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Eco-Evo model on predator prey interactions

Van der Laan and Hogeweg (1995) developed a predator-prey model to study the interaction between ecology and evolution. The model was set up with many phenotypic variants with a probability of consuming each other based on Gaussian distribution around predator phenotypes. Phenotypes are on a wrapped scale (such as time of day being active) and there is no space so that all prey and predators compete globally. There is therefore only shape-space, the space where variants are ordered (nearness). Offspring can be mutants on this shape-space.
Simulations are initialized with monomorphic and symmetrical predator and prey (i.e. same phenotype value). Quickly however there is speciation into two predator-prey pairs (Fig 1). Interestingly the predators are not on top of the prey, but in between them, and some evolutionary wiggles can be observed. Moreover the population dynamics are pretty much stabilized. However, if mutation is stopped then population shows oscillations! It would appear therefore that evolution (in the form of mutation and selection) is affecting ecological population dynamics.

So how to characterize this role of evolution? The most straightforward way is to stop mutations and see what happens. When this is done the system can die out! (This does depend on when mutations are stopped). Moreover, diversity is lost when mutation rates are put to zero. Note that this is not just a quasi-species, but four "true" species. What the wiggles therefore show is that at all times populations are being pushed phenotypically. Although this is a very simple model, this could be an important mechanism for maintaining ecosystem diversity, i.e. continuous adaptation through mutations.

To gain more insight the system is described using ODEs and parameters are fitted. However we know that the system can die out when mutations are stopped, therefore this must be done carefully: parameters are measured over long enough periods and parameters are averaged over time. In that case the ODE system can be found to survive, however there are still differences. In the ODE there are huge cycles (oscillations) and much greater atto-fox problems, i.e. it is still structurally less stable (FIG?). This suggests that:
  • There is a very narrow parameter range to make ODEs viable at an ecological time-scale (i.e. when time scales are separated)
  • The population dynamics of each species (period of system) are much shorter in evo-eco model, which is counterintuitive given that evolution is always considered to be a slow process. Instead evolution speeds up ecological dynamics showing interlocking timescales.
So what about different mutation rates? In a comparison of full simulation rates and the derived ODEs at different mutation rates (Fig 5) it becomes apparent that the effect of evolution on ecological dynamics is strongest for low mutation rates. This counterintuitive finding happens because mutations affect how far predators and prey bands are away from each other: small mutations result in some separation between predator and prey, while high mutation allows close matching and more specialized predators. In ecological models (ODEs) equidistant predators result in chaotic behaviour, and therefore when evolution leads to self-organization of stability, this generates the greatest disparity between the evo-eco model and the eco-ODE model. (Note: different non-wrapped boundary conditions do not significantly change the results, at least in the middle of shape-space).

Changing the width of the interaction of phenotypes (i.e. the width of the Gaussian, σ) also leads to interesting differences. As σ increases the wiggles get longer. Moreover as σ decreases the system changes from a static system to a run-away red queen dynamics.

As a bottom-line we therefore conclude that this model study shows
  • an existence proof of a counter example that ecological processes are necessarily fast and evolutionary ones slow, and
  • that evolution could play an important (continuous) role in stabilizing ecosystems

(Note: although we assumed prey and predators have the same mutation rates, allowing higher mutation rates in predators does not change the results much. This is because evolution is always mediated by competition (selection coefficient). Hence results hold for unequal mutation rates too.)

Eco-evo in space

When the above analysis is repeated in space (Savill & Hogeweg 1997), the most commonly observed behaviour of the system is that of a prey diversifying and predators remaining constant. Such periods are interrupted by predators suddenly switching, leading to a short period of two predator species and then returning to a single predator again. On an ecological level population dynamics show stable predators and oscillating prey.

On a spatial level there are oscillations between small scale patterns and large scale patterns. Small scale patterns correspond to the 2 predator case (i.e. when system switches), where predator and prey are close enough and there is strong predation and large waves. When the prey mutates enough, the strength of predation declines and prey can live amongst predator, which then allows them to specialize again, thus switching the system to larger waves again.

Next: Vesicles and group selection

Van der Laan, J.D. and P. Hogeweg (1995) Predator-prey coevolution: interactions among different time scales. Proc. Royal Soc. London B 259, 35-42. PDF-file
Savill, N. J. and P. Hogeweg (1997) Evolutionary stagnation due to pattern-pattern interactions in a co-evolutionary predator-prey model. Artificial Life 3: 81-100.