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.

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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)

Lotka-Volterra predator-prey system

The Lotka-Volterra equations, which describe a predator-prey system, must be one of the more famous dynamic systems. There are many generalizations and applications outside of biology.

Wikipedia has a nice article, which I used as the basis for this simple model.

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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:

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Fibonacci Rabbits

This is a small, discrete time model that explores the physical interpretation of the Fibonacci sequence. See my blog post about this model for details.

Fibonacci2.vpm This runs with Vensim PLE, but users might want to use the Model Reader in order to load the included .cin file with non-growing eigenvector settings.

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.

sockeye.vmf

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

Logistic Chaos

This is an implementation of the logistic model – a very simple example of discrete time chaotic behavior. It’s sometimes used to illustrate chaotic dynamics of insect populations.

There’s a nice description here, and the other top links on google tend to be good.

Note that this version corrects an equation error in previous versions.

Logistic (Vensim .vpm)

Logistic (Vensim .vmf)

Ultradian Oscillations of Insulin and Glucose

Citation: Jeppe Sturis, Kenneth S. Polonsky, Erik Mosekilde, and Eve van Cauter. Computer Model for Mechanisms Underlying Ultradian Oscillations of Insulin and Glucose. Am. J. Physiol. 260 (Endocrinol. Metab. 23): E801-E809, 1991.

Source: Replicated by Hank Taylor

Units: No

Format: Vensim

Ultradian Oscillations of Insulin and Glucose (Vensim .vpm)