Linear Stability Analysis of Ripa et al. 2009

Part of my thesis work has been looking at non-equilibrial evolutionary dynamics in undersaturated communities. My and my coauthor’s hypothesis is that many non-equilibrial evolutionary dynamics in model systems can be traced back to the system not having enough species to obtain an evolutionarily stable strategy (ESS). If true, then systems at the ESS should show stable evolutionary dynamics while unstable evolutionary dynamics should only be present when a system is not at ESS. We analyzed 4 different models of evolutionary dynamics to see if this conjecture is true, including the one found in Niche co-evolution in consumer-resource communities by Ripa et al., 2009.

The Ripa et al., 2009 paper is a model of competitive and predator-prey dynamics with potential for evolutionary diversification. It has a couple of key parameters that govern this diversification, namely the degree of limiting similarity, available niche space, and specialization of predators. We varied the final two parameters and used linear stability analysis to get the stability of a system with a fixed number of species. The following picture shows the results of a 1 predator, 1 prey system

Here, the colors represent different results of the stability analysis with yellow being a stable, non-oscillatory ESS, green being a stable, oscillatory ESS, blue being a stable, oscillatory non-ESS, purple being a unstable oscillatory non-ESS, and red being a unstable, non-oscillatory non-ESS. As we can see from the results, the ESS is always stable (if oscillatory) while unstable dynamics only appear in non-ESS’s (non-ESS’s can also show stable dynamics). This pattern repeats for an increasing number of species until we get to 3 predator, 3 prey systems which are entirely unstable (seen below).

It remains a question whether this is a true result or whether there was something wrong with the analysis (it was done numerically, not analytically), and if true, the reasons for it.

By thetweedybiologist

Research of theoretical ecology and evolution

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