Abstract spaces for unknown ecologies

Tags

food webs, metanetwork, paleo-food web, Permian-Triassic extinction, Scientific models

A guild-level food web for the Karoo Basin ecosystem.

At the heart of our paper lies a model framework which we devised for analyzing fossil food webs. I stated in the previous post that our main question was “How would those food webs (important parts of the paleoecosystems) have responded to everyday types of disturbances, on the short-term, as the planet was busily falling apart?” We could approach this question in several different ways if we were working with modern food webs. We could conduct manipulative experiments with simple mock-ups of the food web, using for example some of what we believe to be the key species to represent the community. Or, we could conduct large scale manipulations, such as removing a species entirely; but that is very difficult to do, or to obtain permission! Or, we could measure variables such as species population sizes, how species interact with each other, and so forth, to then conduct numerical analyses and simulations. None of those approaches are available to us when dealing with ancient, extinct ecosystems. Therefore, what we did instead was to use the most accurate information that we have for each paleoecosystem, which consisted of categorizing species into “guilds”. Here, a guild is a group of species who shared the same habitat, and  potentially shared the same predators and prey. The “potentially” is based on our best interpretations of the ecologies of those extinct species, because without actually being there to witness their interactions (back to that Tardis again), we cannot be sure. The result is usually something like the box figure above. Even with species lumped, you can see how complex and busy the system would have been! And from this guild-level model, we can then construct many many different food webs, tweaking the specific links between species. An example is shown in the second figure.

Now, the number of different food webs that you could generate based on even a modest number of species, say 20, and a few guilds, is astronomically large (in fact beyond astronomical). The important thing, however, is that all of them would be consistent with the guild scheme. Let me give an example. Say we did a guild scheme for the modern African savannah. We would be justified to some extent to place lions and hyaenas in the same guild. We might not know exactly which antelope species (for example) each predator species was preying on, but we would never draw a food web where antelope were preying on lions, hyaenas or each other! So, what we have done for our food webs is constructed a mathematical space that contains all the food webs which could possibly have existed in our paleoecosystem. In other words, we have taken the full set of food webs that could be constructed for a certain number of species, and constrained ourselves to consider only those that are consistent with our accurate knowledge of the guild structure.

Detail of one possible food web just prior to the mass extinction.

This still doesn’t solve the problem of how those food webs would have responded to various types of disturbances in the distant past. And in fact, we really cannot solve that problem, so we did what we think is the next best thing. We asked if there was anything special about those food webs, compared to any others that were not consistent with our guild structure. In other words, what if the ecosystem had evolved a bit differently, and comprised species a bit different from what we actually observe in the fossil record? We considered a number of such alternative models, differing from the real ecosystem in ways such as moving species around in the guilds, or moving guilds and the interactions between them, or removing guilds altogether. And each time we did that, and generated a food web from the new guild scheme, we examined the stability of the food web. Exactly what we mean by stability, and how we measured it, will be the subject of the next post.

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A new paper on the Permian-Triassic mass extinction

Tags

biodiversity, extinction, food webs, modeling, paleo-food web, paleontology, Permian-Triassic extinction, Scientific models

Dicynodon, an ancient relative of mammals, at the end of the Permian. (Marlene Hill Donnelly).

Yesterday, Ken Angielczyk and I published our most recent paper on the Permian-Triassic mass extinction (PTME) in the journal Science. In a nutshell, we examined a series of paleocommunities spanning the extinction, from the Late Permian to the Middle Triassic, and modelled the stability of their food webs. We compared the models to hypothetical alternatives, where we varied parameters such as how species are divided among guilds, or ecological “jobs”, and the numbers of interactions that species have. One of our very interesting discoveries is that the real food webs were always the most stable, or amongst the most stable of the models, even during the height of the extinction! That’s remarkable, given the devastating loss of species at the end of the Permian. Our other discovery is that the ability to remain highly stable during the extinction stemmed from the more rapid extinction of small, terrestrial vertebrate species. That’s not something we would predict given our experience with modern and ongoing extinctions, where larger vertebrate species are considered to be at greater risk. And finally, our last interesting observation is that the early recovery, the immediate aftermath during the Early Triassic, was an exception to the above. That community was not particularly stable, which seems to have been the result of the rapid evolutionary diversification of the extinction survivors, and the arrival of immigrants from neighbouring regions.

Some aspects of the paper are quite technical, and take advantage of fantastic new paleontological data and recent developments in theoretical ecology. Therefore, over the next few posts I’ll go through what we did, and how we did it, using a more “plain language” approach. In the meanwhile, the paper was covered by a number of news outlets, and here’s my favourite!

“5 things we learned from the mass extinction study that’s “the first of its kind”“, The Irish Examiner.

Simulating a Tragedy of the Commons I

Tags

Code, simulations, tragedy of the commons

I introduced a recent paper, in the previous post, on the Tragedy of the Commons. The paper is published in Sustainability, an open access paper and is therefore available to everyone. One bit that I did not include in the paper, however, is the code that I used for the simulations therein. On the suggestion of a friend, I’ll publish them here in a series of posts. (Thanks Mauro)

The simulations are very simple, being based on difference equations, and were all coded in Octave. The code is not pretty, but I cleaned it up and added some comments. If you have questions, just drop me a comment on the blog. This first simulation is of a basic tragedy involving two users, and corresponds to Figures 1B-C in the paper. A resource is simulated to a steady state based on a Ricker model, for 100 steps, and then users begin to utilize it. They increase their utilization per time step at a steady rate (acceleration of utilization is constant), and the simulation is run until the resource is exhausted. The basic output of resource, one user’s benefit, and total utilization, are plotted in the figure here. The equations simulated are outlined in the excerpt from pg. 754 of the paper:
“We now introduce a term to represent consumption or utilization by our human TOC agents.

where U(t) is the total fraction of R utilized by human users at time t, and ranges from 0 to 1. The term in square brackets on the right hand side of the equation is the modified growth rate of R. The resource is stable when this term is positive or zero, that is, the rate of resource renewal exceeds or is equal to utilization. Since U is the total standardized utilization rate of all users, it may be expanded to

where there are N users and is the standardized utilization rate of the user at time t. The benefit gained from utilization is modelled as

where is the benefit to user i, f is a factor that converts resources utilized into another commodity, for example converting harvested food to energy or currency, and B is the total benefit to all users.”

The code follows, and can be executed by running it in an interactive Octave session. Please be aware that WordPress has somewhat limited options for formatting code. I’ve used Matlab formatting here, but wrapped lines might not be obvious. Single lines are all terminated with a semicolon, “;”, so just look out for those.

#RESOURCE DATA
#initial resource level
R0 = 1;
R = zeros(1,2);
R(1, 1) = 1;
R(1, 2) = 1;
#carrying capacity
K = 50;
#intrinsic growth rate r
r = 1.5;

#USER DATA
#There are 2 users
#initial acceleration of utilization
du0 = 1.05;
#individual user parameters, users 1 and 2
#initialize arrays
u1 = zeros(1,2);
u2 = zeros(1,2);
b1 = zeros(1,2);
b2 = zeros(1,2);
u1(1,1) = 1;
#initial utilization rate
u1(1,2) = .001;
#conversion factor of resource to benefit
f1 = 1;
b1(1,1) = 1;
b1(1,2) = 0;
u2(1,1) = 1;
#initial utilization rate
u2(1,2) = .001;
#conversion factor of resource to benefit
f2 = 1;
b2(1,1) = 1;
b2(1,2) = 0;
U = zeros(1,2);
U(1,1) = 1;
#total utilization
U(1,2) = b1(1,2) + b2(1,2);
du = zeros(1,2);
du(1,1) = 0;
du(1,2) = du0;

#BEGIN SIMULATION

#RUN RESOURCE TO STEADY STATE, NO UTILIZATION
for count2 = 2:100
  R(count2, 1) = count2;
  R(count2, 2) = R(count2-1, 2) * (exp(r*(1-(R(count2-1, 2)/K))));
  u1(count2, 1) = count2;
  u1(count2, 2) = u1(1,2);
  u2(count2, 1) = count2;
  u2(count2, 2) = u2(1,2);
  du(count2,1) = count2;
  du(count2,2) = du0;
endfor

#INITIATE UTILIZATION
for count1 = 101:265
  R(count1, 1) = count1;
  #modified Ricker model with utilization
  R(count1, 2) = R(count1-1, 2) * (exp(r*(1-(R(count1-1, 2)/K))) - (u1(count1-1,2)+u2(count1-1,2)));
  u1(count1, 1) = count1;
  #increase utilization
  u1(count1, 2) = u1(count1-1, 2) * du(count1-1,2);
  #benefits (equal for both users in this case)
  b1(count1, 1) = count1;
  b1(count1, 2) = f1 * u1(count1-1, 2) * R(count1-1, 2);
  b2(count1, 1) = count1;
  b2(count1, 2) = f2 * u2(count1-1, 2) * R(count1-1, 2);
  u2(count1, 1) = count1;
  u2(count1, 2) = u2(count1-1, 2) * du(count1-1,2);
  #total benefits
  U(count1, 1) = count1;
  U(count1, 2) = b1(count1, 2) + b2(count1, 2);
  du(count1,1) = count1;
  du(count1,2) = du(count1-1,2);
endfor

Ecology and the Tragedy of the Commons

Tags

real world networks, Robustness, Scientific models, tragedy of the commons

Well, it’s been quite some time since the last post, but I’ve been busy! This post is just a short notice of a new paper, just published today. The paper is part of a special issue on the Tragedy of the Commons in the journal Sustainability. My paper takes a comparative look at the Tragedy in ecological communities and human societies, and the potential of human mutualisms for avoiding tragedies. The situation is not a very hopeful one, however, given our ever-growing human population. Hardin did note this in his original essay. Finally, this paper was inspired by an earlier paper by myself and Ken Angielczyk.

Here’s a link to the paper, as well as the abstract.

Roopnarine, P. Ecology and the Tragedy of the Commons. Sustainability 2013, 5, 749-773.

Abstract

This paper develops mathematical models of the tragedy of the commons analogous to ecological models of resource consumption. Tragedies differ fundamentally from predator–prey relationships in nature because human consumers of a resource are rarely controlled solely by that resource. Tragedies do occur, however, at the level of the ecosystem, where multiple species interactions are involved. Human resource systems are converging rapidly toward ecosystem-type systems as the number of exploited resources increase, raising the probability of system-wide tragedies in the human world. Nevertheless, common interests exclusive of exploited commons provide feasible options for avoiding tragedy in a converged world.

One more time: Why model?

Tags

modeling, Scientific models, simulations

A wonderful statement by Jay Melosh in a recent interview with Physics Today. It really underscores the approach of much of the research presented in this blog; just replace “physics” with “biology” or “ecology”. And while Melosh is a master of planetary geology and hence the history of the Solar System, we’re dealing here with evolution and the histories of species and ecosystems. Okay, here’s the quote:

“Computer modeling, of course, plays a big role in evaluating the consequences of different hypotheses, which we then compare to observations. While the physics of an individual process may be simple, Nature is messy and computers are one of our principal tools for combining simple processes into the complex fabrics necessary to mimic observations and thus either validate or refute different hypotheses about what we see.“

PNAS: Late Cretaceous restructuring of terrestrial communities facilitated the End-Cretaceous mass extinction in North America

Tags

Cretaceous, dinosaurs, extinction, food webs, mass extinction, modeling, networks, paleo-food web, paleontology, Robustness, Scientific models, simulations, Tipping point, trophic guild

That’s the title of our new paper, hot off the PNAS press. This study was a lot of fun, because it combines my food web work with one of the best known events in the fossil record. The lead author is Jonathan Mitchell, a graduate student at the University of Chicago. Jon became familiar with the food web work via Ken Angielczyk at the Field Museum, also in Chicago, a former post-doctoral researcher in my lab and close collaborator.  Jon wondered what Late Cretaceous, dinosaur-bearing communities would look like when subjected to CEG perturbations (just search this blog for info. on CEG!), and presented his results two years ago at the Annual Meeting of the Geological Society of America. The results were so intriguing that we decided then to explore the question in much greater detail, and ask what sorts of community and ecosystem changes unfolded in the years before the Chicxulub impact, and what role they might have played in the subsequent extinctions. And here are the results! I will list the full reference below, and you can obtain a complete copy of the paper from PNAS (sorry, not open access). Also, here are links to some news websites that have covered the paper, as well as the paper’s abstract. Enjoy!

EurekAlert, Science Daily, Science Codex

Jonathan S. Mitchell, Peter D. Roopnarine, and Kenneth D. Angielczyk. Late Cretaceous restructuring of terrestrial communities facilitated the End-Cretaceous mass extinction in North America. PNAS, October 29, 2012

ABSTRACT

The sudden environmental catastrophe in the wake of the end-
Cretaceous asteroid impact had drastic effects that rippled through
animal communities. To explore how these effects may have been
exacerbated by prior ecological changes, we used a food-web
model to simulate the effects of primary productivity disruptions,
such as those predicted to result from an asteroid impact, on ten
Campanian and seven Maastrichtian terrestrial localities in North
America. Our analysis documents that a shift in trophic structure
between Campanian and Maastrichtian communities in North
America led Maastrichtian communities to experience more second-
ary extinction at lower levels of primary production shutdown and
possess a lower collapse threshold than Campanian communities.
Of particular note is the fact that changes in dinosaur richness had
a negative impact on the robustness of Maastrichtian ecosystems
against environmental perturbations. Therefore, earlier ecological
restructuring may have exacerbated the impact and severity of the
end-Cretaceous extinction, at least in North America.

Biology Letters Special Feature – Models in Palaeontology

Hot off the presses! The “tragedy of the commons” paper with Ken Angielczyk is featured here, along with a truly fantastic collection of other papers. A must-read for any of you modelling-nerds out there. Oh, and the papers are free to access!

Biology Letters Special Feature – Models in Palaeontology.

A species’s tragedy of the commons

Tags

biodiversity, carrying capacity, cascades, competition, extinction, food webs, interaction strength, link distribution, link strength, modeling, networks, paleo-food web, paleontology, Robustness, Scientific models, simulations, Tipping point, trophic guild

My colleague Ken Angielczyk and I have a new paper out in the Royal Society‘s Biology Letters, entitled “The evolutionary palaeoecology of species and the tragedy of the commons“. If you have never read Garrett Hardin’s original paper on the tragedy of the commons, I strongly suggest that you do. It is a principle that I believe has broad application, and would well be worth a re-visit (first visit?!) by today’s leaders and economists. Our paper can be found here or here (first page only). And here is the abstract, as a little teaser!

Abstract

The fossil record presents palaeoecological pat-
terns of rise and fall on multiple scales of time
and biological organization. Here, we argue that
the rise and fall of species can result from a tragedy
of the commons, wherein the pursuit of self-inter-
ests by individual agents in a larger interactive
system is detrimental to the overall performance
or condition of the system. Species evolving
within particular communities may conform to
this situation, affecting the ecological robustness
of their communities. Results from a trophic
network model of Permian–Triassic terrestrial
communities suggest that community perform-
ance on geological timescales may in turn
constrain the evolutionary opportunities and
histories of the species within them.

New paper: Ecological modeling of paleocommunity food webs

Tags

cascades, competition, connectance, edge strength, extinction, food webs, interaction strength, link strength, modeling, Network theory, networks, nonlinear, paleontology, power law, probability, real world networks, Robustness, Scientific models, simulations, small world networks, Tipping point, top-down cascade

Roopnarine, P. D. 2009. Ecological modeling of paleocommunity food webs. in G. Dietl and K. Flessa, eds., Conservation Paleobiology, The Paleontological Society Papers, 15: 195-220.

Find the paper here:
http://zeus.calacademy.org/roopnarine/Selected_Publications/Roopnarine_09.pdf
or here
http://zeus.calacademy.org/publications/

Recipe for building a scientific model

Tags

modeling, Scientific models, simulations

The following is the second part of the excerpt (see previous post) on model-building. Apologies for the paleontological and evolutionary biology jargon to any casual readers who might stumble across this. I’ll try to embellish and explain another time, or just leave me a question. Same for the citations.

“…the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” (Einstein, 1934). There are numerous views and opinions on the purpose for constructing a scientific model, but they all fall roughly between the end-points on a scale of general scientific goals: problem-solving (Grimm and Railsback, 2005) and constructing an underlying theory based on observations (Randall, 2005). For example, Darwin developed the qualitative model of natural selection to explain a theory of evolution that he deduced underlay many of his observations of the natural world. In the process he assumed the existence of a mechanism for the inheritance of traits. Evolutionary biologists of the early 20th century were subsequently faced with reconciling the problem between Mendelian genetics and concepts of natural selection at that time. Models of quantitative and population genetics were developed that went a long way toward reconciliation (Fisher, Wright, Haldane). It is not difficult to appreciate differences between the models and presentations of Darwin, and for example, Fisher, and that brings us to the first guideline.

A model should be appropriate to the problem or question being addressed. Darwin set out to explain patterns which he had observed, and his work is remarkable in its weaving together of so many major themes into the theory of evolution, including inheritance of traits, population growth and the struggle for existence, competition within and among species, and the history of life as recorded in the fossil record. Formulating his theory did not require a detailed understanding of any of these themes comparable to what we have available to us today. Moving beyond Darwin, however, in order to explain observations, test the theory of natural selection, and search for underlying mechanisms, requires the construction of far more specific models, and that is precisely what Fisher did. Darwin presented a convincing case of modification with descent, but it was up to Fisher and others to provide adequate models of natural selection capable of explaining the data. Fisher provided a specific and quantitative model explaining how natural selection can be based on the Mendelian “particulate character of the hereditary elements” (Fisher, 1958). This and related models, plus the later revolution in molecular biology, were crucial to our current understanding of natural selection as an evolutionary mechanism. Nevertheless, the details of these models are not at all essential when incorporating evolutionary theory into many biological models. For example, Sepkoski (1984) did not need them for his kinetic models of Phanerozoic diversity, even though all arguments of taxon origination are based on evolution. How then does one decide on the proper ingredients for a model?

Define the domain of the model. Presumably, all things in nature are connected, but most of these connections are never relevant to understanding the problem immediately at hand. One should define, at the outset, those connections or relationships which are causal, and those that are contingent (Bohm, 1957). To paraphrase an example from Bohm, one can quite accurately deduce a law of gravitation by dropping balls of various sizes and composition. Now drop a sheet of paper. Does the difference in motion suggest a new law, or inadequacy of the old one? The paper does eventually meet the ground, and precisely because of the same law of gravitation to which the balls are subject. Its different motion is a result, of course, of air resistance. If your goal is to deduce a law of gravity, then one realizes that gravity here is causal in the net journey of the paper, but the specific steps taken along the journey are contingent upon the sheet’s air resistance. Air resistance is irrelevant to a law of gravity. If your goal, on the other hand, is to model the motion of sheets of paper, then accounting for gravity alone is not sufficient and air resistance is indeed relevant.

Constrain model specificity. Getting back to the balls for a moment, is air resistance relevant to them? If one’s measurements were precise enough, then you would likely discover differences based on different surface textures of the balls, perhaps variation over time corresponding to air temperature, and so on. But those details should be unimportant when describing the motion of a dropped ball, because mass and sphericity dominate minor differences in ball composition. Therefore, the addition of air temperature as a causal parameter is unnecessary. There is often a temptation and tendency to strive for realism in biological models to the point at which the model becomes too complex. This point can be recognized either when the addition or variation of parameters fail to alter model output in a predictable manner, or variation of model output cannot be explained directly as a result of parameter variation. An accurate simulation of reality is not a model unless one can point to exactly why the simulation reproduces all the aspects that make it realistic. Simulations are not necessarily models.

Having stated that, a major caveat must be given immediately. Ecological systems are mostly complex, that is, they are the result of the interactions of numerous independently acting or semi-independent entities. Simple models of those entities can sometimes produce complicated results, particularly when causal relationships within the model are nonlinear (as we will see later on). The behavior of a system of entities can be modeled as a collection of simple models, or ignore that lower level altogether and treat the entire system as an entity. These approaches are not likely to yield the same results. The former can quickly become a hopelessly realistic simulation, while the latter simply cannot be interpreted at a level relevant to individual entities. A proper model will search for a middle path, and though the search can be as much art as it is science, that’s where the fun is.