A question about sigmoid functions prompted me to collect a lot of small models that I’ve used over the years.
A sigmoid function is just a function with a characteristic S shape. (OK, you have to use your imagination a bit to get the S.) These tend to arise in two different ways:
As a nonlinear response, where increasing the input initially has little effect, then considerable effect, then saturates with little effect. Neurons, and transfer functions in neural networks, behave this way. Advertising is also thought to work like this: too little, and people don’t notice. Too much, and they become immune. Somewhere in the middle, they’re responsive.
Dynamically, as the behavior over time of a system with shifting dominance from growth to saturation. Examples include populations approaching carrying capacity and the Bass diffusion model.
Correspondingly, there are (at least) two modeling situations that commonly require the use of some kind of sigmoid function:
You want to represent the kind of saturating nonlinear effect described above, with some parameters to control the minimum and maximum values, the slope around the central point, and maybe symmetry features.
You want to create a simple scenario generator for some driver of your model that has logistic behavior, but you don’t want to bother with an explicit dynamic structure.
The examples in this model address both needs. They include:
I’m sure there are still a lot of alternatives I omitted. Cubic splines and Bezier curves come to mind. I’d be interested to hear of any others of interest, or just alternative parameterizations of things already here.
Dynamic cohorts: a new approach to managing detail
While it is desirable to minimize the complexity of a model, some problems require the detailed representation of heterogeneous subgroups, where nonlinearities prevent aggregation or explicit chronological aging is needed. It is desirable to have a representation that avoids burdening the modeler or user computationally or cognitively. Eberlein & Thompson (2013) propose continuous cohorting, a novel solution to the cohort blending problem in population modeling, and test it against existing aging chain and cohort-shifting approaches. Continuous cohorting prevents blending of ages and other properties, at at some cost in complexity.
We propose another new solution, dynamic cohorts, that prevents blending with a comparatively low computational burden. More importantly, the approach simplifies the representation of distinct age, period and cohort effects and representation of dynamics other than the aging process, like migration and attribute coflows. By encapsulating the lifecycle of a representative cohort in a single entity, rather than dispersing it across many states over time, it makes it easier to develop and explain the model structure.
Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.
This is a third-order ecological model with juvenile and adult prey and a predator:
I just picked up a copy of Hartmut Bossel’s excellent System Zoo 1, which I’d seen years ago in German, but only recently discovered in English. This is the first of a series of books on modeling – it covers simple systems (integration, exponential growth and decay), logistic growth and variants, oscillations and chaos, and some interesting engineering systems (heat flow, gliders searching for thermals). These are high quality models, with units that balance, well-documented by the book. Every one I’ve tried runs in Vensim PLE so they’re great for teaching.
I haven’t had a chance to work my way through the System Zoo 2 (natural systems – climate, ecosystems, resources) and System Zoo 3 (economy, society, development), but I’m pretty confident that they’re equally interesting.
You can get the models for all three books, in English, from the Uni Kassel Center for Environmental Systems Research – it’s now easy to find a .zip archive of the zoo models for the whole series, in Vensim .mdl format, on CESR’s home page: www2.cesr.de/downloads.
To tantalize you, here are some images of model output from Zoo 1. First, a phase map of a bistable oscillator, which was so interesting that I built one with my kids, using legos and neodymium magnets:
Population sector extracted from the World3 model.
Documented in Dynamics of Growth in a Finite World, by Dennis L. Meadows, William W. Behrens III, Donella H. Meadows, Roger F. Naill, Jorgen Randers, and Erich K.O. Zahn. 1974 ISBN 0-9600294-4-3 . See also Limits to Growth, The 30-Year Update, by Dennis Meadows and Eric Tapley. ISBN 1-931498-85-7 .
We report experiments assessing people’s intuitive understanding of climate change. We presented highly educated graduate students with descriptions of greenhouse warming drawn from the IPCC’s nontechnical reports. Subjects were then asked to identify the likely response to various scenarios for CO2 emissions or concentrations. The tasks require no mathematics, only an understanding of stocks and flows and basic facts about climate change. Overall performance was poor. Subjects often select trajectories that violate conservation of matter. Many believe temperature responds immediately to changes in CO2 emissions or concentrations. Still more believe that stabilizing emissions near current rates would stabilize the climate, when in fact emissions would continue to exceed removal, increasing GHG concentrations and radiative forcing. Such beliefs support wait and see policies, but violate basic laws of physics.
The climate bathtubs are really a chain of stock processes: accumulation of CO2 in the atmosphere, accumulation of heat in the global system, and accumulation of meltwater in the oceans. How we respond to those, i.e. our emissions trajectory, is conditioned by some additional bathtubs: population, capital, and technology. This post is a quick look at the first.
I’ve grabbed the population sector from the World3 model. Regardless of what you think of World3’s economics, there’s not much to complain about in the population sector. It looks like this:
People are categorized into young, reproductive age, working age, and older groups. This 4th order structure doesn’t really capture the low dispersion of the true calendar aging process, but it’s more than enough for understanding the momentum of a population. If you think of the population in aggregate (the sum of the four boxes), it’s a bathtub that fills as long as births exceed deaths. Roughly tuned to history and projections, the bathtub fills until the end of the century, but at a diminishing rate as the gap between births and deaths closes:
Notice that the young (blue) peak in 2030 or so, long before the older groups come into near-equilibrium. An aging chain like this has a lot of momentum. A simple experiment makes that momentum visible. Suppose that, as of 2010, fertility suddenly falls to slightly below replacement levels, about 2.1 children per couple. (This is implemented by changing the total fertility lookup). That requires a dramatic shift in birth rates:
However, that doesn’t translate to an immediate equilibrium in population. Instead,population still grows to the end of the century, but reaching a lower level. Growth continues because the aging chain is internally out of equilibrium (there’s also a small contribution from ongoing extension of life expectancy, but it’s not important here). Because growth has been ongoing, the demographic pyramid is skewed toward the young. So, while fertility is constant per person of child-bearing age, the population of prospective parents grows for a while as the young grow up, and thus births continue to increase. Also, at the time of the experiment, the elderly population has not reached equilibrium given rising life expectancy and growth down the chain.
Achieving immediate equilibrium in population would require a much more radical fall in fertility, in order to bring births immediately in line with deaths. Implementing such a change would require shifting yet another bathtub – culture – in a way that seems unlikely to happen quickly. It would also have economic side effects. Often, you hear calls for more population growth, so that there will be more kids to pay social security and care for the elderly. However, that’s not the first effect of accelerated declines in fertility. If you look at the dependency ratio (the ratio of the very young and old to everyone else), the first effect of declining fertility is actually a net benefit (except to the extent that young children are intrinsically valued, or working in sweatshops making fake Gucci wallets):
The bottom line of all this is that, like other bathtubs, it’s hard to change population quickly, partly because of the physics of accumulation of people, and partly because it’s hard to even talk about the culture of fertility (and the economic factors that influence it). Population isn’t likely to contribute much to meeting 2020 emissions targets, but it’s part of the long game. If you want to win the long game, you have to anticipate long delays, which means getting started now.