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:

1. Define a starting state by selecting a species-level topology from the state space of metanetwork $U$.
2. Define an initial condition as a (stochastic) perturbation of the topology by removing zero or more nodes.
3. 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

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

or

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

I guess the questions at this point are:

1. 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
2. Can $\Omega_{c}$ be explained in the topological secondary extinction model?