Does demography need differential equations

The starting point for this essay is the observation—partly impressionistic—that demography as a discipline has tended to neglect the predator-prey equations in courses, textbooks, compendia, and research
papers. This is surprising, since the equations bear the name of A. J. Lotka, one of the acknowledged founders of modern demography. This relative neglect is unfortunate, since a central fact about the human species is that we are deeply implicated in nature as both predator and prey. Possible explanations for
this situation are discussed, including a general neglect of systematic theory, and of differential equations, a branch of mathematics especially suited to the statement and exploration of theories of demographic processes.


Predator-Prey and Other Differential Equations in Demographic Literature
Predator-prey. A JSTOR search of 24 population studies journals on the terms 'predator-prey' and 'Lotka-Volterra' yields less than two dozen citations. In most of these, the term or concept is discussed only in passing. In a few cases, predator-prey equations are used to study the interactions between two human populations (Keyfitz 1965;Hudson 1970). Keyfitz, in a study of marriage and the two-sex problem in population models, begins with a quote from Volterra to the effect that the study of a population in isolation '... is inadequate, no matter how elaborate the model may be, when the population in question is in effective ecological contact with some other population' (Keyfitz 1965: 276). Hudson uses the predator-prey model to study population growth and migration in a two-region (metropolitan/nonmetropolitan) system. Interestingly, Hudson is a geographer, not a demographer. The JSTOR search reveals virtually no articles dealing at length with the interactions of humans with non-human species.
The neglect of the predator-prey equation in demography is a special case of a broader neglect of the use of differential equations, as is evident from a quick survey of the literature.
Differential equations in texts. Demography is not particularly rich in textbooks, since publishers are reluctant to deal with a relatively small undergraduate market. Nevertheless, a review of a fair sample of recent and older works turns up few instances of the use of differential equations. This is so of substantive texts, such as John Weeks's popular undergraduate text, but it also is true of more technical works. One searches in vain for differential equations in older works, such as Barclay's Techniques of Population Analysis, or Shryock and Siegel's Methods and Materials of Demography.
An authoritative recent work by Preston et al. (Demography: Measuring and Modeling Population Processes, 2001) uses differential equations in only a few places, notably in discussing exponential growth and the force of mortality. Exceptions may be found in specialized works by mathematical demographers, or the journal Mathematical Population Studies. But, as noted earlier, this body of work stands somewhat apart from the demographic mainstream -substantive demography, especially social demography -whereas differential equations appear to be central to substantive exposition in many other disciplines.
Differential equations in journal articles. There are only a handful of mainline demographic papers in which differential equations play a central role, or even appear. A notable exception is a paper by Hernes (1972) on marriage. Hernes presents a differential equation of the first marriage curve, based on behavioral assumptions of: (a) some initial level of 'marriageability;' (b) an exponential decline of marriageability with age; (c) pressure to marry based on the proportion in a cohort already married; and (d) limits to marriage due to declining availability of partners. His behavioral reasoning leads to a relatively simple and easily understood differential equation. The behavioral assumptions have become outmoded by subsequent events (notably the rise of extramarital sex, divorce, and cohabitation), but it was a strong beginning. However, the Hernes model was largely ignored by demographers, with only a few exceptions, and the Coale-McNeil model (Coale and McNeil 1972) became canonical. 3 I compared the two in a 1993 paper (Burch 1993. See also Chap. 6), noting that Hernes's model was more elegant, fit cohort data just as well, and had more behavioral content. The Coale-McNeil model, of course, still has the merit of modeling other aspects of the marriage process, such as culturally defined statuses and waiting times.
Another example is a paper by Rosero-Bixby and Casterline (1993) on fertility decline in Costa Rica. They develop a differential equation model for the diffusion of family planning use, and its impact on fertility over time. It is a compartment model -with women moving from non-motivated, to motivated but not using family planning, to using family planning -with elements of point-source and interaction diffusion in variants of the basic model. Their paper is cited occasionally, but does not seem to have inspired replication or further development.
More examples could be found, but there are not many. As a rule, empirical/ quantitative articles consist of statistical analysis of data, and theoretical articles are not stated in rigorous, formal language; the few that have been are more apt to resort to formal logic than to mathematics and tend to be relatively static.

Lotka's Patrimony
Many demographers, especially mathematical demographers, would agree that Alfred J. Lotka is one of the founders of modern demography. And although we claim to be his intellectual descendants, it is interesting how little of his scientific patrimony we have accepted -stable population theory, reproduction rates, and, more recently, the demography of kinship, inspired by his pioneering work on the prevalence of orphanhood by age. Much else has been left behind. Lotka, of course, was not just a demographer. His early training was in the physical sciences, primarily chemistry and biology. He became active in demographic circles later, including at early meetings of the International Union for the Scientific Study of Population. His work best known to demographers is entitled Demographic Analysis, With Special Reference to the Human Species. This monograph is in fact the second part of a larger work entitled Analytic Theory of Biological Associations. The first part, seldom referenced by demographers, is entitled simply Principles (see Lotka 1934Lotka /1939 On p. 8 of Principles one encounters the differential equation dX i =dt ¼ F i X 1 ; X 2 ; . . . ; X n ; P; Q ½ Lotka introduces it as a general statement of the principle that the rate of increase of any component in a system is a function of the quantity of all other components in the system plus parameters defining the characteristics of each component P, as well as other parameters Q that 'serve to complete the definition of the state of the system' (Lotka 1934: 8).
Later, Lotka begins a chapter entitled 'Biological Stoichiometry' with the following statement, introducing a system of differential equations: 'In asserting that at each instant the rate of growth of each species in the system depends on the size of that species and of all the other species present, as well as on parameters P and Q, we have already noted that the analytic expression of this very general proposition takes the form. . .' -a system of n differential equations follows (see Fig. 5.1). Shortly after, the system is abbreviated by dropping the parameters P and Q, on the grounds that the characteristics of species and of the environment ('climate, topography, etc.') change little over relatively short periods (Lotka 1934: 32-33).
Here, the human species is firmly embedded in a biological system consisting of many other species. His framework is essentially a systems framework. The scope of the systems he envisions is shown in a complex diagram of the interrelations of fish populations and their food supplies (Lotka 1934: 37, Table 2). Lotka was thinking in terms of systems well before the systems concept became popular 30 or so years later.
Later in the chapter, he presents equations for two species in interaction, the 'predator-prey equations,' and develops an expression for the logistic curve, then thought of as a 'law' of population growth. In every case, the development is in terms of differential equations, sometimes leading to an analytic solution, sometimes not. But the differential equations frame the discussion.
In his earlier work, Elements of Physical Biology (1924), some of these ideas are developed in greater detail, including the extension of the two-interacting-species model to three or more, 5 as well as a description of several types of two-species interaction other than as predator-prey. One of his examples relates to humans' relationship to domestic animals such as cattle and poultry, which we breed and nurture to eat. 6 5 Stella, student-oriented systems dynamics software, provides an interesting game in which the student is challenged to bring three interacting populations (deer, wolves, and grass) into equilibrium. The near impossibility of doing so is a powerful demonstration of the effects of non-linearity in systems. A similar challenge is posed in the agent-based modeling software NetLogo, in one of its tutorials. 6 A recent collection of classic papers on mathematical demography (Wachter and LeBras 2013) contains no index entry for predator-prey, nor do the authoritative texts by Preston et al. (2001) and by Wachter (2014). Hanneman's (2005) online text, by contrast, contains a whole section on

Lotka the Human Demographer
In the second part of Analytic Theory..., Lotka begins: Species exist in mutual relationships with one another, such that it is true to say that it would be impossible to make a well-rounded study of a species without taking account of the large number of other species which influence it in one way or another. (Lotka 1939: 5) He continues: However, there exist among the internal factors of a population of living beings (such as natality, mortality, growth, etc.) a large number of relationships which permit and even demand a special study, without the necessity at each step of taking explicit account of other species occupying the same locale. This study, in fact, constitutes a well-defined body of research and of results, which we take up in the present volume, with particular attention to the human species, for which we possess an abundance of data. (Lotka 1939: 5) This comes close to a description of demography as we know it.
Lotka divides the study of human populations into two parts. One he terms demographic analysis, a branch of mathematics (i.e., analysis) applied to human population dynamics, to discover and state necessary relations among demographic variables. He distinguishes this from a second part of demography, which he calls statistical demography, the statistical study of relationships among demographic variables. The two branches seem to be related as theory and empirical research.

Lotka the Theorist
Lotka assumes the importance of empirical research -he is, after all, a scientistbut clearly thinks it is not enough: ... one will find more satisfying to the spirit that knowledge more complete, or at least deeper, which one obtains when one has succeeded in taking account of not only the empirical relationships, whose physical causes and logical reasons escape us ... but also the necessary relationships [imposed by the laws of logic and of physics] among the quantities describing the state of and the changes in a population. (Lotka 1939: 6) the predator-prey model. This is a reminder that systems dynamics software makes it relatively easy to work with systems of differential equations, even for those who are not mathematicians. dX 1 /dt = F 1 [X 1 ,X 2 , … ,X n , P, Q] dX 2 /dt = F 2 [X 1 ,X 2 ,… ,X n , P, Q] … dX i /dt = F i [X 1 ,X 2 , … ,X n , P, Q] … dX n /dt = F n [X 1 ,X 2 , … ,X n , P, Q] It appears that Lotka was at heart a theorist. And he considers differential equations to be a fundamental tool of theory. In Elements of Mathematical Biology, he writes: In the language of the calculus, the differential equations display a certain simplicity of form, and are therefore, in the handling of the theory at least, taken as the starting point, from which the equations relating to the progressive states themselves, as functions of the time, are then derived by integration. (Lotka 1956: 42) He adds in a footnote: 'In experimental observation usually (though not always) the reverse attitude is adopted.' Demography typically uses the integral rather than the original differential equation.
The Hernes model mentioned earlier (see also Chap. 6 below) provides a nice illustration. The differential equation is simple and transparent. Its integral, giving proportion married by age in a cohort, is more complicated and harder to intuit, but more useful for fitting cohort data on proportions married by age.
One wonders whether Lotka contemplated a third part to Analytic Theory..., which would have revisited his system of equations, discussed at length in the earlier monograph, to study relationships between human populations and other species. But clearly demography has focused on the more limited study of human population as defined above. In doing so, we have left behind a large part of Lotka's intellectual heritage, including: (a) a strong emphasis on theory as well as empirical, statistical research; (b) regular use of differential equations as a natural tool for the theoretical study of process; and (c) the study of the interrelationship between human and non-human populations.
A student of demography could go far in the field without ever being taught to think of the human species as both predator and prey. We study diseases as causes of death, not as a manifestation of micro-organisms using human bodies as habitat. We study fish as a natural resource, not so much as a population on which we prey, although this is changing with the disappearance of many stocks. Joel Cohen notes in a paper on population projections: 'Other species are recognized explicitly only in the recent innovation of quantifying the devastating impacts of HIV and AIDS' (2003: 1172).
As noted above, Hernes and Rosero-Bixby and Casterline used differential equations to study processes and systems that demographers study regularlycohort behavior, multi-state systems, and diffusion. Lotka and others (notably biological ecologists) use them to study processes and systems that we have largely ignored.
In other cases, we have studied certain systems, but only in a limited, technical way. The logistic model is a case in point. In ecology and in differential equations texts, it is introduced as a differential equation. In demography, it typically is presented simply as a mathematical curve (the analytic solution of the differential equation), invariably identified as a technique for population projection. As such, it is rejected in favor of the standard cohort-component technique, partly because it deals only with total population, not with the components of growth. In an obvious sense this is partly so, but in another sense it is not. When ecologists (e.g., Wilson and Bossert 1971) discuss the logistic curve, it is derived from assumptions relating 5.5 Lotka the Theorist to the relationships among population density, fertility, and mortality. In demography, the logistic is simply a population projection tool. In ecology, it is a theoretical model.
What is the explanation for our relative lack of interest in multi-species models? Human ethnocentrism, perhaps? There is a large element of exceptionalism in our view of our place in the natural world. The introduction to Demographic Analysis and Synthesis: A Treatise in Population (2005), a four-volume work of nearly 3000 pages, states that the treatment is limited to human demography, partly because the material on those is already so vast, but also '. . .to highlight its singularity. Humans are not just statistical units, simple living beings, or merely social creatures like bees and ants. . .. In Aristotle's phrase, man is a political animal, thinking and influencing his or her own individual and collective destiny, which situates the study of population dynamics clearly in the field of social science rather than biology. ' (p. xxvi, emphasis added) Granted there is room for different emphases in the study of bees and ants versus humans, there is ever-increasing evidence of overlap between biology and the sciences of human behavior. Humans are a biological species.
In 'Population dynamics of humans and other animals' (1987), Ronald Lee struck a better balance, arguing that density-dependent phenomena that affect most non-human species also affect humans, although indirectly. 7 Another possible explanation for a neglect of differential equations is the fact that a large proportion of practicing demographers, especially social demographers, simply do not know the mathematics of differential equations, even the low level of knowledge necessary to understand the predator-prey equations.
Other disciplines, notably biological ecology or population biology, have continued to develop Lotka's insights and equations, including the systematic study of inter-species relationships. Gotelli's introductory text (1998), for example, devotes about 50 out of 200 pages to the topic.

Abbot on Coleman vs. Blalock
Andrew Abbot, in his stimulating paper 'Transcending general linear reality' (1988) suggests a similar neglect of differential equations as a tool in empirical sociology. He notes the domination of quantitative sociology by multivariate statistical analysis based on the general linear model. In a footnote, he compares citations to Blalock's 1960 text Social Statistics (featuring the use of regression) to Coleman's (1964) text Introduction to Mathematical Sociology [featuring the use of differential equations]. In the period 1966-1970, there were 162 citations of Blalock vs. 117 of Coleman;by 1980, it was 117 vs. 24, andby 1984, it was 104 vs. 15. He notes that Coleman's work has never been reprinted. He attributes the dominance of regression analysis to its 'commodification' in easy-to-use packages.
Abbot makes the useful distinction between the 'representational' interpretation of regression models ('My model represents the social system') and the 'entailment' interpretation ('If my theory is correct, then I should get certain results in my regression model'). One is largely descriptive of relationships among measured variables; the other is oriented towards testing theory. Abbot considers the representational interpretation a case of reification, the positing of a 'general linear reality' based on a highly abstract empirical model.
Blalock presented regression clearly as a tool of empirical research, although, in keeping with the logical empiricist doctrine of the time, he viewed the resulting empirical generalizations as 'laws,' and therefore as the essential foundation blocks for theory. Coleman tends to see differential equations as a theoretical tool used to 'represent' dynamic systems. Either tool can be used in Abbot's entailment mode.
The impact of 'commodification' is difficult to judge. But it is worth noting that the commodification of differential equations also occurred relative early -Dynamo was developed in the 1960s and became commercially available soon after. In the same year that Abbott wrote, Robert Hanneman published a book urging sociologists to consider Dynamo as a tool for modeling dynamic social systems (Hanneman 1988). 8 And the major mathematical software packages (Mathematica, Maple, Mathcad, Matlab) regularly expanded their utilities for solving differential equations. Why did empirical sociology and demography buy so much of the one commodity and not the other?
Several possible answers to the first question suggest themselves: 1. Differential equations are not necessary or particularly useful for the study of most issues of greatest interest to demographers. Other analytic methods have been more fruitful. 2. The average demographer has little competence in the use of differential equations. That level of mathematics has not been required for entrance into, or successful completion of, most graduate programs. 3. Demography has avoided substantive areas that essentially require the use of differential equations, including non-linear equations. 4. Differential equations are more a theoretical than an empirical tool, and demographers have never given high priority to theory, as opposed to data and techniques. 8 Hanneman has continued the use of systems dynamics software in an online work entitled Spatial Dynamics of Human Populations: Some Basic Models, (2005). (http://faculty.ucr.edu/~hanneman/ spatial/index.html). He has switched from the now obsolete Dynamo software to Berkeley Madonna, developed with U.S. government support. An interesting but rare use of a systems dynamics approach by economist-demographers is to be found in the Wonderland project of Sanderson et al. See Sanderson (1994).
I would argue that #1 is questionable. Why should a tool that has proven so fruitful in other sciences be of little use to demography? Answer #2 lies at the heart of the problem: Demographers generally were not schooled in differential equations, so we didn't try to use them, and avoided topics that required their use even at the most elementary level (as with predator-prey).

Systems Dynamics Software
In light of (2) in the previous section, one can ask a second question: Why has demography not taken advantage of systems dynamics software? It enables the 'mathematically challenged' to construct and work with models of complex systems with feedbacks, in effect, systems of differential equations. And it necessarily orients thinking towards dynamics and process, not just cross-sectional recursive relationships.
The invention of systems dynamics software is generally attributed to an engineer, Jay Forrester, who applied engineering principles of feedback and control to social systems. His first work, Industrial Dynamics, was published in 1961. World Dynamics appeared in 1971 and became the basis for the influential and controversial book The Limits to Growth by Meadows et al. (1972). The MIT systems dynamics school has generated a large literature, both general works and simulations of particular systems, and has stimulated the development of other software packages with similar structure and aims. 9 It is characteristic of much of the literature of the MIT group that more attention is paid to the building of models than to their relationship to the real world. A basic hardback text from the MIT group (Roberts et al. 1983), for example -a work of over 500 pages -contains no chapter on testing, validation, parameter estimation, or goodness of fit; indeed, these words don't even appear in the index. This exclusion apparently is deliberate. The authors include 'model evaluation' as one of the phases in the model-building process, and comment: [N]umerous tests must be performed on the model to evaluate its quality and validity. These tests range from checking for logical consistency to matching model output against observed data collected over time, to more formal statistical tests of parameters used within the simulation. Although a complete discussion of model evaluation is beyond the scope of the book, some of the important issues involved are presented in the case examples. (Roberts et al. 1983: 9) The main technique of model evaluation is the demonstration that the model fits one or more empirical time series of outputs. If the model can generate the output reasonably closely, then it is considered a good model. But it is not 'proven,' of course. To assume so is to commit the fallacy of affirming the antecedent.
Whatever the intent, it is hard for the reader to avoid the impression that evaluating a model's fit to real world, or at least to data, is less interesting and less important than building the model. An earlier work from the same group makes clear that the emphasis on model building rather than model estimation or testing goodness of fit reflects a deepseated attitude towards scientific and policy analysis, one somewhat at odds with traditional statistical methodology: The systems dynamics approach to complex problems takes the philosophical position that feedback structures are responsible for the changes we experience over time. The premise is that dynamic behavior is the consequence of system structure. (Richardson and Pugh 1981, p. 15) That is, if one has the structure right, the details (for example, specific parameter values) don't matter so much. And later: . . .experience with feedback models will convince the reader that model behavior really is more a consequence of structure than parameter values. One should therefore be more concerned with developing the arts of conceptualization and formulation than finding ultimate parameter selection methods. Our advice for beginners would be to estimate parameters with good statistics (data) but not Statistics (mathematical methods). In the systems dynamics context the latter are a collection of power tools that just might cut off your intuition. (Richardson and Pugh 1981, p. 240) In general, they are skeptical about the value of correlational approaches and standard regression techniques, especially when dealing with dynamic models with feedback (Richardson and Pugh 1981, pp. 238-239).
Validating a model in this tradition, as noted above, is achieved primarily by comparison of model output of key variables with 'reference behavior modes,' essentially observed time-series measures of the phenomena of interest. But still, the greater emphasis is placed on causal understanding: how does the process really work? Regression equations, with coefficients attached to a set of distinct factors to reflect their relative importance, are viewed as uninformative, at least as a representation of process in an underlying system. In Abbott's terms, they reject a 'representational' approach to linear regression models in favor of an approach that they feel accords better with our intuition of how a system actually works.
A later example in this tradition criticizes an econometric analysis of milk production, expressed as a function of GNP, interest rates, etc., because the model nowhere mentions cows; and a model of human births (as a function of birth control, education, income, health, religion, etc.) because the model nowhere mentions mothers (HPS 1996: 25-8). Much of these early texts seemed almost hostile to statistical research in the social sciences.
Substantive research using the systems dynamics approach was heavily criticized by social scientists and others. A special target was Forrester's 'world model,' the basis for The Limits to Growth. The model was so large and complex that some questioned whether it could be meaningful. 10 It went beyond what could be intuited, and was so large that there was high risk of programming errors, functional misspecification, and wrong parameters. Despite the size of the model, as my former colleague Tom Wonnacott constantly reminded me, the resource module contained no variable for price. Although a best-seller, The Limits to Growth was dismissed by many economists, demographers, and others.
A recent re-evaluation of The Limits to Growth studies (Bardi 2011) suggests that much of the early criticism was misplaced, based on misunderstanding of the purpose of the simulations or, in some cases, outright errors in describing the models or their results. And, he notes that some of central projections of the studies have been borne out by subsequent events.
There is a special reason why demographers might criticize this body of work. Population projections are done in an unconventional way and use unconventional language. The absolute numbers of births and deaths flowing into and out of a population per unit of time are referred to as rates (per unit of time) -a perfectly good usage in calculus and common in ecology, but at odds with demographic usage. The relative numbers of births and deaths are referred to as fractional rates. And the number of deaths is calculated by dividing population by average lifetime (life expectation at birth from a life table), instead of using the crude death rate (the rough equivalence obtains, of course, only in the stationary population model). Instead of surviving an age group to the next older age group using survival ratios, age-groups remain in place, as it were, with deaths being subtracted, and population 'aging in' from the age group below, and 'aging out' to the age group above. For a 5-year age group, for example, it is assumed that, apart from deaths, one fifth will move to the next-highest age group, with one fifth of the next-lowest age group moving in. The language and procedure strike the average demographer as improper, and suggest a lack of understanding of population dynamics. By convention, they are indeed incorrect, and a student who used this approach on a demographic techniques exam probably would get a failing grade. But in fact, given identical input, the systems dynamics procedure can generate projections by age and sex that do not differ appreciably from those produced by the standard cohort-component projection technique. Both approaches, of course, contain approximations.
The intellectual history of systems dynamics remains to be written. But my impression is that some early excesses and some disciplinary rivalries (Forrester, after all, was an engineer who did not 'convert' to economics or demography) gave a perfectly sound approach and its associated software a bad name. In talking to colleagues about Dynamo, I remember getting a distinct impression that reputationally it was 'lower-class' software. But I think we may have thrown the baby out with the bathwater.
Although still generally ignored by demographers and many other quantitative social scientists, the systems dynamics approach has continued to develop, and is now widely taught and used in other circles. Nearly 50 years of practice have led to greater balance and sophistication, such that many earlier critiques -including some of my comments above -have less relevance or force.
One indication that systems dynamics has come of age is its inclusion in the recently released Wolfram System Modeler software package. While relying heavily on Modelica, it also provides a separate systems dynamics utility, with most of the main features of older software, such as the graphic interface for the initial definition of a system. Clearly, Wolfram thinks that various forms of systems modeling, including the relatively accessible systems dynamics approach, are as important to contemporary scientific research and policy analysis as mathematics and programming, both of which are covered in their older software, Mathematica.
More direct evidence that systems dynamics has come of age is found in the work of John D. Sterman, Professor of Management at the Sloan School of Management, Massachesetts Institute of Technology. His 900+ page textbook -Business Dynamics: Systems Thinking and Modeling for a Complex Worlddevelops the systems dynamics approach in detail, and with great common sense and balance. And while oriented to business, it includes scientific examples, including some demographic models.
Sterman's central argument is not that systems dynamics models can represent real-world systems perfectly, but only that they can often do so better than the 'mental models' that we inevitably develop and use in analysis and practice. Our mental models typically are overly simple, linear, relatively static, and unable to think effectively about feedback and delays. Similar limitations affect many of our multivariate statistical, econometric, and demographic models, which are single equation, linear or log-linear, static, without feedback or delays.
But unlike some earlier proponents of systems dynamics discussed above, Sterman has a healthy respect for statistics. In discussing the estimation of model parameters, he notes: 'The basic choice is formal statistical estimation from numerical data, or judgmental estimation' (Sterman 2000, p. 867). He continues: 'Systems dynamics modelers are well-advised to study econometrics and other approaches to formal parameter estimation. It is essential to know how the important regression techniques work, what their maintained hypotheses and limitations are, and when each tool is appropriate' (p. 868). This is a far cry from the complaint that a regression equation on milk production fails to mention cows. Judgement comes into play when there are no reliable statistical measures, direct or indirect, on a variable thought to be important. The systems dynamics tradition prefers to make an informed guess rather than to omit that variable altogether, relegating it to the error term.
So why has demography -or sociology, for that matter -not taken greater advantage of these tools? In addition to the possible answers given earlier to the more general question, the following come to mind: 1. There are inherent flaws in the systems dynamics approach and associated software 11 ; 2. Demographers were put off by the exaggerated claims of early systems dynamics modelers, and by their seeming indifference, and even hostility, towards statistical research in the social sciences; 3. We dismissed their population models because they did not use the 'correct' approach and terminology, that is, the canonical approach in demography; 4. Quantitative social scientists in general often viewed systems modeling as second-rate empirical work, dealing with made-up numbers instead of hard data. Social theorists, on the other hand, assumed it was 'number crunching,' since it relied on the computer and dealt with numbers and quantitative relationships. Thus, a valuable tool fell through the cracks.

Concluding Comment
Demography is a wonderful discipline. I have come to think of it as a better discipline than is generally recognized, because we have not codified and presented it in the most effective way. And clearly, I think it might be an even stronger discipline if it had assimilated the regular use of differential equations in general, and systems dynamics software in particular. The latter would have allowed those of us who lack a thorough grounding in mathematics to work with relatively complex systems of differential equations. This is not just for the sake of using them, but to help us with thought processes that need help. Their use would encourage us to think more about dynamics and process, and not just cross-sectional relationships and equilibria. They could help us think better about complex social and demographic systems containing non-linear relationships and feedbacks. They could help us introduce more clarity in our theoretical models (for example, transition theory) typically stated in words and manipulated by everyday logic. And they could introduce these intellectual habits to our students -even sociology undergraduates who typically know little mathematics. The use of differential equations could also help us to extend our discipline to consider topics previously neglected. It would help us to develop a richer portfolio of population growth models, beyond the exponential, stable, and projection models. We might begin to renew a serious interest in the logistic, which in the very long term may apply to human population after all (see Lee 1987). We could learn about the Allee effect 12 from our biological cousins in ecology; in almost 60 years in the field, I had never heard about this in demography, yet it would seem to have relevance to our past and future. We would be better equipped to study interactions among humans and other species, to finally recognize and accept the fact that we are both predator and prey.
In all of this, we must get over a common confusion referred to several times above, a confusion of differential equation models with empirical work. They are not a substitute for statistical investigation, qualitative description, or other forms of empirical study. Rather they are a tool for the construction and exploration of the theory and theoretical models that attempt to explain our empirical findings. Demography is generally thought to be rich in data and technique, and poor in theory. I have suggested elsewhere (Burch 2003a, b) that we have more and better theory than is generally thought. But our body of theory could be richer still if we were to take advantage of both classic (differential equations) and contemporary (systems dynamics software) tools for the statement and manipulation of theoretical ideas about demographic processes.