Topological extinction « Roopnarine’s Food Weblog

New paper: Ecological modeling of paleocommunity food webs

October 30, 2009 proopnarine Leave a comment

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/

Tipping point II

January 8, 2009 proopnarine 1 comment

Red ellipses represent the two threshold points.

The story so far: We have a food web of a shallow coastal marine community from the Late Miocene of the Dominican republic. The metanetwork comprises 29 guilds, 139 guild-level links, and 130 species. A perturbation of the system, where all three primary producer guilds plus detritus were systematically and incrementally removed from derived species-level networks, results in the typical CEG result: that is, a relatively flat and low level of secondary extinction ( $\Psi$ ) over a broad range of perturbation magnitude ( $\Omega$ ), succeeded by a rapid transition to a state of high secondary extinction. In fact, for this community, there are two transitions. The first occurs at $\Omega=0.51$ , and represents a very minor but secular increase in $\Psi$ . The second transition occurs at $\Omega=0.67$ and represents a catastrophic increase in $\Psi$ . Topological-only perturbation of the system makes it very clear that these transitions correspond exactly to two stages of the perturbation: First, the complete extinction or removal of benthic autotrophs and complete disruption of the particulate detritus supply. The second and greater transition occurs at the complete extinction of the benthic macroalgae and macrophytes. Accompanying the second transition is the complete extinction of the benthic herbivore guilds which specialize on the macroalgae and macrophytes (and derived detritus), comprising families such as the Phasianellidae, Cerithiidae, Vitrinellidae, Haminoeidae and Retusidae. This is accompanied by extinction of species in other more generalist guilds that include macroalgae in their diet.

Extinction of those heterotrophic taxa is not itself the cause of the major tipping point though. Simulations where the perturbation is specifically removal of these herbivores result in very low levels of secondary extinction, with no tipping point or threshold. The obvious question then is, why does extinction of the macroalgae drive the system to a new state? The qualitative answer is that the complete loss of this resource, and the bottom-up propagation to the herbivores, in turn cause intense top-down cascades of compensatory responses from higher level consumers. These cascades propagate throughout the network, even to the remaining source of production, the phytoplankton. The result is a tremendous loss of species. A very curious thing, however, is that phytoplankton productivity in the network is almost 3 times greater than macroalgal productivity, reflecting the much greater diversity of planktivores. So why does the collapse coincide with loss of the macroalgae?

Perturbation of macroalgae (top) and phytoplankton (bottom), and resulting loss of autotrophic productivity because of top-down effects (left), and secondary extinction of heterotrophic species (right).

I performed two separate perturbations to answer this question. First, I perturbed the system by removing macroalgae only, and second by removing phytoplankton only. The top row of the second figure shows the results of the first experiment. Secondary loss of autotrophic resources (left column) as a result of top-down effects is effectively zero. Secondary extinction of heterotrophs (right column) is significant but not dramatic. There is a mild increase in the region of $\Omega=0.6-0.7$ , which represents the loss of the specialized herbivore guilds. Removing phytoplankton had a more dramatic impact, reflecting the greater overall dependence of the community on phyloplankton resources. There is a clear threshold, occurring at approximately $\Omega=0.5$ . At this point, resource loss to the community is great enough to trigger the catastrophic top-down cascades and feedback within the network. Therefore, it seems that in the previous experiment, where all resource guilds were perturbed, the complete loss of macroalgae triggers the top-down cascades and compensatory feedback that in turn deplete phytoplankton resources to the point where the system transitions to a higher state of secondary extinction. This conclusion is supported by the fact that when all producer guilds are perturbed, the contribution or perturbation of phytoplankton at the tipping point is 38%, whereas when only phytoplankton are perturbedm the tipping point occurs at 50%.

Some closing observations:

Topological analyses of network vulnerabilities are likely to underestimate the severity of link losses when those links have variable interaction strengths, and the nodes have varying properties. In the case of a biological community, species could and are likely to alter interaction strengths to compensate for lost resources (i.e. links). Topological vulnerability analyses should be well suited for networks with static properties, perhaps such as power grids and the internet (though I’m no expert here!), but are ill-suited for dynamic networks, such as those describing transportation, metabolic/physiologic and ecologic systems.
An hierarchically structured, directed network such as an ecological community should be resistant to a broad array of random perturbations. This is a function of both the underlying link distributions (as already understood in the case of static networks or graphs), as well as the compensatory abilities of consumer species, and the variance of dietary breadth. The network is, however, vulnerable to the loss of highly linked nodes. Here I am referring specifically to basal, autotrophic nodes, and not necessarily keystone consumer species. Not all autotrophic nodes are equal, however, as shown in the above results. Nevertheless, because of the complexity of the species interactions and the hierarchical divisions of ecological functions, there should be strong nonlinearities in the network responses. This is borne out by the differences between the topological-only and fully dynamic simulation results. The nonlinearities are expressed as two or more alternative states of secondary extinction, separated by rather sharply defined thresholds of perturbation. I can think of no way in which to analytically predict the threshold points, but heuristically I would argue that they should exist in every ecological community.
Perturbation of top-level consumers are observed in nature to often result in top-down cascading effects, compatible with such notions as keystone predators. I will show in later results that the CEG model captures all this. The results will also show, however, that while top-down effects can be locally catastrophic, i.e. for individual species or groups of closely linked species, they are never globally catastrophic in the manner in which bottom-up perturbations are. This conclusion has implications for understanding the role of ecological collapse in large scale extinctions observed in the fossil record. It also has implications for the ongoing biodiversity crisis, where species far removed from the “tops” of food webs are increasingly threatened by climate change and habitat destruction.
An close examination of many of the results presented in this blog will show apparent “bifurcation” of the results, e.g. beyond the threshold point in the lower right graph above. These observations suggest that there is more than one type of species-level network that can be derived from the same metanetwork. So, while the higher-level organization of the community is the same, networks are being generated that vary enough in their interspecific link topologies to yield very different responses to the same level of perturbation. I believe that this is a statistical property of the underlying trophic link distributions and the resulting multinomial probabilities from which the species-level networks are drawn stochastically. In the case of the above results, where one set of networks is significantly more resistant than the other (i.e. they have a much higher tipping point), this mathematical feature of the model is not likely to be of great relevance ecologically. That is because the lower threshold is already so high, in this case, 50% shutdown of primary productivity. Those are catastrophic environmental conditions and would occur with very low frequency in nature on a large scale. There are cases, however, such as the Early Triassic Lystrosaurus zone community, where there seem to be multiple alternative states at very low perturbation levels. Those communities would very likely have experienced frequent low-level perturbations, and then one has to consider whether: (1) this feature of the model is a mathematical artifact, in which case one wonders about the constraints necessary to prohibit it in nature, or (2) the feature is real, and then one wonders how species within a community cope with such a situation.

Categories: CEG theory, Tipping point, Topological extinction Tags: extinction, networks, food webs, nonlinear, simulations, cascades, top-down cascade, network theory, Tipping point

Dichotomous topological extinction

December 23, 2008 proopnarine Leave a comment

Probability of topological secondary extinction. Prey guild diversity is 150, and consumer in-degree ranges from 1 to 10.

I suspect that the dichotomous topological extinction results have to do with the presence of two distinct types of species-level networks in the simulations. I’ll explore the reasons for that later, but right now, I want to know if this explanation is feasible. The networks would differ in having very different types of species composing the guilds, different in their in-degrees and hence resistance to topological secondary extinction. A clue is given by how the probability of extinction varies with in-degree. The first figure illustrates this, although it is not the probability surface for our particular 3-guild community (calculations performed with GNUPlot, which is unfortunately limited in its ability to calculate factorials in the necessary range). We see that the probability of extinction declines nonlinearly as in-degree increases.

If we focus on the primary consumer guild only, the suggestion of distinct classes of networks becomes a bit more obvious. Plotted here (second figure) are the simulation results for this guild only. The expected levels of secondary extinction are overlaid again, as previously. There is apparently one set of networks that conform reasonably well to the expected results, but another set with very high extinction at even low levels of perturbation. I hypothesize that these latter networks comprise consumer species of very low in-degree. The likelihood of getting such networks must be fairly high; this can be determined, I think, by examining the multinomial probabilities of such networks. It is quite possible that at low guild diversities and metanetwork complexity, the multinomial likelihood surface is relatively flat. We’ll see.

Secondary extinction results for primary consumer guild only. The simulations results are connected by lines, and the odd "radiating" lines are simply the connecting points between consecutive simulations.

Categories: Topological extinction Tags: networks, simulations

Estimating topological extinction III

December 17, 2008 proopnarine Leave a comment

Let’s apply some network thinking to this problem now. First, the network, or food web, is perturbed by the removal (extinction) of several nodes from one or more guild. If enough nodes, or the “right” ones are removed, this could in turn cause secondary extinctions of species that consume the extinct ones. Whether that actually happens or not depends on the probabilities of extinction. Therefore, to estimate topological extinction in our network, after perturbing it, we have to follow the paths of propagation and estimate the levels of resulting secondary extinction from the probabilities of extinction. The formula for those probabilities was given in an earlier post. Here we restate it in the following framework.

Guild $G_{j}$ comprises species of different in-degrees, $y_{j}$ and hence probabilities of secondary extinction. If $x_{i}$ is a species that potentially preys upon species in $G_{j}$ , then its probability of extinction, given a measured level of extinction of $y_{j}$ , denoted $\psi_{yj}$ , is

$\mathrm{pr}(e, x_{i}|\psi_{yj}) = \psi_{yj}!\left( |y_{j}|-r_{xy}\right) ! \left[ |y_{j}|!\left( \psi_{yj}-r_{xy}\right) !\right]^{-1}$

where

$\mathrm{E}(r_{xy}) = \frac{|y_{j}|}{b_{i}}r_{x}a_{ij}$

is the expected number of $x_{i}$ ’s links which come from $y_{j}$ -type species. The total expected level of secondary extinction of species of in-degree $x_{i}$ is therefore

$\mathrm{E}(\psi_{xi}) = |x_{i}|\prod_{j=1}^{|U|}\prod_{r_{y}=1}^{b_{j}} \mathrm{pr}(e, x_{i}|\psi_{yj})$

Incorporating the estimates of secondary extinction, $\psi$ for each class (in-degree) of species in each guild, we can see how an iterative estimate of secondary extinction can be made for the entire network. Say that the perturbation was a disruption of primary productivity and that guild $G_{j}$ is a guild of primary consumers. Then $\psi_{yj}$ is an estimate of the level of topological secondary extinction of $y_{j}$ species. If guild $G_{i}$ is a guild of secondary consumers, carnivores, with species that prey on those in $G_{j}$ , then we see why topological secondary extinction of species in $G_{i}$ , $x_{i}$ , is a function of $\psi_{yj}$ .

The actual order in which the calculations is made is important, because the perturbation will propagate along paths in an order defined by the metanetwork and food web topologies. I’ll cover the determination of that ordering in the next post.

Categories: CEG theory, Topological extinction Tags: Add new tag, extinction, food webs, networks

Estimating topological extinction II

December 16, 2008 proopnarine Leave a comment

The topological extinction of a species $x_{i}$ requires extinction of all its prey resources, or incoming links. Since the probability of extinction is a function of in-degree, it is helpful to distinguish among prey of different in-degrees. Also, prey guild membership is also a necessary parameter, as the manner in which a perturbation propagates through the network is a function of metanetwork topology and hence guild linkages.

The expected number of links between $x_{i}$ and a particular class of prey, $y_{j} \in G_{j}$ , is

$E(|x_{i}\leftarrow y_{j}|) = \frac{|y_{j}|}{b_{i}}r_{x}$

(we’ll ignore our integer links for now). The probability of topological secondary extinction of $x_{i}$ is then a function of the probabilities of extinction of all its prey, those prey being distinguished by guild membership and in-degree. This may be written as

$\mathrm{pr}(e,x_{i}|\omega) = \prod_{j=1}^{|U|}\prod_{r_{y}=1}^{b_{j}}\mathrm{pr}(e,y_{j}|r_{y})^{\frac{|y_{j}|r_{x}a_{ij}}{b_{i}}}$

where $|U|$ is the number of guilds in metanetwork $U$ , $b_{j}$ is the number of potential prey species (in-links) of $y_{j}$ , and $a_{ij}$ is the $ij^{th}$ element of $U$ ’s adjacency matrix. The use of the adjacency matrix allows us to generalize the formula to all guilds and species in the community, regardless of the metanetwork’s topology.

Categories: CEG theory, Topological extinction Tags: extinction, networks

Estimating topological extinction I

December 16, 2008 proopnarine 1 comment

What I mean here, when I say “analytical approach”, is basically a non-simulation approach to the problem. I adopt an ensemble approach to estimate the level of topological secondary extinction, where an ensemble consists of all species $x_{i}$ of in-degree $r_{x}$ , where $x_{i} \in G_{i} \forall x_{i}$ . That is, all species, of a particular in-degree or dietary breadth, belonging to a specific guild. Let $|G_{i}|$ be the species richness of that guild. Then, on the basis of the guild’s trophic link distribution $P(r,i)$ , the number of species of in-degree $r_{x}$ , or $|x_{i}|$ , is estimated as

$|x_{i}| = \frac{\int_{r_{x}-1}^{r_{x}}P(r,i)}{\int_{0}^{b_{i}}P(r,i)} |G_{i}|$

The fraction expresses the relative frequency of species of degree $r_{x}$ in the guild, and hence the function is an estimate of the number of such species, given the guild species richness. The numerator integral is taken over an integer interval of course since species interactions only occur as whole numbers . The range of dietary breadth, $r$ is $0 \rightarrow b_{i}$ . The minimum recognizes that a species must have at least one other species which it consumes, but no more than the maximum number of species with which its guild-dictated ecology can interact given guild species richnesses.

The probability of topological extinction of a species $x_{i}$ is equal to the probability that, given a perturbation to the community (network), all species (nodes) to which it is linked are lost. The basic formula for this, ignoring considerations of multiple metanetwork connections, is given by a hypergeometric probability. Stated simply, given $b_{i}$ prey, an extinction or perturbation of magnitude $\omega$ , what is the probability that $n_{x}$ out of $r_{x}$ links will be lost?

$\mathrm{pr}(n_{x}\mid \omega) = \left( \begin{array}{c}r_{x}\\n_{x}\end{array}\right) \left( \begin{array}{c}b_{i}-r_{x}\\\omega -n_{x}\end{array}\right) \left( \begin{array}{c}b_{i}\\ \omega \end{array}\right)^{-1}$

Topological secondary extinction occurs when $n_{x}=r_{x}$ , and the above formula then yields the probability of secondary extinction as

$\mathrm{pr}(e,x_{i}\mid \omega) = \left( \begin{array}{c}b_{i}-r_{x}\\ \omega -r_{x}\end{array}\right) \left( \begin{array}{c} b_{i}\\ \omega \end{array}\right)^{-1} = \frac{\omega !(b_{i}-r_{x})!}{b_{i}!(\omega -r_{x})!}$

This formulation must be elaborated to account for links to multiple different guilds, and consumed/prey species of different in-degrees, and hence their own varying probabilities of secondary extinction.

Categories: CEG theory, Topological extinction Tags: extinction, networks

First topological extinction results

December 13, 2008 proopnarine 2 comments

Program topo_CEG needed a bit of re-writing. The adjacency matrices generated from real communities are very large, due to high species richnesses. The matrices are so large that they cannot be initialized as simple arrays in C++, at least not on the stack. Had to use the Boost MultiArray function.

Comparison of full CEG (red) and topological-only results.

I ran 10 simulations of the Dicynodon Assemblage Zone (DAZ) community. Topological secondary extinction increases slowly, and then somewhat exponentially, as a result of increasing bottom-up perturbation. This is very encouraging in that the results are similar to the analytical results that can be obtained by using the combinatoric version of the model presented in Roopnarine (2006); results of application of this model to the DAZ were presented in Roopnarine et al. (2007). The main difference is that the simulations capture the effect of the stochasticity of the perturbation vector. Now, if we compare these results to those obtained with the full CEG simulation model applied to the DAZ, there are two obvious differences:

First, the full model yields higher levels of secondary extinction.
Second, the full model yields the “typical” CEG result, which means that there is, at some level of perturbation, a rapid increase (threshold) in the level of secondary extinction.

Therefore, topological extinction cannot account for the CEG results.

What else is there? The obvious missing feature are the top-down cascades that are initiated as a result of compensation for lost links/resources. And there is also link strength variance. These two features apparently generate a lot of the nonlinearity of the model. Exactly how much can be measured by basically subtracting the topological results from the full model results. This will require combining the full simulation program and topo-CEG. Going to need a bit of parallelization here!

Categories: Topological extinction Tags: paleontology, extinction, networks, food webs, nonlinear, simulations

topo_CEG

December 12, 2008 proopnarine Leave a comment

Whew. Wrote the initial program to examine the topological extinction. It’s just a skeleton right now, but it works! It’s a fork of the main CEG program, but additionally generates the adjacency matrix, applies the matrix manipulation outlined in an earlier post, and discards link strengths and trophic cascades from the system. It’s therefore looking only at topological secondary extinction.

Categories: Topological extinction

Next step

December 11, 2008 proopnarine Leave a comment

I’ve been reading one of Wolfram’s earliest papers on cellular automata, “Statistical mechanics of cellular automata“, and there are some striking similarities to themes and approaches that he utilized to explore elementary (and beyond) automata, and my attempts to understand some aspects of the CEG model, namely variance and criticality. The CEG model, however, while founded on necessary and sufficient ecological first principles, generates a level of complexity that does not allow the level of insight that Wolfram gets into the CAs. For example, I’ve looked at sensitivity to initial network configurations using an estimate of Hamming distance, but the relationship between Hamming distance and $\Psi$ remains unclear. So, I’m going to begin again by deconstructing CEG and re-building incrementally. The first step is to implement the iterative matrix approach to topological extinction, to see how much of the full model is reproducible.

Categories: Topological extinction, Uncategorized Tags: cellular automata, networks

Topological extinction II

December 10, 2008 proopnarine Leave a comment

Topological secondary extinction has been explored by a number of workers, such as Dunne et al. The CEG simulation approach incorporates topological extinction. Written in C++, it treats each species (and guilds) as individual objects, which gives the program a lot of flexibility and power in dealing with various networks. But I think that a much simpler approach can be taken for topological extinction alone, and it could open up some interesting new directions.

We take $A_{0}$ as the binary adjacency matrix of a directed species-level network derived stochastically from metanetwork $U$ . Rows are species or nodes of productivity. Row elements represent in-links, so a producer node has entries all equal to 0, except the diagonal element which equals 1. Consumer node rows are also binary, with element $ij=1$ for species $i$ if it is a predator of species $j$ . As an example, say that we have 6 species (numbered 1-6). Species 1-3 are producers, and 4-6 are consumers. Species 4 preys on 1 and 2, 5 preys on 2 and 3, and 6 is a top predator, specializing on species 4. The unperturbed topology matrix will therefore be

$A_{0} = \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right)$

Now we apply a perturbation equal to the removal (extinction) of a species $i$ . I do this by replacing all elements in the $i^{th}$ row and column with zeros. Continuing with the example, if species 1 and 2 are removed, the matrix now becomes

$A_{1} = \left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right)$

Notice that as a consequence, the 4th row now consists of all 0, meaning that species 4 has lost all its prey resources. This species thus becomes a topological secondary extinction. The next step in this iterative process is to therefore replace all elements of the 4th row and column with zeros, resulting in

$A_{2} = \left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right)$

resulting in the secondary extinction of species 6, the specialist predator of species 4. This is a very intuitive, and computationally rapid approach to examining topological extinction on networks! Essentially:

Define a starting state by selecting a species-level topology from the state space of metanetwork $U$ .
Define an initial condition as a (stochastic) perturbation of the topology by removing zero or more nodes.
Iterate as above until no more changes occur in the network’s topology.

I wonder if the final state can be predicted? And what is the impact of stochasticity of the perturbation when it is drawn from a large, species-rich guild such as the producers? This iteration is, I think, akin to a multidimensional cellular automaton. This will be an interesting direction to explore.

There is a significant difference between this however, and true cellular automata: the rules here are essentially one-way. By that I mean that you can always have the transition $1\rightarrow 0$ , but never the other way around. In other words, because we are treating the community in ecological time, and all its possible interactions are specified, we can only lose links; we cannot gain them. Therefore, the system will always settle to a stable state. As noted in an earlier post, there are two sets of stable states. First, there is minimal secondary extinction, and second there is very high to complete secondary extinction. An unstable state separates the two. Secondary extinction in the system could therefore be written as

$\Psi_{L} = f(\Omega \mid 0 \leq \Omega < \Omega_{c})$

$\Psi_{H} = f(\Omega \mid \Omega_{c} < \Omega \leq \Omega_{max})$

I guess the questions at this point are:

Is it possible to distinguish, from the initial topology and $\Omega$ whether secondary extinction will be low (L) or high (H), using topological secondary extinction only? And
Can $\Omega_{c}$ be explained in the topological secondary extinction model?

Categories: Topological extinction Tags: cellular automata, extinction, food webs, networks

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