DRAFT: Summer, 2000

TYPE OF ARTICLE: Research Paper

An early draft of part 3 appears as ``Excess female mortality and the Lotka-Volterra predator-prey interaction: An exercise in demographic modelling." Pp. 298-305 in S. K. Dey, J. P. Ziebarth, and J. M. Ferrandiz, editors. 1998. Special proceedings of IMACS ‘98, Alicante, Spain. Charleston, IL: Eastern Illinois University.

Michael A. Faia,
College of William & Mary

Address for correspondence:
Michael A. Faia,
327 Richmond Road,
Box 8795,
College of William & Mary,
Williamsburg, VA 23187-8795

mafaia@wm.edu
(757) 221-2593

Filename: ode-2.mws

Abstract
 

This paper expands on an argument made by Nielsen and Rosenfeld (1981:161), who say that in interpreting differential equations one must recognize, first, that the process of solving such equations typically results not in a specific number, but in a time-series trajectory of Y values, the orbit of Y, taken to represent the behavior of a dependent variable; secondly, that the most edifying interpretations of differential equations focus on the parameters that describe trajectories rather than directly on the coefficients of the equations themselves.

Title: Differential Equation Modeling as a Source of Theoretical Insight: Four Disparate Examples

Introduction: Search strategies for mathematical applications
 

---Mario Bunge (1997:419)

Metaphysical conceits: Sonnets, templates, and mad-libs

In a book about poetry by Main and Seng (1961, 94-95) we find this:
 

Notice that at least one of these outrageous images has evolved in such a way that it is now an everyday expression. Students of metaphor and simile say that these images change predictably: At their inception they are fresh, exciting, perhaps shocking; later on, they have become easily replicated cliches.

SELFREF develops the idea that the functionalist template---a.k.a. the functionalist paradigm---behaves in the same way as a standard set of operating procedures for writing sonnets or Japanese haiku. The question is: How is this sort of procedural pattern, this deus ex machina, put together? How does it work? How does it focus our minds? How does it enable us to penetrate the mist without becoming lost in fog that rolls in forever from every side?

Dillon and St. Vincent Millay (1936) wrote a fascinating translation of the poetry of Baudelaire. In the preface of this book, St. Vincent Millay explains in detail how she developed the templates necessary for translating French alexandrines into forms of English verse that would capture the structure, the rhythms, and the essential meanings of the original poetry. This was a tough challenge. She tells us (1936:xi) that ``poetry should not, and indeed cannot properly be translated except by poets. But there is more to it than that; it is as complicated as blood-transfusion." And regarding our current concerns: Who is best qualified to state a word problem for a mathematician? The mathematician herself, or somebody who works in a realm offering mathematical applications, who cultivates language skills more generally, and who does so with great imagination? How complicated are these sorts of ``transfusions," and for whom are they most forbidding? For the French or the English poet? For the mathematician or the non-mathematician? For the mathematician or the social scientist?

Where, in other words, do we find people skilled at seeing and stating problems similar to the following ecological conundrum (Bauldry and Fiedler, 1996:43)?
 

... You are an employee of the Otsego Electric Company. You have just left a meeting in which you were assigned the following task:

The Otsego Electric Company must run a power line from Power Station 23 in Section 34 to the sawmill under construction in Section 26. The sawmill is 2.3 km north and 5.2 km east of the power station; the intervening private land is 1,300 m wide (east-west). The power line must cross private land. The cost of an easement is 0.75 dollars per meter through the state forest and 2.25 dollars per meter on the private land.

This particular problem is a fairly well worn template; a ``standard application." When we enter the realm of non-standard applications, things seem to be considerably more tricky (Israel, 1996:46). What accounts for this difference, for this trickiness? In a study by Saxenian (1994:134-41), one finds an excellent metaphysical conceit drawing an analogy between the Silicon Valley and a Sun workstation, and the comparative narratives come together like a lengthy, complicated, multidimensional novel. At first the analogy seems implausible, but the utility of it becomes much more clear when we read Saxenian, and when we see precisely the same analogy translated into a valuable mathematical application (www.orie ..., n.d.). In this latter instance, the author is discussing the famous travelling salesman problem---i.e., if a travelling salesman has to visit a large number of cities, more or less randomly selected, how does she create an itinerary that will minimize the amount of time spent travelling, the costs, and the mileage, while returning her to the point of origin?
 

The traveling salesman problem is an important computational model ... It forms a central component of more complicated models, such as routing a fleet of vehicles for pickups and deliveries ..., the sort of problem that a UPS depot might solve on a daily basis. It can also be used to model a number of problems that, at first glance, do not seem to be related.

Consider the following problem that arises in the process of manufacturing VLSI [very large-scale integration] computer chips ... There is a square (a silicon wafer) on which one is going to etch a sequence of lines ... The machine doing the etching first moves to the correct position where the line starts, [and] etches the particular line ... Then it must move to the correct position for the next line, and so forth. The lines may be etched in any order ... The etching machine must start and end at the upper left-hand corner of the square ... The VLSI etching optimization problem is to select an order for the lines ... so that as little time as possible is taken.

If you wish to play with a machine that generates metaphysical conceits the way an etching machine generates circuitry, learn the child's game called ``mad libs." Keep a statistical tabulation of results. Count the proportion of instances---perhaps small---in which a lib does not appear to be totally mad. Compare your results with those of Platt and Barker, described below. Compare mad libs against other forms of translation. In mad libs, kids exchange nouns, verbs, adjectives, and other parts of speech. In stating a mathematical word problem, we are more likely to change nouns (``terms") only, retaining the verbs and other parts of speech---mathematical operators---as an essential part of a given model (Nielsen and Rosenfeld, 1981).

Left brain, right brain, and hints of helices

In The Act of Creation(1964), Koestler devotes a large section to ``the sage"; remaining sections of the book deal with ``the jester" and ``the artist"---it is an impressive metaphysical conceit in itself for Koestler to say that all these creative modes, and all the forces they muster, are essentially the same. The discussion of the sage is filled with instances in which geometrical and/or artistic imagery---we're often not sure---stimulates an insight of a rationalistic, algebraic nature. Or vice versa. In a way, these are all instances of Hilbert vindicated, instances in which left brain and right brain work together. They are all illustrations of the Hughes-Hallet et al. (1994:vii) two-part ``prescription" for restoring both ``mathematical content" and ``practical understanding" to the study of calculus. First, the ``rule of three": All topics should be presented by means of geometry, algebra, and numerical examples. Second, the ``way of Archimedes": Understanding evolves from the study of practical applications.

Koestler (1964:118) tells the famous story of Kekule's dream---a dream that led to basic insights into the geometry of a class of chemical compounds. In this dream, Kekule saw serpents swallowing their own tails, a vision that helped him to comprehend the hexagonal structure of benzene rings. If the ability of a (perhaps atypical) hollowed-out serpent to swallow its own tail is limited only by the distance from the mouth to the interior surface of the very same tail, and if the serpent is indeed swallowing its own tail deeply and enthusiastically and with no limiting factor except the tail itself---well, this is an extraordinary geometry with seemingly infinite prospects: How can a singular tail contain itself? But while this speculation, this elaboration of Kekule, is slightly off the wall, it may, as Watson (1968:75,77) suggests in recounting the discovery of DNA, retain the plausibility of a comparable architecture:
 

Again an extraordinary geometry, and the reader is left wondering whether there had been some sort of subconscious process---perhaps a dream or a daydream, during the bus ride itself---that the author never recognized: One cannot always report these events as if they had occurred inside a virtual-reality headset. Furthermore, one may have excellent imagery available---say, Linus Pauling's ``tinker-toy-like models" of chemical structures---and yet be a little tardy in realizing the significance and importance of it (Watson, 1968:51).

ANOVA as the classic left brain/right brain model

In the mathematical/statistical realm, Porter (1986) and Dawkins (1986; cf. SELFREF) make it clear that analysis of variance is not merely a statistical technique. In Porter's view (1986:316) ANOVA modeling provided essential insights that helped biologists work out the synthesis of Mendelian genetics with natural selection. Dawkins (1986:41) shows clearly how the theory fits the mathematical model, and vice versa: ``[Genetic] mutation," he concludes, ``is random; natural selection is the very opposite of random." ANOVA was made for precisely this sort of reality, and it provides help of inestimable value in our efforts to perceive such realities. Statistics (or biostatistics) teachers have a big advantage when they know how to explain ANOVA both algebraically and by means of the graphic features of Tukey's Exploratory Data Analysis (EDA) offered, say, by the Minitab statistical package.

Quantiphobes in the social sciences doubtless would accuse both Dawkins and Porter of allowing the methodological tail to wag the theoretical corpus---we hesitate to invoke the usual image. The truth is that one cannot see most theoretical problems clearly unless one has the appropriate quantitative models in mind, and one cannot get quantitative models in mind unless one understands the basic logic of abstract quantitative (and non-quantitative) theory. In short, we're dealing with reciprocal causation between theory and method (Platt, 1996).

Don't forget to duck

However, there is a caveat. New ideas---certainly including metaphysical conceits---are like the genetic mutations elucidated by ANOVA: Most of them are bad. From Koestler (1964:212):

A stimulating inquiry by the American chemists Platt and Barker showed that among those scientists who answered their questionnaire eighty-three per cent claimed frequent or occasional assistance from unconscious intuitions. But at the same time only seven per cent among them asserted that their intuitions were always correct; the remainder estimated the percentage of their ``false intuitions" variously at ten to ninety per cent.

If our eventual self-assessments discourage us to this degree, just imagine the devastating impact---not to mention the deterrent impact---of a thorough peer review. We don't know whether mathematicians have more or fewer failures than other scholars, but Silverman (1997:3), in a discussion of number theory, captures an attitude that must be commonly held among productive scholars in any field: ``Most experiments in any science are failures. It is only by persevering through each setback that one eventually achieves success, and the satisfaction of success is made all the more sweet by the intense effort leading up to it." One must, then, have a thick skin if one wishes to do stimulating work, and scholars who have fantastic ideas have a lot in common with accomplished home run hitters who often strike out. Get accustomed to walking sadly back to the dugout.

Fisher (1973:1115-16) speaks of the factors that keep mathematicians from revealing their efforts to resolve the ``Poincare conjecture." These factors are dangerous:
 

As I say, a thick skin. Now, let me Donne mine.

The paradox of ``hitting bottom"

For many years the idea of ``hitting bottom," as an essential element of a successful recovery from alcoholism, has survived as a sort of metaphysical construct: Hitting bottom is said to be an inexorable manifestation of an underlying, somewhat mysterious transition. As such, this idea generates skepticism comparable to that entertained by Malinowski in his famous debate against the Freudian Oedipus complex as the bedrock supporting unhealthy behaviors occurring cross-culturally. By now, the phenomenon of hitting bottom has evolved into a highly elaborate theory. It is a theory not about etiology (as in the case of the Oedipus complex), but rather about prognosis and recovery. It does not see the recovery process for alcoholics as emanating largely from within the alcoholic, but rather as dependent on an array of external forces: the nature and availability of the drug itself, the alcoholic's family dynamics, his/her interactions with co-alcoholics, interactions with the legalistic/therapeutic community, and interactions with factors and forces of a religious or spiritual nature. Accordingly, the differential equation written below, representing the recovery process, will not embody feedback from the dependent variable---the recovery process---to itself.

The recovery process also is thought to have an inherent, underlying, cyclical character, giving it ``cybernetic" properties (O'Reilly, 1997: Chapter 1) and creating a complex series of ``slips" and ``relapses"; the latter are said to be ``... at the heart of the recovery process" (Denzin, 1987:153). It is assumed that recovery, at least in terms of externalities, typically is already under way (Brown and Lewis, 1999:5; Denzin, 1987:143, 152) at the point (or points) where an alcoholic and/or co-alcoholics reach the most disturbing depths of malfunction, degradation, disintegration. For Brown and Lewis (1999:108), hitting bottom is ``... a very hard time, as the drinking family and the drinking world collapse and the foundation blocks for recovery are set in place"; ultimately, in this view, ``... everyone needs to hit bottom" (Brown and Lewis, 1999:184). Following a major crisis, further slips may become, for alcoholics or co-alcoholics, ``... a part of their A. A. identity."

Throughout the recovery process there are thought to be constantly operating factors, such as the alcoholic's dreams, that have a positive impact in restoring health (Denzin, 1987:148-51). Finally, the role of the therapist is to keep drinkers and their families focussed on the realities of the problem (Brown and Lewis, 1999:182-83), by cleverly and constructively attacking self-delusional systems.

The differential equation deq1, presented below (cf. Ellis et al., 1997:106), is written on the assumption that the process of hitting bottom, often experienced by severe alcoholics and often anticipated with a surprising degree of optimism by their significant others, is actually a manifestation of a lengthier, more complex, underlying process of recovery. This equation captures what may be the latent meaning of the popular and paradoxical claim that ``s/he won't turn things around and get better till s/he hits bottom." The rate of change of the alcoholic's underlying recovery process is taken to have three components: First, it involves a sinusoid function of time, because presumably there are cyclical ups and downs along the way; second, it involves steady, linear improvements over time due to the existence of a latent recovery process independent of the cyclical pattern; third, it incorporates a final term representing the ability of medical and psychological interventions to contribute to the recovery process, or at least to make matters slightly less severe and distressing as the process unfolds.

We begin by clearing registers for the computer algebra system used by this article, and by invoking necessary subroutines.

restart;

with(DEtools):

First, we set up three constants representing the coefficients of the three essential terms. It seems appropriate that the term for highly disruptive cyclical relapses, , should have the highest coefficient, dominating and perhaps obscuring the linear trend toward recovery.

k1 := 4; k2 := 1/3; k3 := 1/3;

Next, we write the differential equation [note 2]:

deq1 := diff(y(t),t) = k1*sin(t) + k2*t + k3;

The dependent variable, for reasons stated, does not appear on the rightward side of the equation, and therefore we obtain a solution in the form of a relatively simple integral. The integral curves, defined as sol1, will define the pattern of a recovering alcoholic's observable trajectory through time.

sol1 := dsolve(deq1, y(t));

The _C1 term is an arbitrary constant.

In this relatively simple example, it will prove helpful to plot both the differential equation and a few of its solutions. First, we isolate the righthand side of deq1:

rh1 := rhs(deq1);

Now, working toward a plot that will contain both the differential equation and its solution for a specific type of case, isolate the righthand side of the solution :

rh2 := rhs(sol1);

The expression , representing a constant, enables us to select several illustrative trajectories for . If _C1 were set at 10, then at time the cosine term would lower the value of the solution function by 4 units. The constant _C1enables us to distinguish, in Figure 1, between the differential equation and the specific solution shown. Clearly, if , then the solution, presumably depicting observable behavior, will have a y-intercept of .

_C1 := 10:

The following several plots, then, show typical developmental sequences. In Figure 1, we observe the trajectory of the specific type of case defined above; the second plot (Figure 2) shows several comparable trajectories for different values of _C1; the third plot (Figure 3) shows direction fields for several trajectories. Clearly the big ``dives" or ``binges" take place as a given case moves toward the bottoming-out point; although they still occur as relapses or slips, they are less severe beyond that point. Eventually---to mix metaphors in a way that produces a happy outcome---slips may become the sort of relapse that is actually a plateau, perhaps a brief resting place on the long, arduous climb toward recovery. Notice that in the neighborhood of time (Figure 3), the differential equation generates a plateau in all trajectories, a uniformity across all cases that may be amenable to empirical test. If it were indeed a uniformity, such a temporal pattern would support the ``... underlying premise of A. A., which is that all alcoholics are the same" (Denzin, 1987:152).

Returning to Figure 1, then, we have seen that the lower function is the differential equation itself, while the upper function is its solution for the current value of _C1; the differential equation defines the rates of change that govern the behavior of the particular solution that passes through a fixed value for at time . The observable trajectory of the solution is that of the severe alcoholic's bottoming-out process. Paradoxically, this process occurs despite the fact that the underlying dynamic, captured by the differential equation, moves consistently toward improvement with only an occasional setback. Very popular among those who work with alcoholics, the paradox implies that what may appear on the surface to be a continuing series of setbacks or even disasters may actually result from a consistent underlying recovery process. Recall, again, that it is generally assumed in the literature that hitting bottom occurs after alcoholics and co-alcoholics have made a sincere commitment to serious treatment (Brown and Lewis, 1999; Denzin, 1987).

The values k11and k12define the boundaries for time, in Figures 1 through 3.

k11 := -10; k12 := 15;

plot({rh1, rh2}, t=k11..k12, thickness=3, color=black, title=`Figure 1: Alcoholic trajectory with latent recovery`);

_C1 := '_C1';

Figure 2 suggests the possibility that relatively simple latent recovery processes create alcoholic trajectories that may differ tremendously in terms of the depths to which alcoholics and co-alcoholics descend (Brown and Lewis, 1999:108; Denzin, 1987:210). In this instance---as in the instance of Richardson arms races, to be considered later---we see the dangers of overreaction to observed patterns of change. All trajectories of Figure 2 are manifestations of the same underlying recovery process, despite the fact that the bottom for one case may differ considerably from that of a comparable case.

plot({subs(_C1=-2, rh2), subs(_C1=-1, rh2), subs(_C1=0, rh2), subs(_C1=3, rh2), subs(_C1=10, rh2)}, t=k11..k12, y=-10..15, thickness=3, color=black, title=`Figure 2: A family of trajectories`);

Direction fields for many trajectories will make the same point. Derivatives are uniform for any given value of time; yet, a given constant is capable of generating a distinctive outcome.

DEplot(deq1, y(t), t=k11..k12, y=-5..15, thickness=1, color=black, title=`Figure 3: Direction field`);

If this differential equation, along with its solutions, is more meaningful than the popular platitude about hitting bottom, it is because the equation says much more than the platitude: The differential equation explains a socially constructed paradox by showing that the recovery process has three distinct, reasonably well-defined, and yet entirely latent components that manifest themselves in unique and alarming ways, ways that often correspond to reported observations. And given that the equation allows the properties of these components to vary according to plausible hypotheses, it does not assume that all recovering alcoholics follow a fixed trajectory even though there may be an essential sameness to the basic pattern. Within the framework established by a given differential equation, the number of unique trajectories is indefinitely large.

Thus far this discussion has said nothing about instrumentation, about measures that would generate empirical data pertaining to the recovery process of alcoholics. If the quantitative model developed above reflects the widely proclaimed bottoming-out process as a way of healing, it implies that measures of manifest, socially prominent behaviors should be supplemented by subjective perceptions on the part of alcoholics themselves and by measures of their physiological functioning. In general, the socially perceived behaviors involved in hitting bottom may follow the solution trajectories of our equations, while the underlying subjective and physiological indicators may conform more closely to the stabilities of the differential equations themselves.

The social dynamics of the Peter Principle

restart;

with(DEtools):

Although paralleling Nielsen and Rosenfeld (1981), this section illustrates the Peter Principle (Peter, 1969), according to which resistance to upward mobility increases as one moves to higher positions in a hierarchy---an instance of negative feedback. For a hypothetical upwardly-mobile person, then, the current level of status s(t) will be multiplied by resist to obtain a measure of the growing resistance to one's further ascent.

resist := 90/100;

M is motivation, persistence, deferred gratification, etc., making for continued social mobility; its value, a constant, is for the moment unknown and will be inferred from our basic differential equation model. The constant called skills is taken to be a measure of general educational attainment, specific work skills, ``social" skills, and the like. Social mobility is defined as the rate of change of s(t), or . (Recall that notation is used for all derivatives.)

skills := 110/2;

Here, then, is a differential equation that shows how social mobility is thought to be influenced by skills, motivation, and resistance. The impetus for social mobility is taken to be the product of skills and motivation. For a measure of resistance, as stated, we multiply the current level of status attainment by the constant, resist.

deq1 := diff(s(t),(t)) = (skills*M) - (resist*s(t));

Now we obtain a solution, sol1, that enables us to project future status attainments. Consider the case of a person who has achieved a relatively low early-career status score, approximately 35 on the NORC occupational status scale. If this level of occupational attainment were taken to be a starting point where time then the following equation would enable us to project this individual's occupational attainments into the future.

sol1 := dsolve({deq1, s(0)=35}, s(t));

All we lack now is information about M, motivation.

Before we proceed further, it would be a good idea to check sol1. We do so by substituting it back into the original differential equation deq1, and then making sure that the two sides of the equation are indeed equal. We first isolate the right side of sol1, giving it a new name, and then we make the substitution. (The % symbol causes the preceding result to be substituted into the current expression.)

testsol := rhs(%);

Now we substitute testsol for s(t) in deq1, make an appropriate simplification, and verify that the two sides are equal.

subs(s(t) = testsol, deq1);

simplify(%);

Q.E.D.

We shall project the trajectory defined by sol1 over a period of several years, inferring the level of motivation required over a fixed period of time to propel this individual to a high status score---somewhere in the 60's or 70's on the NORC scale---in the face of increasing and increasingly effective resistance. (Units of time are decades, so that time , for instance, would represent 25 years.)

trajec := subs(t=23/10, rhs(sol1));

M := solve(trajec=62, M); evalf(%, 7);

In this instance, M has a value slightly in excess of one.

Within this model, then, what is the limit to which occupational status s(t) arrives over time, the limit that may define the stopping point for s(t) when, according to the Peter Principle, negative feedback becomes maximal and eliminates further upward mobility? We find this limit,

,

by taking the righthand side of sol1, as modified by the numerical changes introduced above, and raising the value of t to infinity thereby causing the second term below to vanish.

sol1;

limit(rhs(sol1), t=infinity); evalf(%, 4);

We see, then, that as the typical upwardly mobile person approaches a status level that is relatively high on the NORC scale, there ensues a sort of massive resistance against his/her further ascent. Peter implies that resistance increases rapidly because, as one moves upward in a hierarchy, one's limitations---I believe he uses the term incompetence---become manifest.

Having incorporated the value , the trajectory defined by sol1is now almost entirely numerical, and it is a function of time. Here is a plot (Figure 4) representing a social mobility process of the form just described, with NORC occupational status defining the vertical axis:

plot(rhs(sol1), t=0..5, thickness=3, color=black, title=`Figure 4: A trajectory representing upward social mobility with resistance`);

Now we undefine the current value of M and experiment with several values of it through substitution (Figure 5).

M := 'M';

Repeat, as a reminder:

sol1;

Sequence through several values of M:

patterns := seq(subs(M=i, rhs(sol1)), i = [.9, 1, 1.1, 1.2]):

plot({patterns}, t=0..5, s=15..75, thickness=3, color=black,
title=`Figure 5: The impact of M on social mobility trajectories`);

Trajectories of Figure 5 differ only because of variations in levels of motivation attributed to individuals. If we examine several of these trajectories together we see that they define career patterns that diverge, in social status, over time. If, along with status attainments, there were also divergence in access to resources and in levels of productivity we would observe the kinds of career patterns that allegedly occur among scientists who experience accumulative advantage (Merton, 1973:439-59). The accumulative advantage hypothesis involves a snowballing effect in which high social status, extensive work resources, and high productivity develop a mutually supportive momentum that Merton refers to as ``Matthew Effect," defined by an elderly and pious neighbor of ours as the pattern whereby ``them as has, gits, and them as ain't, don't." Where accumulative advantage exists, therefore, it is likely to create increasing divergence of status, resources, productivity, and power among individuals (or ``institutions"), along the lines of what we do indeed observe in Figure 5 (Allison and Stewart, 1974; SELFREF; Bentley and Blackburn. 1990). However, the differential equations that generated Figure 5 did not involve accumulative advantage; rather, a spurious appearance of accumulative advantage has been created by differences in motivation, while skill and other dynamics of the Peter Principle remain constant. Something akin to the Peter Principle should be added to accumulative advantage, political influence, and merit as potential factors explaining increasing divergence of status and related attainments (Mitroff and Chubin, 1979).

Excess female mortality

In the following analysis we apply Lotka-Volterra predator/prey interaction models (Abell and Braselton, 1994:425-29; Redfern and Chandler, 1996:96-97; Shone, 1997:432-36) to the problem of excess female mortality (EFM) as discussed by Harris and Ross (1987) and others. Regarding the contemporary magnitude of the EFM problem, Soroka and Bryjak (1995:244, 371; cf. Weeks, 1999:201-202) cite a recent United Nations estimate of 100 million ``missing" females throughout the world; presumably, historical conditions were even more severe. Harris and Ross emphasize the long-term historical development of EFM, focussing especially on the perinatal forms of mortality---feticide, infanticide, and pedicide---and arguing that, under conditions of intense population pressure, both infanticide and pedicide tend to select against females.

While population pressure, in the Harris-Ross view, has a powerful impact on perinatal mortality, it also generates ``harsh treatment" of adolescent and adult women. Such treatment may produce considerable EFM directly, but it also may generate ``reproductive selection," i.e., lower fertility accomplished by means short of death. If, in such a situation, lower fertility involved high levels of feticide and ``spontaneous" abortion, then selectivity against females probably would be diminished to some degree; therefore, we cannot always expect that population pressure will raise sex ratios consistently, although that particular tendency is probably strong even in contemporary societies (www.nytimes ...).

While the role of population pressure is not at all clear among the earliest sedentary agricultural societies (Smith, 1995:209-11), EFM often does appear to increase under straitened conditions in this type of society (Harris and Ross, 1987:69-70). All such generalizations, however, are vitiated by special methodological problems. As cultural materialists, Harris and Ross (1987:6, 13, 104-5, 161-64) consider the -cide suffix for the perinatal forms of death to be misleading: They claim, for instance, that of the seven forms of infanticide only one involves deliberate killing [note 3], and that infanticide like abortion has often been regarded ``... not as killing, but as terminating a biosocial gestation process." To Harris and Ross, these sorts of considerations preclude any strong possibility of learning about EFM through interviews, through personal testimonials of any kind, or through historical documents. On the contrary, we must rely on ``impersonal" ethno-demographic data such as formal and informal censuses and whatever information they may produce about sex ratios; or, as in the present study, on estimates of sex ratios inferred from population trajectories produced by differential equations.

Harris and Ross (1987:32-33) have misgivings about various simulation studies of female infanticide. They claim that several such studies have erred in regarding the rate of infanticide as a constant, not as a variable, over long periods of time: If the rate of infanticide were set at a high level, and held there for a long period of time, it would indeed wipe out a given population. Suppose, however, that the various perinatal forms of mortality, along with EFM imposed on adolescent and adult women, were regarded as highly variable population-adjustment mechanisms diminished in their intensity only after the occurrence of sharp declines in the female segment of a population. Then, given the strong likelihood that these mechanisms, when activated, would tend to select against females, we might find evidence in the archaeological and ethnographic record that, again from time to time, population pressure is associated with relatively high or increasing sex ratios.

This is precisely the sort of evidence cited by Harris and Ross (1987:32): In paleolithic times, for instance, female infanticide was widely practiced as the ``... most effective method of population control," and the supportive data are mainly sex ratios. Further, the high sex ratios of 19th-century India, China, and Japan are taken as evidence of widespread female infanticide induced by pressure on subsistence. Incidentally, the Harris-Ross thesis about the role of population regulation by means of perinatal mortality is held to be a major challenge to ``transition theory," which has a central place in contemporary demography (Harris and Ross, 1987:100). Harris and Ross maintain essentially that contemporary demography does not give adequate attention to variations in the ``vital revolution" by social class, thereby implying that historical demography also should emphasize social class more strongly.

In the analysis following, we develop a mathematical model that assumes that historical population pressure creates Darwinian competition between males and females involving both lethal and reproductive selection, that males have a survival advantage in this competition (Collins, 1975:228-59) for all forms of mortality except feticide and ``spontaneous abortion," and that this competition, in its impact on male and female populations considered separately, has the dynamics of the Lotka-Volterra predator-prey interaction equations. We propose to show how the Lotka-Volterra model would produce the cybernetic, equilibrium-seeking sex-ratio disparities hypothesized by Harris and Ross.

restart; Digits := 5:

with(DEtools): with(plots):

Warning, the name changecoords has been redefined

We now write the Lotka-Volterra equations. The female population---analogous to the prey---has positive growth (Keyfitz, 1968:271) captured by , and it is then suppressed in proportion to the size of the predator population; this suppression is captured by the coefficient.

If we assume a high level of internecine warfare resulting from Malthusian pressure on subsistence (SELFREF, 1986: Chapter 2), then the male population---analogous to the predator---has a tendency toward decline () but may grow due to exploitative interaction with the female population; the positive growth rate is captured by the coefficient.

e1 := diff(x(t),t) = alpha*x(t) - eta*y(t);

e2 := diff(y(t),t) = - beta*y(t) + theta*x(t);

Discussed by Keyfitz (1968:271-76) as the essential pattern for predator/prey interaction, the following assigned values for the four coefficients emerged from a lengthy series of experiments with the basic model. Each of these coefficients has plausibility as an annual vital rate. The value for implies a pattern of high demographic duress; notice, however, that appears to be well below the ``rate above 8 percent" that, according to Harris and Ross (1987:33), might well lead to ``genocidal extinction within five hundred years ..." The population orbits given below persist for five hundred years, and they do not lead to extinction.

alpha := 11/1000; beta := 2/100; eta := 5/100; theta := 2/100;

We now repeat the equations above, with appropriate substitutions:

e1; e2;

Observe the equilibrium points, as shown below in the phase plane (Figure 6). Any male/female population arriving at these points, representing a population optimum for a given area (SELFREF, 1999), would stabilize. Departures from these optimal values should be interpreted as percentage increases or decreases.

solve({rhs(e1)=0, rhs(e2)=0}, {x(t), y(t)});

Next, we select the all-important starting values: First, a starting population for x, the female population, stated as a percentage excess:

k1 := 30;

Then a comparable starting size for the male population, y:

k2 := 20;

We then solve the system of equations as an initial-value problem:

sol := dsolve({e1, e2, x(0)=k1, y(0)=k2}, {x(t), y(t)}, numeric):

The phase plane shows that the solution system moves toward an equilibrium, delightful to functionalists (SELFREF, 1986:111), in which an oscillating sinusoid pattern emerges. This is essentially what happens with the current solution model; it oscillates for a long period of time while it is damped.

We check the solution for time , making sure that it transforms the starting populations appropriately.

sol(1);

Set the upper limit of (in years) at 500, for the following calculations:

k3 := 500;

Now we examine the phase plane (Figure 6) for the interaction of the two population segments; it tells us about the amplitudes of this oscillating interaction over many data points. Clearly, the amplitude for females is considerably larger than that for males. As suggested by the Harris-Ross theory, females are the ``adjustment mechanism."

odeplot(sol,[x(t), y(t)], 0..k3, numpoints=75, axes=boxed, labels=[x,y], thickness=3, color=black, title=`Figure 6: Phase portrait: Male and female populations`);

The plot below (Figure 7) shows the current model, the phase plane for a large number of possible male/female population interactions, and the equilibrium populations. These equilibrium populations, as we have seen, would result from regression to the long-term mean size for both populations. As in the instances of alcoholism and social mobility, the phase plane shows a large number of specific situations evolving according to the same underlying dynamic.

DEplot({e1,e2}, {x(t), y(t)}, t = 0..k3, [[x(0)=k1, y(0)=k2]], thickness=2, color=black, title=`Figure 7: Phase plane: Two interacting populations`);

Below are selected numerical results, as the current interaction model moves through time. Notice that we have in this output a dataset from which one might derive sex ratios, thereby creating an opportunity for testing the Harris-Ross hypotheses about EFM under occasionally harsh environmental conditions. What would we observe as the pattern of change for the sex ratios? The fictitious numerical values given below, the sort of information that might appear in a series of ethnographic reports, represent the way in which each population departs from its long-term optimum. If these values were translated into actual population counts and the sex ratios were derived, it appears that the typical cross-sectional view would show a relatively high sex ratio, i.e., a preponderance of males. The situation, however, is perceived more clearly by means of plots and the calculation of an integral, as we show below.

sol(0);sol(50);sol(75);
sol(100);sol(200);
sol(300);sol(350);
sol(490);

Regarding most of the data points listed above for time and beyond, we would report a relatively high sex ratio, with . As anticipated, however, there are many exceptions, especially as we move through time and encounter a relatively recent phenomenon in which fluctuations of male and female population size around the optimum become smaller. Given the behavior of these sinusoid trajectories, it is entirely possible that a series of ethnographic analyses, especially if conducted in recent times, might miss the compelling evidence of EFM inherent in these data.

Now, we plot the two populations and encounter a time-series process that has much in common with the sinusoid patterns implied by Stinchcombe (1968:89) and made explicit in SELFREF (1986:106-18):

plotX := DEplot({e1,e2}, {x(t), y(t)},
t = 0..k3, [[x(0)=k1, y(0)=k2]], scene=[t,x], arrows=NONE, stepsize=35/10, linecolor=BLACK):

plotY := DEplot({e1,e2}, {x(t), y(t)},
t = 0..k3, [[x(0)=k1, y(0)=k2]], scene=[t,y], arrows=NONE, stepsize=35/10, linecolor=BLACK):

plots[display]([plotX,plotY], thickness=3, color=BLACK, title=`Figure 8: Time-series orbits: Male and female populations`);

This result appears to be highly plausible, and to provide considerable support for the Harris-Ross thesis. As implied by these authors, the female population is generally more variable than the male population; yet, over time, it appears to have about the same average size as the male population. The female population starts with a 30 per cent excess above its own mean. Presumably, when population pressure occurs it causes the female population to drop sharply, as it does throughout nearly the first 100 time units, and the sex ratio increases as predicted by Harris and Ross (1987). Later the sex ratio drops again, as the female population recovers rapidly from socially imposed EFM. Periodicities for both populations appear to be nearly uniform, and amplitudes, as noted, become smaller over time.

In general, the pattern of change of the two populations is such that, if we integrate the differences between them, there are many points in time when the male population is larger than the female population and relatively few instances in which this difference reverses itself. It is also arguable that, over time, the trend toward damped oscillation reflects improved mortality conditions and the associated tendency for mortality variability to decline in modernizing populations (Dorn, 1959).

Let us, therefore, review a few matters, obtain solutions for the Lotka-Volterra equations, and then obtain the difference between the long-term male and female population orbits. First, re-insert the starting values k1 and k2 into the differential equations e1 and e2, and then obtain time-series trajectories for the two resultant populations. We now define the two equations as a system,

sys := e1, e2;

so that we can isolate specific solutions for initial values k1 and k2. Specific solutions for x(t) and y(t) are then assigned to the two populations, prey and predators respectively.

k1, k2;

Output is suppressed for the following command, because the two solution equations are re-assigned immediately.

populations := dsolve({sys, x(0)=k1, y(0)=k2}, {x(t), y(t)}):

prey := subs(populations, x(t));

pred := subs(populations, y(t));

For these two sinusoid orbits we obtain the lower and upper boundaries of two complete cycles of the predator-prey model shown in Figure 8 (Ellis et al., 1997:16); we thereby assess the general value of the sex ratio within this series of cycles. By inspection of Figure 8, one sees that the lower and upper boundaries occur near the endpoints of the time-series plot, where the two functions cross.

leftcut := fsolve(pred = prey, t, 0..40);

rightcut := fsolve(pred = prey, t, 460..500);

Examining the following plot (Figure 9), we see that the predator population, despite its lower amplitude, is often larger than the prey population---sometimes by a substantial amount approaching 25 per cent---so that for most data points the sex ratio will be relatively high. These mixed and variable results typify the sort of data cited by Harris and Ross (1987) in support of their major hypotheses, and such results probably constitute a reasonable replication of historical patterns.

plot(pred - prey, t=leftcut..rightcut, thickness=3, color=black, title=`Figure 9: Time-series variation of the sex ratio, two cycles`);

Now obtain the integral of the difference---males minus females from the left to the right boundary. The expression %1 is substituted for the repeated expression .

Int(pred - prey, t=leftcut..rightcut) = int(pred - prey, t=leftcut..rightcut);

The elaborate calculations involved in solving this integral give us a difference in percentage-years. This result suggests that, while the sex ratio of these two populations at any given time may be at, above, or below 100 (males per hundred females), there is an excess male population of more than 900 percentage-years. Presumably, these are instances in which EFM has been most severe.

As a check (Israel, 1996: 242-45), we take the solution curve for, say, the prey population, substitute into it time , and then make sure that the result corresponds with that shown above for the numerical solution, in which the prey population after 50 years is 10.6 per cent below its optimum while the predator population has an excess of 11.3 per cent. We first repeat the equation for prey, assigning it a new name,

checkval1 := prey;

and then we make the appropriate substitution:

checkval2 := subs(t=50, checkval1);

Finally, we examine the result:

evalf(checkval2, 10);

Q.E.D.

Sensitive symmetries of trust: The social dynamics of a Richardson arms race

In the following analysis we develop a model for a Richardson arms race. For a comparable analysis, see Zill (1997:352ff.).

restart;

with(DEtools): with(plots):

Warning, the name changecoords has been redefined

In writing the Richardson equations, the and coefficients represent mutual mistrust, creating an impetus toward arms acquisition. The coefficients and represent the costs of arms production. The values for and refer to ``bellicosity" as a cultural tradition: They are sometimes positive, sometimes negative. In the present instance they are negative.

deq1 := diff(x(t),t) = alpha*y(t) - eta*x(t) + mu;

deq2 := diff(y(t),t) = beta*x(t) - theta*y(t) + nu;

Here we set up the numerical parameters for the Richardson system and re-write it with substitutions. It is edifying to experiment with the four cases found in Olinick (1978:34). [note 4]

alpha := 13/1000; beta := 13/1000;

eta := 10/1000; theta := 10/1000;

mu:= 3/1000; nu := 4/1000;

Make substitutions:

deq1 := diff(x(t),t) = alpha*y(t) - eta*x(t) - mu;

deq2 := diff(y(t),t) = beta*x(t) - theta*y(t) - nu;

Next, we select the starting values: First, a starting arms level for nation (or bloc) x:

k1 := 70/100;

Then a starting arms level for nation (or bloc) y.

k2 := 95/100;

Assume that the starting values differ because of the impact of an immediately preceding war, and that such a large arms discrepancy may have the psychological effect of an early, extremely violent confrontation. It is then comparable, say, to the Battle of Shiloh in its ability to create strong expectations of further conflict (McDonough, 1977). If these highly discrepant starting values for armament levels are allowed to dictate future responses---i.e. if, in the famous phrase of Hans Kelsen, ``the cause of war is war itself"---then this arms race is not likely to be controllable; movement toward the high equilibrium would be a benign outcome. If, on the other hand, we have patience and rationality, then the underlying dynamic of this specific Richardson model does indeed make for evolution toward a relatively low level of armament.

Let's now calculate the anticipated future of this armaments competition by solving the above system.

sol1 := dsolve({deq1, deq2, x(0)=k1, y(0)=k2}, {x(t), y(t)});

The plots below suggest that this particular Richardson system may move toward an equilibrium (SELFREF, 1986:111) with low levels of armament. This is precisely what happens with the current model: It generates dangerously different arms levels for a long period of time, but throughout the entire process the difference is rapidly damped as is the general level of armament.

We set the upper limit of time (in months) for the following calculations:

k3 := 250;

Now we examine graphic representations of what turns out, if it is not disrupted, to be a disarmament process. The problem is to recognize this series of changes, early, for what it is, and then to avoid disruptions of it as it develops, presumably governed by the basic equations.

The plot below (Figure 10) shows the current model, along with the phase plane for a large number of comparable, relatively benign interactions.

DEplot({deq1, deq2}, {x(t), y(t)}, t = 0..k3, [[x(0)=k1, y(0)=k2]], x=0..1, y=0..1, thickness=2, color=black, title=`Figure 10: Phase plane: Bi-polar arms race`);

Now, we plot results (Figure 11) that show the disarmament process for this specific instance:

plotX := DEplot({deq1, deq2}, {x(t), y(t)},
t = 0..k3, [[x(0)=k1, y(0)=k2]], scene=[t,x], arrows=NONE, stepsize=35/10, linecolor=BLACK):

plotY := DEplot({deq1, deq2}, {x(t), y(t)},
t = 0..k3, [[x(0)=k1, y(0)=k2]], scene=[t,y], arrows=NONE, stepsize=35/10, linecolor=BLACK):

p1 := textplot([40, (85/100), `orbit for nation y`], align={ABOVE,RIGHT}):

p2 := textplot([10, (68/100), `orbit for nation x`], align={ABOVE,RIGHT}):

plots[display]([p1, p2, plotX, plotY], thickness=3, color=black, title=`Figure 11: Time-series orbits: A highly sensitive disarmament process`);

Once again we have a result that appears to be highly plausible. In the aftermath of a preceding war, nation y has a substantial arms advantage over nation x. The immediate reaction of nation x---and here ``immediate" refers to a period of several years---is to increase its arms levels at a rate that appears initially to be dangerously high and potentially destabilizing. But this appearance is deceptive: As in the case of recovering alcoholics, the underlying differential equations tell us that a latent ``healing" process is already under way. It is important to observe that during this early phase, nation y responds by signalling its clear intention, i.e., by reducing arms at a rate that, in absolute value, exceeds the positive rate of increase of armaments for y. This high-risk phase is perhaps the most crucial: Nation y dissipates its arms advantages rapidly, but in a sense it is exploiting these advantages in a way that seems to induce x toward abandoning a one-sided arms competition after approximately 40 time units. With trust reasonably well established, y appears to be willing to lead x toward further disarmament, albeit only by a small amount.

Conclusions

Of the four examples preceding, none was guided primarily by theory and none was guided primarily by methodology. Each was guided by the interaction of theory and method---books, courses, discussions, thought, research going back several decades. And each, in large part, was inspired by luck: If we had not had the good fortune to run across Harris and Ross' Death, sex, and fertility, the EFM example would not exist and its corresponding mathematical model, if ever looked at, would not have been seen. We know that theory is generally considered paramount, and that those who give equal emphasis to methods are accused often of allowing the methodological tail to wag the theoretical dog. But the central issue has nothing to do with whether or not tail wags dog. The real question is this: How do we make sure that our two finest stalking dogs---theory and method---have an opportunity to interact in ways that are mutually inspirational?

In my estimation, Bunge is not entirely convincing in the epigram that opens this article. If sociologists were entirely closed-minded, we would not have the benefit of the perpetual Comtean festival that spreads across a huge intellectual landscape. On the other hand, Bunge is right in suggesting that we do not respond quickly, audaciously, to intellectual challenges. Computer mathematical systems are one such challenge, an inspiring innovation that has been enticing us, for time enough, from the bushes at the edge of the festival park. The major value of these programs is that, due to their highly flexible experimental capabilities, they give us many algebras and many images to look at. In order to see, it is then incumbent on us to use the sociological imagination, to ask ourselves what sort of social process would embody a given mathematical model, where and when would such a process occur, and how and why would it assume what may have been revealed as a distinctive form.

Notes:

(1) See the comment regarding Bunge at the end of this paper.

(2) In the software package used for this paper, all derivatives---not merely partial derivatives---contain Greek characters.

(3) The remaining six: placing infants in dangerous situations; abandonment; ``accidents"; excessive physical punishment; low biological support; low emotional support. For a discussion of forms of killing that often are not ``deliberate," see Philips (1993).

(4) For an instructive discussion of stability conditions for Richardson models, see Olinick's (1978:33-34) intriguing demonstration of ``the bad effect of good will," in which system properties destroy the best of intentions.

References:

Abell, Martha L. and James P. Braselton. 1994. Maple V by example. Boston: Academic Press.

Allison, Paul D. and John A. Stewart. 1974. ``Productivity differences among scientists: Evidence for accumulative advantage." American Sociological Review 39:596-606.

Bauldry, William C. and Joseph R. Fiedler. 1996. Calculus projects with Maple. Pacific Grove, CA: Brooks/Cole.

Bentley, Richard and Robert Blackburn. 1990. ``Changes in academic research performance over time: A study of institutional accumulative advantage." Research in Higher Education 31:327-45.

Bronson, Richard. 1994. Theory and problems of differential equations. New York: McGraw-Hill.

Brown, Stephanie and Virginia Lewis. 1999. The alcoholic family in recovery: A developmental model. New York: Guilford.

Bunge, Mario. 1997. ``Mechanism and explanation." Philosophy of the social sciences 27:410-65.

Collins, Randall. 1975. Conflict sociology: Toward an explanatory science. New York: Academic Press.

Dawkins, Richard. 1986. The blind watchmaker. New York: W.W. Norton.

Denzin, Norman K. 1987. The recovering alcoholic. Sage: Newbury Park, CA.

Dillon, George and Edna St. Vincent Millay. 1936. Flowers of evil. New York: Harper and Brothers.

Dorn, Harold F. 1959. ``Mortality." Pp. 437-71 in Hauser, Philip M. and Otis D. Duncan, editors. The study of population: An inventory and appraisal. Chicago: University of Chicago Press.

Dowling, Edward T. 1990. Theory and problems of calculus for business, economics, and the social sciences. New York: McGraw-Hill.

Ellis, Wade Jr. et al. 1997. Maple V flight manual, Release 4: Tutorials for calculus, linear algebra, and differential equations. Pacific Grove: Brooks/Cole.

Fisher, Charles S. 1973. ``Some social characteristics of mathematicians and their work." American journal of sociology 78:1094-1118.

Harris, Marvin and Eric B. Ross. 1987. Death, sex, and fertility: Population regulation in preindustrial and developing societies. New York: Columbia University Press.

Hughes-Hallett, Deborah et al. 1994. Calculus. New York: Wiley.

Israel, Robert B. 1996. Calculus the Maple way. Don Mills, Ontario: Addison-Wesley.

Keyfitz, Nathan. 1968. Introduction to the mathematics of population. Reading, MA: Addison-Wesley.

Koestler, Arthur. 1964. The act of creation. New York: Macmillan.

Main, C. F. and Peter J. Seng. 1961. Poems. San Francisco: Wadsworth.

McDonough, James L. 1977. Shiloh---in hell before night. Knoxville: University of Tennessee.

Merton, Robert K. 1973. The sociology of science: Theoretical and empirical investigations. Chicago: University of Chicago Press.

Mitroff, Ian I. and Daryl E. Chubin. 1979. ``Peer review at the NSF: A dialectical policy analysis." Social studies of science 9:199-232.

Nielsen, Francois and Rachel A. Rosenfeld. 1981. ``Substantive interpretations of differential equation models." American sociological review 46:159-74.

Olinick, Michael. 1978. An introduction to mathematical models in the social and life sciences. Reading, MA: Addison-Wesley.

O'Reilly, Edmund B. 1997. Sobering tales: Narratives of alcoholism and recovery. Amherst: University of Massachusetts Press.

Peter, Laurence J. and Raymond Hull. 1969. The Peter principle. New York: W. Morrow.

Phillips, David P. 1993. ``Psychology and survival." Lancet 342:1142-45.

Platt, Jennifer. 1996. A history of sociological research methods in America, 1920-1960. New York: Cambridge University Press.

Porter, Theodore M. 1986. The rise of statistical thinking: 1820-1900. Princeton University Press.

Redfern, Darien and Edgar Chandler. 1996. The Maple O.D.E. lab book. New York: Springer-Verlag.

Richardson, Lewis F. 1960. Arms and insecurity: A mathematical study of the causes and origins of war. Pittsburgh: Boxwood Press.

Saxenian, AnnaLee. 1994. Regional advantage: Culture and competition in Silicon Valley and Route 128. Cambridge: Harvard University Press.

SELFREF, 1986.

SELFREF, 1999 (hassan-1.mws).

Shone, Ronald. 1997. Economic dynamics: Phase diagrams and their economic application. Cambridge: Cambridge University Press.

Silverman, Joseph H. 1997. A friendly introduction to number theory. Upper Saddle River, NJ: Prentice Hall.

Smith, Bruce D. 1995. The emergence of agriculture. New York: W. H. Freeman.

Soroka Michael P. and George J. Bryjak. 1995. Social problems: A world at risk. Boston: Allyn and Bacon.

Stinchcombe, Arthur L. 1968. Constructing social theories. New York: Harcourt Brace.

Sulloway, Frank J. 1998. Born to rebel: Birth order, family dynamics, and creative lives. London: Abacus.

Watson, James D. 1968. The double helix. New York: Atheneum.

Weeks, John R. 1999. Population. Belmont: CA: Wadsworth.

www.nytimes.com/library/world/asia/061198asia-econ.html.

www.orie.cornell.edu/~or115/handouts/tsp/tsp.html.

Zill, Dennis G. 1997. A first course in differential equations with modeling applications. Pacific Grove, CA: Brooks/Cole.