed in this paper calculates limiting adult sex ratios without these explicit solutions. The underlying assumptions that led to these models are quite general but a few limitations are worth discussing. Perhaps the most serious limitation is that adults are put into two mature classes, one for each sex, with no provision for non-reproductive adults or for age-dependent reproduction rates. The obvious answer is to include more adult age classes. However, the new delay-difference equations that emerge become unmanageable. Secondly, the reproduction functions and, through it, the actual polygamous mating system were assumed to depend only on adult population sizes M and F. Instead, individual characteristics might be important in any discussion of the degree of polygamy. Such considerations quickly go beyond the population dynamics of this paper into areas such as selective mating and generic effects. We hasten to add that these limitations do not diminish the importance of the results presented in this paper. As in most biological models, the power
A. T. DASH AND R. CRESSMAN
64
of mathematics lies not only is specific computations structural unity brought to a collection of models. APPENDIX
but more often in the
A
Consider the polynomial P(x) = x”+’ - uxn - b where n is a positive integer and a, b are positive constants. The graph of P(x) is shown below for x > 0. Properties of P(x) that we wish to note are (1) P(a) = P(0) and P has one extremum-a minimum at x =
&
,
(2) P(x) has one positive root x0 which is larger than a, (3)
All other (possibly complex) roots have modules less than x,,.
Proof: (1) (3)
and (2) are obvious from the graph and P'(x). If P(z)= 0 where z = reie and 0 < 0 < 2n, then rn+lei(n+l)B
=
urneinO
+
b.
(A.11
Thus p+l = Iurn@@ + bl Q ur” + b with equality if and only if eine is positive. If erne is positive and 0 < 0 < 2n, then e’(“+‘)’ is not positive and (A.l) cannot be satisfied. That is, rn+’ < ur” + b. But rn+’ > urn + b for t-ax,, bythegraph.Thus r
y=P (x>
I
65
POLYGAMY IN HUMAN AND ANIMAL SPECIES
APPENDIX
B
To discuss stability in Section 5 we need the determinant of a matrix of the form
-b
0
..
0
. .
0
-a
0
-b
x x
-s,“-,
X-SM --c
-x$
-d x
x -
Expansion along the first row yields
xlM1IIM21+(-1) “+‘~l~~II~21+(-~)~+B~l~~ll~~I where -“s,
x
MI =
X
I
.Y
-s,“-,
x-SM
-d
P2l= X
SaFl x-s’ =xa(x-S~)+(-1)8+1d(-1)PS,F.....S~1 =x8(x-SF)-ddgF
=(-l)“s~.
. . . .s,“_l=(-l)aaM
$7
x-SF
66
A. T. DASH AND R. CRESSMAN
and x -
s,F
l&l =
= ( -l)p+lcaF. That is, the required determinant is [X”(X-s~)+(-l)a+1a(-l)ao~][X~(X-s~)-A7~] +(-1)
“++I( -l)“lP(
-l)p+*CoF
= [ xy x - S”) - aaM] [ xfi( x - SF) - daF] - bca”uF.
REFERENCES T. T. Agnew, Stability and exploitation in two-species discrete time population models with delay, Ecolog. Model. 15:235-249 (1982). S. T. Emelen and L. W. Oring, Ecology, sexual selection, and the evolution of mating system, Science 197:215-223 (1977). D. G. Kendall, Stochastic processes and population growth, J. Roy. Stat. Sot. B(11):230-284 (1949). G. P. Murdock, World ethonographic sample, Amer. Anthropol. 59~664-687 (1957). K. Rosen, Mathematical models for polygamous mating systems, Math. Model. 4:27-39 (1983). E. Seneta,
Non-negative
New York, 1980. D. Smith and N. Keyfitz, Vol. 6, Springer-Verlag,
Matrices and Markov Chains, Second Edition, Springer-Verlag, Mathematical
Demography,
Selectedpapers.
Biomathematics,
Berlin, 1977.
P. E. Smouse, The evolutionary advantages of sexual dimorphism, Theoret. Pop. Biol. 2~469-481 (1971). G. W. Stewart, Introduction to Matrix Computations, Academic Press, 1973.
R. CRESSh4AN Department of Mathematics,
Wiifrid Laker
University,
Waterloo, Ontario, Canada N2L 3C5 Received
10 April, 1987; revised 9 September 1987
ABSTRACT An age-structured discrete-time population dent survival and fertility rates. Polygamous
model is developed that includes sex-depenas well as monogamous mating systems are
considered. If adult survival rates are sex independent, it is shown that optimum species growth is attained when the sex ratio at maturity balances the degree of polygamy of the species. Furthermore, if a positive equilibrium occurs when growth rates are density dependent, then stability criteria are established using either Perron-Frobenius theory for non-negative Leslie-like matrices or the Gershgorin Disc Theorem in more general settings.
1.
INTRODUCTION
Mathematical models are widely used to describe the population dynamics of human and animal species. The usual models in both ecology and demography are replete with many assumptions and restrictions. One of the common assumptions in the models of both these subjects (that describe the dynamics of a single species and of human populations, respectively) is that a population may be modelled by only considering one sex. That is, either the entire population is considered unisexual (asexual) or one of the sexes (usually the male) is completely ignored. However, as it became clear to several researchers in both disciplines that unisexual mathematical models in population dynamics are often unrealistic, various two-sex models started to be developed. In the last forty-five years, literature in demography has contained many two-sex models (see [7] for detailed discussions and further references). In all of these models, the mating system of human populations has been assumed to be monogamous. In fact, this is the customary mating system of most MATHEMATICAL OElsevier
Science
52 Vanderbilt
BIOSCIENCES Publishing
88:49-66
49
(1988)
Co., Inc., 1988
Ave., New York, NY 10017
0025-5564/88/$03.50
50
A. T. DASH AND R. CRESSMAN
present-day societies. Nevertheless, Murdock [4] reports that at least seventy percent of all societies exhibit some degree of polygamy. Smouse [8] has introduced population models for sexually-dimorphic animal species by considering intra and inter specific competitions (based on Lotka-Volterra equations) between the males and the females. He used these models to study the advantage of a dioecious over a monoecius mating system. However, Smouse assumed the mating system to be monogamous. This is clearly unreasonable for those animal species that exhibit a high degree of sexual dimorphism. Indeed, polygamous mating is quite common among animal species such as mammals and birds. The above discussions led Rosen [5] to develop mathematical models that describe the dynamics of both human and animal polygamous mating populations. He implicitly assumed that either the birth and death rates are independent of age or that population changes occur in such a way that the age distribution remains unaltered. However, many species contain various year classes, and recruitment to the reproductive population takes place several years after birth. From this point of view, Rosen’s polygamous mating model is somewhat unrealistic. In this paper, mathematical models will be developed to describe the dynamics of polygamous mating populations taking into account age structure and sex-dependent age of maturity. This type of model can be reduced to a non-linear delay system of difference equations. In Section 2, we introduce age-structured, sex-dependent population models that include such factors as survival and fertility rates. We establish the connection between adult sex-ratio, survival rates, and age of maturity for general reproductive functions. Section 3 examines particular reproductive functions that are tied closely to the rarer sex in the polygamous mating system. The next section analyzes the resulting dynamics. To attain the greatest species growth, the sex-ratio at maturity must be in balance with the degree of polygamy when adult males and females are equally likely to survive. If adult survival is sex-dependent, it is shown that the sex-ratio shifts to favour the “weaker” sex. Section 5 introduces density dependent factors into the age-structured polygamous-mating model. The conditions for local stability at a positive equilibrium of the age-structured dynamics are established and interpreted biologically. 2.
AGE-STRUCTURED,
SEX-DEPENDENT
POPULATION
MODEL
Consider a species where male and female offspring may be divided into distinct year classes until they reach reproductive age (i.e. maturity). Suppose the species breeds seasonally with male offspring maturing in a! years and females in /I years. Although these ,assumptions are not strictly met by human species, we will show the results qualitatively relate to human demography.
POLYGAMY IN HUMAN AND ANIMAL SPECIES
0
=:
z 20
. ..O@O
B
c
0
0
$j
2’
op
51
52
A. T. DASH AND It. CRESSh4AN
Let Mi (t)( 4 (t)) be the number of immature males (females) of age i in the population at year t. The index i varies over O,l, . . , a - 1 for males and O,l,..., j3 - 1 for females. Also, let M( t)(F( t)) be the number of mature males (females) in year t. The difference equation relating the population structure from one year to the next is given by the modified Leslie matrix of (2.1). Here Si” and Si” are the density independent survival rates of immature offspring of age i whereas SM and SF refer to adult survival rates. Notice that there is no class for adults that are no longer reproductive-they are not important in the dynamics. The functions f and g, the density dependent fertility functions, play an important role in our model. We rewrite them in the form
R&f(t)&)) =f(M(t)2’(t))M(t) R.(M(t),F(d) =dM(t),F(d)F(t)
(2.2)
where R,(R,) give the total number of male (female) offspring born in a particular year to a population with M mature males and F mature females. From Equation (2.2), the system (2.1) reduces to the following two delay-difference equations: M(t+l)
=S’+‘M(t)+
d’&(M(t-a),F(t-a)) (2.3)
F(t+l)
=SFF(t)+aFRF(M(t-P),F(t-P)).
The constants u~=S~~,.S~~. me. *SF and aF=S$?,.S$?,+ ... *SC are the respective proportions of male and female offspring that survive until adulthood, We will call these constants the pre-adult surviva.l rates. For the rest of the paper we will assume that the sex-ratio r of male versus female offspring is fixed at birth by the species. However, the total number of offspring is not fixed and indeed depends on both M(t) and F(t) in general. Thus &,(M(t),F(t))
=r&(M(t),F(t))
=rR(M(t),F(t))
where we define the number of female offspring R(M(t), reproductive function. The system (2.3) now becomes M(t+l)
=S”M(t)+
(2.4)
F(t)) as the
u%R(M(t-a),F(t-a))
F(t+l)=SFF(t)+uFR(M(t-@,F(t--8)).
(2.5)
POLYGAMY
IN HUMAN
AND ANIMAL
53
SPECIES
In the final analyses of this section, we make general observations on the limiting behaviour of adult sex ratios and interpret these results biologically. We divide the observations into two cases based on the relative magnitudes of adult survival fractions. Case A: SM=SF=S We can eliminate the reproductive
function
a”rF(t+/3+1)-aFM(t+cr+1)
=S(uMrFit+P)-uFM(t+a)).
from (2.5) to obtain
(2.6) This is very similar to the differential equation approach without time delay used in Rosen [5]. The solution to (2.6) is the geometric sequence u”rF(r+j3)-uFM(r+a)=AS’ for some constant A. Since O
F(r+P)
z----r
uM uF
(2.7)
That is, the sex-ratio r at birth when adjusted by the pre-adult survival rates and the maturity ages yields the adult sex-ratio in the long run (i.e. as r gets large). Case B: SM # SF. Assume SF > SM. Equation (2.6) changes to u”rF( r + j3 + 1) - u FM( r + CY+ 1) = SFuMrF( r + j3) - S”uFM( r + a). This new equation cannot be solved explicitly without knowing R but it can be rearranged to yield a qualitative result: u”rF(r+j3+1)-uFM(r+a+1)=SF(uMrF(r+/3)-uFM(r+a)) +(SF-S”)uFM(r+a). Since (SF - S”)aFM( r + a) is positive, solution of (2.8) to that of (2.6), that
b t-co
M(r+a) sup F(r+B)
one can show by comparing
uM
GTr.
(2.8) the
(2.9)
(The limit supremum of a sequence { u, } is lim, _ m(sup{ a,, a, + 1,. . . }).) In biological terms, formula (2.9) states that the adult sex-ratio of males to females when adult females have a higher survival rate is no larger than
A. T. DASH AND R. CRESSMAN
54 the same ratio when adults Clearly, a desirable result.
3.
of both
REPRODUCTIVE FUNCTIONS MATING SYSTEMS
sexes are equally
likely to survive.
OF POLYGAMOUS
To proceed further, we need some assumptions on the reproductive function R. If there are M(F) mature males (females) and the species has a monogamous mating system, then it is reasonable that R( M, F) should depend on the number of mature individuals of the rarer sex. With this in mind, Kendall [3] suggested R( M, F) = c min( M, F) where c is a constant for the species and min( M, F) is the minimum value of M and F. For polygamous mating where each male on the average has k mates each reproductive season, Rosen [5] has considered R(M,F)
=cmin(kM,F).
(3.1)
When k = 1 in this equation, we clearly have Kendall’s monogamous mating. The case of one male mating on average with more than one female (i.e. polygyny, k > 1) is extremely common in human societies while polyandry (k < 1) is quite rare in humans but does occur more often in animal species [2,4]. It should be emphasized that a reproductive function like (3.1) is density independent in the sense that if one doubles the number of males and females, then the total number of offspring also doubles. (In mathematical terms, such R are called homogeneous functions of two variables with order 1.) These types of reproductive functions cannot yield a positive limiting population size in the long run. As in a single-sex, density-independent population model, either the species becomes extinct or grows infinitely large in total number. The next section analyzes the exponential behaviour when R is given by (3.1). Section 5 on the other hand allows density dependence in the reproductive function and examines behaviour near resulting equilibria. 4.
DENSITY
Consider namely,
INDEPENDENT
POLYGAMOUS
the system that results
MODELS
from the Equations
M(t+a+l)
=S”M(f+a)+aMrcmin(kM(t),F(f))
F(t+B+l)
=SFF(t+/3)+aFcmin(kM(t),F(t)).
(2.5) and (3.1);
(4.1)
We will examine
the two cases in Section 2 but now with respect to (4.1).
POLYGAMY
Case A:
IN HUMAN
AND ANIMAL
SPECIES
55
SM = SF= s.
From (2.7), it is not difficult to solve (4.1) for large values of r. With this asymptotic behaviour, we will determine what sex ratio r maximizes the species growth rate. More explicitly, if the survival rates S, u M, aF; the degree of polygamy k; and the total number of offspring for a given M and F are all independent of r, we answer the question, for what r does the total adult population size grow the fastest? Although the discussion leading to (4.5) below assumes CY = /3, a similar conclusion holds in the general case except that adult population sire must be taken as M(t + a) + F(t + /I) to adjust for different maturity ages. We have taken maturity ages equal since the mathematical complications of the general case unnecessarily obscure the main point. The total number of offspring is (a”rc + aFc)min( M, F). If this is to depend only on M and F, then c must change with r according to b C= (u"r+uF)
for some constant b. Add the two equations of (4.1) to obtain M(t+cr+l)+F(t+~~+l)
=S(M(r+a)+F(t+a)) +bmin(kM(t),F(t)).
Asymptotically, M(t)
= ( u”r/uF) F( t), so that F(t+a)+bmin
That is, F(t+a+l)
=SF(t+a)+
uM;;uFmin
(4.2)
This is a linear, homogeneous difference equation of order Q + 1 whose solution must be a linear combination of solutions of the form t”x’ where x is a root of the polynomial equation P( x; r) = x4+l - Sx” -
buF
u”r + uF
(4.3)
and rr is a nonnegative integer less than the multiplicity of the root x. By Appendix A, there is exactly one simple (i.e. multiplicity l), positive root x0 larger than the absolute value of all other roots, real or complex. In
56
A. I-. DASH AND R. CRESSMAN
the long run, the term involving ~6 will dominate and F(t) will behave like Ax; asymptotically (F(t) - Ax;). In particular, if x0 is less than 1, F(t) is asymptotically zero and hence the species will become extinct. Although our analysis includes this possibility, we are more interested in situations where X,>l. We proceed to find r for which this positive root is as large as possible. Suppose
%
(4.4)
Then P(x; r) = xa+l - Sx” -(bku”r/aMr
dP(x;r) = Jr
bku”( -
+ uF) and
u”r + u”) - bka”ruM ( u”r + u”)’
1
bku”uF
=-
( u”r + IJ”)”
-=c0.
With x fixed, P(x; r) decreases with r. That is, P( x; ri) > P( x; r2) whenever r, < r,. If P(x,; rl) = 0, then P(x,; r2) < 0. From the shape of pendix A, the positive root of P(x; r2) must be larger than means is that r should be as large as possible while still have total population size growing fastest. The analysis for ( ku”r/uF) > 1 proceeds analogously.
the graph in Apx,,. What all this satisfying (4.4) to From (4.3),
baF
P(x;r)=x*+‘-Sx”-
u”r+uF’
Since the partial derivative is now positive, we conclude that r should be as small as possible. From these two factors, we now know that population size will increase at its fastest rate in the long run if
kaMr=l uF
.
(4.5)
The biological significance of (4.5) is that the sex ratio of birth should be in perfect balance with the degree of polygamy of those offspring that survive to adulthood. This balance is very clear if we let uM = uF for the moment. If one male mates with k females on average, from (4.5) the growth is maximized when k females are born for each male. Alternatively, when k = 1 as in most modem societies, result (4.5) says the sex-ratio should be the same as female to male pre-adult survival rates. This
POLYGAMY
IN HUMAN
AND ANIMAL
57
SPECIES
validates the argument that the higher male infant mortality rate of humans explains why r is slightly larger than 1 in modern western societies. The interesting possibility of significant shifts in r for human societies exhibiting a higher degree of polygamy will not be pursued here. Rather, we offer another possible explanation why r > 1 in humans. If adult females survive longer than adult males, a careful analysis of Case B below will show that the adult sex-ratio L of (4.6) is less than that in Case A. This implies that r must now be larger to maximize the growth of population size. Since both SF and uF are larger than SM and u”, respectively, for humans, these parameters of the model both act to make the offspring sex-ratio larger than 1. Case B: SF > SM. As before, take (Y= /3, but now assume that there exists an L such that
M(t)
h
r+m
EL
F(f)
(4.6)
.
That is, we assume that an adult sex-ratio eventually emerges. We will see that for some parameters such an L is impossible. In these cases, it will also be shown the species becomes extinct. With (4.6) the asymptotic behaviour of (4.1) reduces to M(t+a+l)=S”M(t+cy)+aMrcmin(kL,l)F(~) (4.7) F(t+ol+1)=S~F(t+cr)+aknin(kL,1)F(t). From the last equation
F(t) - Ax; where x,, is the positive root of
P,( x; r; L) = xa+l - SFxa - uFc min( kL , 1) . Substitution of
(4.8a)
of M(t) - A Lx; into (4.7) implies that x,, must also be a root
P,(x;r;
L) =xa+l
- S”xa - u”rcmin(
k,l/L).
(4.8b)
As an aside to help clarify the role of L, notice that Equations (4.8), when applied to Case A, can only have the same root if uFc min( kL,l) = u”rc min( k, l/L) or u’c = u”rc/L. That is, the only possible adult sex-ratio L is simply (u”/uF)r, a result we already knew from (2.7). Moreover, when SF> S”, then
S”x”0 + +min(
kL,l)
= SFx; + uFcmin( kL,l),
58
A. T. DASH AND R. CRESSMAN
and this implies
a”rcmin( kL,l) L=
(S~-S~)X~+&min(kL,1)
<%
which is consistent with (2.9). There is no guarantee that Equations (4.8) will have a common root for some adult sex-ratio L. Indeed we prove the following: There is an L for which Equations (4.8) have a common root if and only if
P,( SF; r;O) -e0. Furthermore,
(4.9)
when (4.9) holds, this L is unique.
Proof. For large I, the positive root of P,(x; r; Z) is approximately S”; and as I decreases, the positive root increases (see the shape of these functions in Appendix A), On the other hand, PF( x; r; 0) has root SF and the root of PF(x; r; I) decreases as 1 decreases. Thus the smallest root of PF( x; r; I) is SF and this occurs when I = 0. Since SM < SF, Equations (4.8) can both be satisfied if and only if the positive root of PM(x; r; 0) is larger than SF. This is equivalent to condition (4.9) by Appendix A. Furthermore, as I changes, the positive root of either (4.8a) or (4.8b) also changes. This implies there can be at most one I where (4.8a) and (4.8b) have the same root. The failure of condition (4.9) has an interesting biological consequence. If (4.9) is not true, then (SF)a” - S”(SF)o - a”rck > 0. Since SF< 1, we see from Appendix A that (4.10)
l-S”-u”rck>O.
Choose 6 < 1 such that SM + a”rck < S from (4.10). From (4.1), we have M(t+a+l)~S~M(t+a)+u~rckM(t) Q ( SM + u”rck)max(
M( t + a), M(t))
+8max(M(t+a),M(t)). It is not difficult to see from this inequality that lim,, ,M(t) = 0. This, in turn, implies lim,, mF(t) = 0. In summary, if there is no L for which (4.8a,b) are satisfied (i.e. (4.9) fails), then the species must tend to extinction in the long run.
59
POLYGAMY IN HUMAN AND ANIMAL SPECIES
5.
DENSITY
DEPENDENT
MODELS
In this section, we return to the system of difference equations in Section 2,
developed
M(t+ol+l)=S~M(t+a)+a%R(M(t),F(t)) (5.1) F(t+/3+1)
=sFF(t+p)+uFR(M(t),F(t)),
to study the local stability at a positive equilibrium point (M*, F*). That is, M* and F* are both positive and satisfy M* = S”M* + a”rR( M*, F*), F* = SFF* + aFR( M*, F*). As noted previously, such an equilibrium can exist only if the reproductive function displays some density dependence, in contrast to the situation in Sections 3 and 4. Stability is determined through the Jacobian matrix J corresponding to the matrix equation (2.1) but with the reproductive function (2.4) substituted appropriately. J is a square matrix of order (Y+ /I + 2 whose ij th entry is J._= aZ;(t+l) IJ aZJ(r)
where the partial
derivatives
are evaluated
=
,..., cx i=ar+l
M(t) Fr_ca+9(r)
i=cr+2,...,a+j?+l’ i=a+j?+2
F( t) The calculation
at (M*, F*) and
i=l
M,-,(t) ‘i(‘)
’
yields 0
...
0
C?R
r-gjq
0
.”
0
aR
‘37
GM
J= 0
g
0
L
ss”-1
SF
60
A. T. DASH AND R. CRESSMAN
By standard theory, (M*, F*) will be locally stable if all eigenvalues of J have modulus less than 1 and unstable if any eigenvalue has modulus greater than 1. From Appendix B, the characteristic equation is
det(xZ-J)
= [xa(x-~~)--~~~I~][x~(x-~f)--(rf~] _ &feF?!?
_?!? b’M dF’
(5.2)
Eigenvalues of J are the roots of (5.2). If we assume that the total number of offspring increases if either the number of mature males or females increases then all entries of J are nonnegative and, in fact, J is a positive matrix. Such matrices, called primitive in Perron-Frobenius theory [8], have a positive eigenvalue x0 that is larger than the modulus of any other root of (5.2). Stability will occur if x0 < 1 but not if x0 > 1. In other words, we want to know if det(xZ-J)>Oforallx>l. This is true if and only if all three of the following satisfied: (1)
l-S”-ru”--->O
(2)
1-SF-uF-->O
(3)
(l-S”-
conditions
are
8R dM dR aF
r+&)(
(5.3) I-SF-&+
>r#~~. 1
Proof. Suppose (5.3) is true. From Appendix A, both factors inside the square bracket of (5.2) are increasing for x > 1. By the last condition of (5.3), det(xZ- J) > 0 for x al. Now suppose one of the three conditions of (5.3) is not satisfied. We want to show x,, > 1. This is clear if (3) is not true. Consider the case where (1) is not true. Now there is some x1 > 1 with xF(xi - s”) - ra”( dR/ilM) = 0. For this value, det(x,Z - J) < 0. Since det( xZ - J) is eventually positive, x,, > xi > 1. A similar argument is used if (2) is not true. Condition (5.3) can be further reduced to a form that is quite easy to check for a given reproduction function. At equilibrium, (1- S”)M = a”rR(M, F) and (1- SF)F = uFR(M, F). With this substitution, the three
POLYGAMY
IN HUMAN
AND ANIMAL
61
SPECIES
parts of (5.3) simplify to R(M, F) > M( aR/aM), R(M, F) > F( aR/aF), and R( M, F) > M( aR/aM) + F( aR/aF), respectively. Since the partials are positive, the last of these inequalities implies the other two. We have proven Theorem: An equilibrium (M*, F*) of the system (5.1) is stable if and only if aR R(M,F)>Mm+FTF.
aR
(5.4)
Species that have, at equilibrium, either a large number of offspring or a reproduction function that remains relatively constant at nearby adult population sizes will exhibit the most stability by (5.4). A surprising aspect of this analysis is that the stability condition (5.4) does not involve any survival rate parameters or the sex-ratio at birth. Let us briefly consider the possibility that R (M, F) does not increase for both M and F increasing. For instance, suppose the number of males over a certain threshold has no effect on total number of offspring. Since the Jacobian is no longer primitive, we will use Gerschgorin’s Disc Theorem [9] to test for stability. Applied to the columns of matrix J, the theorem states that the eigenvalues of J are contained in the union of the following discs in the complex plane: a+p+2
Dj=
x: Ix-J;~I<
c i-l
i
i#j
Jlj
, j=1,2
,..., a+/3+2.
i
Unless j = LY+ 1 or OL+ /I + 2, it is clear Dj is a subset of the unit disc. Thus the equilibrium point will be stable if both Du+1= Da+/?+2
=
(x:
Ix-SMIC(I+1)$
{x:
Ix-S’,<(r-l)%)
are also subsets. We conclude all eigenvalues
of J are in the unit disc if
(5.5) We wish to emphasize that these new sufficient conditions (5.5) for stability are certainly not necessary. If, in fact, J is primitive, the inequalities of (5.3)
62
A. T. DASH AND R. CRESSMAN
are
satisfied in many cases when those of (5.5) are not. The problem is that the converse of Gerschgorin’s Theorem is not very useful here. Both Da +1 could be outside the unit disc and the system remain stable. We and L,uz will not pursue this possibility further except to refer the interested reader to a paper by Agnew [l] where delay-difference equations of the form (2.3) are examined using the approach of this paragraph. We argue instead that the natural approach to stability questions for our sex-dependent, age-structured populations is through nonnegative matrices that lead to condition (5.4). Thus the test for stability at a typical equilibrium does not depend on either the population’s survival or sex-ratio parameters. 6.
DISCUSSION
AND EXAMPLE
The techniques and conclusions of Sections 4 and 5 are quite different but there is a tenuous connection between the limiting adult sex-ratio of both sections that should be mentioned. At an equilibrium of (5.1)
M* F*
-=I-
CTM1-P uF l-SM.
(6.1)
With
unbounded growth in Section 4 we have L = r(u”/u’) or L G when S”= SF or SF> S”, respectively. When S”= SF, the limiting sex-ratios of both sections are the same. One might conjecture r(u”/uF)
uM l-SF L=r(l,
I_SM
for all cases of Section 4. A numerical example will show density independent growth is more subtle than this. Consider a species with equal sex-ratio at birth (r = l), no time delay to maturity (a = p = 0 and uM = uF= 1) and with SF> SM. From (4.1), the difference equations are
M(t+l)
=S”M(t)+cmin(M(t),F(t))
F(t+l)=SFF(t)+cmin(M(t),F(t)).
Eventually
(i.e. for all f > to) F( t) will be greater than M(t)
since SF > SM.
POLYGAMY
63
IN HUMAN AND ANIMAL SPECIES
Take t, = 0 and solve to obtain
M(r) =(S"+c)'M(O)
for some constant
Em I--trn
and
F(t)
=A(SF)‘+
CM(O) SM+c_SSF
(SM+c)t
A. Then
M(O)
M(t)= F(t)
+
CM(O) s”+c-SF
0
ifSF>SM+c
1-p SF-SM
= i
ifSF
C
It is only the last case SM + c > 1 > SF that we are interested the species become extinct. Comparison with (6.1) shows 1_
in. Otherwise,
SF-SM =- l-SF C
l-SM
if and only if SM + c =l. Since SM + c > 1, the adult sex ratio for unbounded growth is somewhere between 1 and the equilibrium ratio of (6.1). The more realistic time-delay models are much more difficult to solve completely for M(t) and F(t) but the analysis developed in this paper calculates limiting adult sex ratios without these explicit solutions. The underlying assumptions that led to these models are quite general but a few limitations are worth discussing. Perhaps the most serious limitation is that adults are put into two mature classes, one for each sex, with no provision for non-reproductive adults or for age-dependent reproduction rates. The obvious answer is to include more adult age classes. However, the new delay-difference equations that emerge become unmanageable. Secondly, the reproduction functions and, through it, the actual polygamous mating system were assumed to depend only on adult population sizes M and F. Instead, individual characteristics might be important in any discussion of the degree of polygamy. Such considerations quickly go beyond the population dynamics of this paper into areas such as selective mating and generic effects. We hasten to add that these limitations do not diminish the importance of the results presented in this paper. As in most biological models, the power
A. T. DASH AND R. CRESSMAN
64
of mathematics lies not only is specific computations structural unity brought to a collection of models. APPENDIX
but more often in the
A
Consider the polynomial P(x) = x”+’ - uxn - b where n is a positive integer and a, b are positive constants. The graph of P(x) is shown below for x > 0. Properties of P(x) that we wish to note are (1) P(a) = P(0) and P has one extremum-a minimum at x =
&
,
(2) P(x) has one positive root x0 which is larger than a, (3)
All other (possibly complex) roots have modules less than x,,.
Proof: (1) (3)
and (2) are obvious from the graph and P'(x). If P(z)= 0 where z = reie and 0 < 0 < 2n, then rn+lei(n+l)B
=
urneinO
+
b.
(A.11
Thus p+l = Iurn@@ + bl Q ur” + b with equality if and only if eine is positive. If erne is positive and 0 < 0 < 2n, then e’(“+‘)’ is not positive and (A.l) cannot be satisfied. That is, rn+’ < ur” + b. But rn+’ > urn + b for t-ax,, bythegraph.Thus r
y=P (x>
I
65
POLYGAMY IN HUMAN AND ANIMAL SPECIES
APPENDIX
B
To discuss stability in Section 5 we need the determinant of a matrix of the form
-b
0
..
0
. .
0
-a
0
-b
x x
-s,“-,
X-SM --c
-x$
-d x
x -
Expansion along the first row yields
xlM1IIM21+(-1) “+‘~l~~II~21+(-~)~+B~l~~ll~~I where -“s,
x
MI =
X
I
.Y
-s,“-,
x-SM
-d
P2l= X
SaFl x-s’ =xa(x-S~)+(-1)8+1d(-1)PS,F.....S~1 =x8(x-SF)-ddgF
=(-l)“s~.
. . . .s,“_l=(-l)aaM
$7
x-SF
66
A. T. DASH AND R. CRESSMAN
and x -
s,F
l&l =
= ( -l)p+lcaF. That is, the required determinant is [X”(X-s~)+(-l)a+1a(-l)ao~][X~(X-s~)-A7~] +(-1)
“++I( -l)“lP(
-l)p+*CoF
= [ xy x - S”) - aaM] [ xfi( x - SF) - daF] - bca”uF.
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Non-negative
New York, 1980. D. Smith and N. Keyfitz, Vol. 6, Springer-Verlag,
Matrices and Markov Chains, Second Edition, Springer-Verlag, Mathematical
Demography,
Selectedpapers.
Biomathematics,
Berlin, 1977.
P. E. Smouse, The evolutionary advantages of sexual dimorphism, Theoret. Pop. Biol. 2~469-481 (1971). G. W. Stewart, Introduction to Matrix Computations, Academic Press, 1973.