Environmental Homeostasis

Replicated from

The Emergence of Environmental Homeostasis in Complex Ecosystems

The Earth, with its core-driven magnetic field, convective mantle, mobile lid tectonics, oceans of liquid water, dynamic climate and abundant life is arguably the most complex system in the known universe. This system has exhibited stability in the sense of, bar a number of notable exceptions, surface temperature remaining within the bounds required for liquid water and so a significant biosphere. Explanations for this range from anthropic principles in which the Earth was essentially lucky, to homeostatic Gaia in which the abiotic and biotic components of the Earth system self-organise into homeostatic states that are robust to a wide range of external perturbations. Here we present results from a conceptual model that demonstrates the emergence of homeostasis as a consequence of the feedback loop operating between life and its environment. Formulating the model in terms of Gaussian processes allows the development of novel computational methods in order to provide solutions. We find that the stability of this system will typically increase then remain constant with an increase in biological diversity and that the number of attractors within the phase space exponentially increases with the number of environmental variables while the probability of the system being in an attractor that lies within prescribed boundaries decreases approximately linearly. We argue that the cybernetic concept of rein control provides insights into how this model system, and potentially any system that is comprised of biological to environmental feedback loops, self-organises into homeostatic states.

See my related blog post for details.

Continue reading “Environmental Homeostasis”


Wonderland model by Sanderson et al.; see Alexandra Milik, Alexia Prskawetz, Gustav Feichtinger, and Warren C. Sanderson, “Slow-fast Dynamics in Wonderland,” Environmental Modeling and Assessment 1 (1996) 3-17.

Here’s an excerpt from my 1998 critique of this model: Continue reading “Wonderland”

Early warnings of catastrophe

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models (PLOS ONE – open access)

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:

See my related blog post on the topic, in which I also mention a generic model of critical slowing down.

The model, with changes files (.cin) implementing some of the experiments: CatastropheWarning.zip

Forest Cover Tipping Points

This is a model of forest stability and transitions, inspired by:

Global Resilience of Tropical Forest and Savanna to Critical Transitions

Marina Hirota, Milena Holmgren, Egbert H. Van Nes, Marten Scheffer

It has been suggested that tropical forest and savanna could represent alternative stable states, implying critical transitions at tipping points in response to altered climate or other drivers. So far, evidence for this idea has remained elusive, and integrated climate models assume smooth vegetation responses. We analyzed data on the distribution of tree cover in Africa, Australia, and South America to reveal strong evidence for the existence of three distinct attractors: forest, savanna, and a treeless state. Empirical reconstruction of the basins of attraction indicates that the resilience of the states varies in a universal way with precipitation. These results allow the identification of regions where forest or savanna may most easily tip into an alternative state, and they pave the way to a new generation of coupled climate models.

The paper is worth a read. It doesn’t present an explicit simulation model, but it does describe the concept nicely. I built the following toy model as a loose interpretation of the dynamics.

Some things to try:

Use a Synthesim override to replace Forest Cover with a ramp from 0 to 1 to see potentials and vector fields (rates of change), then vary the precipitation index to see how the stability of the forest, savanna and treeless states changes:

Start the system at different levels of forest cover (varying init forest cover), with default precipitation, to see the three stable attractors at zero trees, savanna (20% tree cover) and forest (90% tree cover):

Start with a stable forest, and a bit of noise (noise sd = .2 to .3), then gradually reduce precipitation (override the precipitation index with a ramp from 1 to 0) to see abrupt transitions in state:

There’s a more detailed discussion on my blog.

forest savanna treeless 1f.mdl (requires an advanced version of Vensim, or the free Model Reader)

forest savanna treeless 1f.vpm (ditto; includes a sensitivity file for varying the initial forest cover)

A System Zoo

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, http://www.usf.uni-kassel.de/cesr/. Follow the Archiv(e) link on the home page and enter the downloads Archiv(e). This will put you in a file browser. Choose the Software folder, then the Zoo folder to obtain a .zip archive of the zoo models for the whole series, in Vensim .mdl format.

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:

Continue reading “A System Zoo”

Bifurcating Salmon

A nifty paper on nonlinear dynamics of salmon populations caught my eye on ArXiv.org today. The math is straightforward and elegant, so I replicated the model in Vensim.

A three-species model explaining cyclic dominance of pacific salmon

Authors: Christian Guill, Barbara Drossel, Wolfram Just, Eddy Carmack

Abstract: The four-year oscillations of the number of spawning sockeye salmon (Oncorhynchus nerka) that return to their native stream within the Fraser River basin in Canada are a striking example of population oscillations. The period of the oscillation corresponds to the dominant generation time of these fish. Various – not fully convincing – explanations for these oscillations have been proposed, including stochastic influences, depensatory fishing, or genetic effects. Here, we show that the oscillations can be explained as a stable dynamical attractor of the population dynamics, resulting from a strong resonance near a Neimark Sacker bifurcation. This explains not only the long-term persistence of these oscillations, but also reproduces correctly the empirical sequence of salmon abundance within one period of the oscillations. Furthermore, it explains the observation that these oscillations occur only in sockeye stocks originating from large oligotrophic lakes, and that they are usually not observed in salmon species that have a longer generation time.

The paper does a nice job of connecting behavior to structure, and of relating the emergence of oscillations to eigenvalues in the linearized system.

Units balance, though I had to add a couple implicit scale factors to do so.

The general results are qualitatitively replicable. I haven’t tried to precisely reproduce the authors’ bifurcation diagram and other experiments, in part because I couldn’t find a precise specification of numerical methods used (time step, integration method), so I wouldn’t expect to succeed.

Unlike most SD models, this is a hybrid discrete-continuous system. Salmon, predator and zooplankton populations evolve continuously during a growing season, but with discrete transitions between seasons.

The model uses SAMPLE IF TRUE, so you need an advanced version of Vensim to run it, or the free Model Reader. (It should be possible to replace the SAMPLE IF TRUE if an enterprising person wanted a PLE version). It would also be a good candidate for an application of SHIFT IF TRUE if someone wanted to experiment with the cohort age structure.


For a more policy-oriented take on salmon, check out Andy Ford’s work on smolt migration.


This is the latest instance of the WORLD3 model, as in Limits to Growth – the 30 year update, from the standard Vensim distribution. It’s not much changed from the 1972 original used in Limits to Growth, which is documented in great detail in Dynamics of Growth in a Finite World (half off at Pegasus as of this moment).

There have been many critiques of this model, including the fairly famous Models of Doom. Most are ideological screeds that miss the point, and many modern critics do not appear to even have read the book. The only good, comprehensive technical critique of World3 that I’m aware of is Wil Thissen’s thesis, Investigations into the Club of Rome’s WORLD3 model: lessons for understanding complicated models (Eindhoven, 1978). Portions appeared in IEEE Transactions.

My take on the more sensible critiques is that they show two things:

  • WORLD3 is an imperfect expression of the underlying ideas in Limits to Growth.
  • WORLD3 doesn’t have the policy space to capture competing viewpoints about the global situation; in particular it does not represent markets and technology as many see them.

It doesn’t necessarily follow from those facts that the underlying ideas of Limits are wrong. We still have to grapple with the consequences of exponential growth confronting finite planetary boundaries with long perception and action delays.

I’ve written some other material on limits here.

Files: WORLD3-03 (zipped archive of Vensim models and constant changes)

Payments for Environmental Services

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Model Name: payments, penalties, and environmental ethic

Citation: Dudley, R. 2007. Payments, penalties, payouts, and environmental ethics: a system dynamics examination Sustainability: Science, Practice, & Policy 3(2):24-35. http://ejournal.nbii.org/archives/vol3iss2/0706-013.dudley.html.

Source: Richard G. Dudley

Copyright: Richard G. Dudley (2007)

License: Gnu GPL

Peer reviewed: Yes (probably when submitted for publication?)

Units balance: Yes

Format: Vensim

Target audience: People interested in the concept of payments for environmental services as a means of improving land use and conservation of natural resources.

Questions answered: How might land users’ environmental ethic be influenced by, and influence, payments for environmental services.

Software: Vensim



Models in the Special Issue of the System Dynamics Review on Environmental and Resource Systems

Models in the Special Issue of the System Dynamics Review on Environmental and Resource Systems, Andrew Ford & Robert Cavana, Editors. System Dynamics Review, Volume 20, Number 2, Summer of 2004.

  • Modeling the Effects of a Log Export Ban in Indonesia by Richard G. Dudley
  • The Dynamics of Water Scarcity in Irrigated Landscapes: Mazarron and Aguilas in South-eastern Spain by Julia Martinez Fernandez & Angel Esteve Selma
  • Misperceptions of Basic Dynamics: The Case of Renewable Resource Management by Erling Moxnes
  • Models for Management of Wildlife Populations: Lessons from Spectacle Bears in Zoos and Gizzly Bears in Yellowstone by Lisa Faust, Rosemary Jackson, Andrew Ford, Joanne Earnhardt and Steven Thompson
  • Modeling a Blue-Green Algae Bloom by Steven Arquitt & Ron Johnstone

See the following web site for article summaries and downloadable models described in this special issue:  http://www.wsu.edu/~forda/SIOpen.html

Submitted by Richard Dudley, 23 April 2008