Propagation – for the want of a nail…

09 Tuesday Mar 2010

Posted by proopnarine in CEG theory, Topological extinction

Tags

extinction, Network theory, paleontology, Robustness

Understanding the dual role of in-degree on resistance (previous post) makes it possible to examine the structural robustness of an entire metanetwork in response to a specific perturbation. First, the network is perturbed by the removal (extinction) of several species from one or more guilds. If enough species, or the “right” ones are removed, this could in turn cause topological 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 secondary 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.

Let guild $G_{j}$ comprise species of different in-degrees, $y_{j}$ , and hence probabilities of secondary extinction. If $x_{i} (x_{i}\in G_{i})$ 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 $\psi_{j}^{y}$ , is
$p(e_{i}^{x}\vert \psi_{j}^{y}) = \frac{\psi_{j}^{y}!(\vert y_{j}\vert - r_{ij}^{x})!}{\vert y_{j}\vert! (\psi_{j}^{y}-r_{ij}^{x})!}$
Note that this is a simple re-statement of the earlier formula given for the probability of topological secondary extinction, with two differences. First, $r_{ij}^{x}$ is the expected number of $x_{i}$ ‘s links that come from $y_{j}$ -type species. It is estimated simply as the proportion of $y_{j}$ -type species out of the total number of species available as prey to $x_{i}$ :
$E(r_{ij}^{x}) = a_{ij}\frac{\vert y_{j}\vert}{b_{i}} r_{i}^{x}$
$\vert y_{j}\vert$ itself can be estimated from the trophic in-link distribution of guild $G_{j}$ as
$\vert y_{j}\vert \approx \lfloor \frac{\int_{r_{j}^{y}-1}^{r_{j}^{y}}P(r_{j})}{\int_{0}^{\vert G_{j}\vert}P(r_{j})} \rfloor$
Second, $\psi_{j}^{y}$ is substituted for $\omega$ as a generalization. Whereas $\omega$ referred specifically to the primary perturbation of the network, $\psi_{j}^{y}$ refers to any species removal, including secondary extinction in guild $G_{j}$ caused by perturbation elsewhere in the network.

The total expected level of secondary extinction of species of in-degree $x_{i}$ can now be estimated by applying the above formula to all types of prey species in all guilds with in-links leading to $G_{i}$ .
$E(\psi_{i}^{x}) = \vert x_{i}\vert \prod_{j=1}^{\vert U\vert} \prod_{r_{j}^{y}=1}^{b_{j}}p(e_{i}^{x}\vert \psi_{j}^{y}\vert)$
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 $\psi_{j}^{y}$ 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 $\psi_{j}^{y}$ . The actual order in which the calculations is made is important, because the perturbation will propagate along paths in an order defined and constrained by the metanetwork and food web topologies.

The formulas for $p(e_{i}^{x}\vert \psi_{j}^{y})$ and $E(\psi_{i}^{x})$ are significant, because they enable the estimation of the magnitude of extinction in a paleocommunity in response to the extinction of any single species or guild.

The Probability of Extinction

07 Sunday Mar 2010

Posted by proopnarine in CEG theory, Topological extinction

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Tags

connectance, extinction, link distribution, metanetwork, paleo-food web, probability, Robustness

p(extinction) when number of prey species=10

The CEG model asserts that food web structure plays a role in extinction. The intricate patterns of relationships among species in a community distribute the effects of changes in one species to others in its community. Therefore, while the ultimate causes of increased extinction in an interval of time may be abiotic, and might affect only some species directly, the effects could be felt more broadly.

Topographic secondary extinction.–The narrow definition of secondary extinction, where a species becomes extinct because it has lost all its resources, is termed topological secondary extinction (Roopnarine, 2009). Topological refers to the dependence of extinction solely upon the topology (pattern) of the network. Note that topological secondary extinction affects the network only in a bottom-up fashion, that is, in the direction of energy flow from producers to consumers of increasing trophic level. Measuring or estimating topological secondary extinction in a food web, in response to a particular perturbation, can be approached in three ways, again depending on whether one assumes accuracy of a higher-level representation of the food web (e.g. a metanetwork) or precision of a species-level network. Here I will outline a probabilistic approach using metanetworks.

Let a perturbation of magnitude $\omega$ be equal to the number of species removed randomly from the network. The probability that a species $x_{i}$ will become secondarily extinct is the probability that all its links are to species that are a subset of the $\omega$ set. This is determined from a hypergeometric distribution, where we can first ask: Given an in-degree of $r_{i}^{x}$ , what is the probability that $n_{i}^{x}$ of them will be lost?
$p(n_{i}^{x}\vert \omega) = \binom{r_{i}^{x}}{n_{i}^{x}} \binom{S-1-r_{i}^{x}}{\omega - n_{i}^{x}} \binom{S-1}{\omega}^{-1}$
where there are S-1 other species in the network and $0\leq\omega\leq S-1$ . The probability of $x_{i}$ becoming extinct occurs when $n_{i}^{x}=r_{i}^{x}$ ,
$p(e_{i}^{x}\vert \omega) = \binom{r_{i}^{x}}{r_{i}^{x}} \binom{S-1-r_{i}^{x}}{\omega - n_{i}^{x}} \binom{S-1}{\omega}^{-1} = \frac{\omega ! (S-1-r_{i}^{x})!}{(S-1)!(\omega -r_{i}^{x})!}$
The formula can be re-stated interestingly as
$\boldmath{p(\text{extinction of } x_{i}) = \frac{(\text{perturbation magnitude)}!(\text{species richness - trophic breadth)}!}{(\text{species richness})!(\text{perturbation magnitude - trophic breadth})!} }$
where trophic breadth is the number of species consumed by species $x_{i}$ , out of a pool of potential prey (species richness). For example, in a network of 10 species, $1\leq r_{i}^{x}\leq 9$ and $0\leq\omega\leq 9$ , and the probability of extinction increases as $\omega\rightarrow \text{S-1}$ , and decreases as $r_{i}^{x}\rightarrow\text{S-1}$ (see figure). Species with more resources are thus more resistant to topological secondary extinction. This is the same as saying that the more trophically generalized a species, the greater its resistance to extinction.

This explains in part the suggestion that food webs of greater connectance (C) are more resistant to topological secondary extinction (Dunne et al., 2002). Overall food web or network resistance to this type of extinction has been termed structural robustness (Dunne and Williams, 2009). Food web connectance does not increase, however, because of uniform increases in the in-degrees of all species in the network, but increases instead because of the presence of highly linked species. The skewed, long-tailed in-link distributions discussed earlier indicate the non-uniformity of in-degrees within real food webs. The above formula for extinction shows that $latex p(e_{i}^{x})

r_{i}^{y}$, that is, $x_{i}$ is of greater in-degree than $y_{i}$ . This will also be the case if the species on which $x_{i}$ preys are more resistant to extinction, even if $r_{i}^{x}=r_{i}^{y}$ . The presence of generalist consumers therefore enhances robustness both because of their own greater resistance, and the resistance which they confer upon their consumers.

New paper: Ecological modeling of paleocommunity food webs

30 Friday Oct 2009

Posted by proopnarine in CEG theory, Scientific models, Tipping point, Topological extinction

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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/

Tipping point II

08 Thursday Jan 2009

Posted by proopnarine in CEG theory, Tipping point, Topological extinction

≈ 1 Comment

Tags

cascades, extinction, food webs, Network theory, networks, nonlinear, simulations, Tipping point, top-down cascade

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.

Dichotomous topological extinction

23 Tuesday Dec 2008

Posted by proopnarine in Topological extinction

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networks, simulations

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.

Estimating topological extinction III

17 Wednesday Dec 2008

Posted by proopnarine in CEG theory, Topological extinction

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Add new tag, extinction, food webs, networks

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.

Estimating topological extinction II

16 Tuesday Dec 2008

Posted by proopnarine in CEG theory, Topological extinction

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extinction, networks

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.

Estimating topological extinction I

16 Tuesday Dec 2008

Posted by proopnarine in CEG theory, Topological extinction

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extinction, networks

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.

First topological extinction results

13 Saturday Dec 2008

Posted by proopnarine in Topological extinction

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Tags

extinction, food webs, networks, nonlinear, paleontology, simulations

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!

topo_CEG

12 Friday Dec 2008

Posted by proopnarine in Topological extinction

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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.

Roopnarine's Food Weblog

~ Ramblings and musings in evolutionary paleoecology

Category Archives: Topological extinction

Propagation – for the want of a nail…

The Probability of Extinction

New paper: Ecological modeling of paleocommunity food webs

Tipping point II

Dichotomous topological extinction

Estimating topological extinction III

Estimating topological extinction II

Estimating topological extinction I

First topological extinction results

topo_CEG