2008 December « Roopnarine’s Food Weblog

Nonlinear cascades

December 26, 2008 proopnarine Leave a comment

All the simulations described so far are of bottom-up perturbations to the basal level producer guilds. Network theory predicts that given networks such as food webs, with power law-like link distributions, the networks should be robust against random removal (extinction) of nodes, while being highly vulnerable to the removal (perhaps targeted) of highly linked (hub) nodes. This of course is a topological prediction, since it in no way incorporates dynamics of link strengths, compensatory modification of link strengths, or extinction thresholds (e.g. those Allee effects). Then, why do the CEG predictions and simulations of topological effects have a gradually, mildly exponential, rate of increase of secondary extinctions as the number of nodes removed is increased? The answer is two-part:

The probability of link loss increases with the in-degree of the consumer. Therefore most of the links being lost at any given level of perturbation are lost by highly linked species. But those species are also the most resistant to extinction.
The probability of secondary extinction increases almost linearly for consumers of very low degree, but almost not at all for the most highly linked species, until levels of perturbation are very high. Therefore most of the extinctions that occur at low to mid- levels of perturbation are of poorly connected species.

The nonlinear increase seen in the CEG simulations is therefore likely a response to a threshold being reached where highly linked consumers, though still robust to topological extinction, initiate significantly devastating top-down cascades because of compensatory increases of link or interaction strengths.

It is also therefore reasonable to hypothesize that very high levels of secondary extinction at low perturbation levels is the result of having a few highly connected, upper-level consumers. This could explain the great difference, at those perturbation levels, between the topological expectations and the simulations. Should low diversity communities, or communities with low diversities of high trophic level consumers, then be limited to those consumers being very specialized?

Categories: CEG theory Tags: cascades, connectance, extinction, food webs, networks, nonlinear, simulations

Allee effects

December 23, 2008 proopnarine Leave a comment

This is a very nice and detailed explanation of Allee Effects, a summary term for (mostly) stochastic events that heighten the probability of population extinction when population levels become very low. These effects are included in the CEG simulations as the MVP parameter, or Minimum Viable Population. MVP is taken as a proportion of the initial K (carrying capacity), and if realizable K falls to or below this level, the population is considered to be functionally extinction.

Categories: CEG theory Tags: carrying capacity, extinction, simulations

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

The Mystery Deepens…

December 19, 2008 proopnarine Leave a comment

3 guild simple metanetwork chain.

Just when I thought that I was getting a handle on this. I’ve decided to begin exploring topological extinction with a simple 3 guild metanetwork.This example shows three guilds, a producer, primary consumer and secondary consumer. The guild diversities (species richnesses and producer nodes) are 250, 25 and 5 respectively. I then proceeded to do four things.

CEG simulation results

FIrst, I conducted 100 CEG simulations of bottom-up, primary producer disruption of the system. The results are shown in the second figure, and illustrate the typical CEG response. Note, however, that past the threshold point that there is a subset of networks which continue to exhibit high levels of resistance to secondary extinction.

Next, I conducted simulations of topological extinction only. The results are shown in the third figure. The differences are quite dramatic. There are definitely two different types of networks here. One set seems to show the typical CEG result, and another set exhibits high secondary extinction at low levels of perturbation. It seems that the removal of top-down cascades and compensatory effects removes some sort of control or enhancement of resistance! And finally, superimposed on these results are results of estimating topological extinction as outlined in the previous posts. The two curves show two different ways of doing it. The lower curve (green) is a continuous interpretation of the link distributions, while the upper curve simply rounds results to integers, so that links (and lost links) come in integers only. BUT, the good news is that the analytical estimates of topological extinction provide a very nice match to the lower subset of the topological simulation results.

Topological-only simulation results. Green and blue curves are analytical estimates.

Why, however, are the simulation results divided into two subsets?

Categories: CEG theory Tags: extinction, food webs, 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

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