DRAFT: Summer, 2001

TYPE OF ARTICLE: Research Paper

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-4.mws

Abstract
 

We explore differential equations involving alcoholism, social mobility, excess female mortality, and international arms competition. In each of these instances we show that the initial equation, or system of equations, has a sociological plausibility comparable to that of the associated solutions; the solutions do indeed describe time-series trajectories that seem to represent important and unique social processes. We argue that the central challenge of differential equation modeling is to use experimentation to clarify relationships between, on the one hand, the equations and their coefficients and, on the other, the solutions and the time-series orbits created by them. Such feedback interaction of differential equations and their solutions appears to be the basis for further theoretical insight, and rapid assessments of these interactions are now possible largely because modern software encourages experimentation with many combinations of input coefficients.

This paper expands on an argument made by Nielsen and Rosenfeld (1981:161), who recommend that differential equations be interpreted in a way that emphasizes their solutions, i.e., the time-series trajectories of Y values, the orbits of Y, taken to represent behavior of dependent variables through time. We conclude that the most edifying interpretations of differential equations focus on the equations themselves, the resulting trajectories, the relationships between equations and trajectories, and the theoretical significance of all three.

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 an intellectual hybrid of folk wisdom and metaphysical construct: Hitting bottom is said to be a near universal and 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 that occur 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.

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 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 assorted 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 ``she won't turn things around and get better till she hits bottom." The rate of change of the alcoholic's underlyingrecovery 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 sometimes obscuring the linear trend toward recovery.

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

Next, we write the differential equation [note 1]:

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

The dependent variable, for reasons stated earlier, 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 constant _C1 enables us to distinguish, in Figure 1, between the differential equation and the specific solution shown. With , the solution, presumably depicting observable behavior, will have a y-intercept of . See Figure 1 to verify this result.

> _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, an important uniformity across all cases that may be amenable to empirical test. If it were indeed a uniformity, such a temporal pattern would provide an operational definition of 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 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. 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), which suggests that at some level of perception they already discern strong prospects of success, so that the continued relapses, whether consciously or not, are the sort of charade which in alcoholics is likely to be misinterpreted.

The values k11 and k12 define 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 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.

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, which we shall take to be slightly above an average of unity.

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 increasingly manifest.

Having incorporated the value , the trajectory defined by sol1 is 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; Faia, 1975; 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).

This example, then, has important analogues to that of recovering alcoholics: Again, surface variations obscure an underlying uniformity.

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, earlier 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. In modern times, feticide through induced abortion is also likely 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 through ``spontaneous" abortion, then selectivity against females probably would be diminished to a considerable degree; therefore---and perhaps contrary to Harris and Ross---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 what appears to be deliberate killing [note 2], 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.

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 ``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. 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 (Faia, 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, 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 (Faia, 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.

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, the female population is likely to be 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 become smaller around the optimum. 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 (Figure 8) and encounter a time-series process that has much in common with the sinusoid patterns implied by Stinchcombe (1968:89) and made explicit in Faia (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):

> p1 := textplot([60, 10, `male population`], align={ABOVE,RIGHT}):

> p2 := textplot([220, 14, `female population`], align={ABOVE,RIGHT}):

> plots[display]([p1,p2,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 presumably has 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 immediately 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 will then 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. Here we obtain the precise values.

> 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 arguably 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.

> 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.

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 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 3]

> 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 in such a situation 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, contending nations have patience and rationality, then the underlying dynamic of this specific Richardson process does indeed make for evolution toward a relatively low level of armament.

Let's now calculate the anticipated future of 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 (Faia, 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; it may be a reasonable representation of armaments competition between the United States and the Soviet Union/Russia. 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 small steps.

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, to cite one instance, 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, or with any comparable canine conundrum. 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?

Notes:

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

(2) 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).

(3) 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.

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