I started this project in 1999, when I was interested in how different species of creatures living together in an ecological community adapt to each other, and what that change does to the community as a whole. I knew that Lotka-Volterra models are the simple, traditional standard for describing ecological interactions, and predator-prey interactions in particular, and I was interested in how I might use them to ask questions about how coevolution changes food webs.
I was also interested in the idea of ecological affordances, and Stuart Kauffman's ideas that you can't predict before the fact what features of an organism or habitat will be important to natural selection at later times. I was wondering how a model could have creatures develop a new feature at one time that turns out to be important to another creature later. I could just consider each creature to have a changing set of "features", but I also imagined what would happen if there were really a lot of "features", with randomly chosen "impacts" on others: the net impact would end up being normally distributed, following the law of large numbers.
That led me to consider a simplified model scenario: what if every time there's a mutation in population x, it changes each of that population's interactions by a small, normally distributed amount? This is a departure from the accepted ways of modeling mutations and selection: we normally model mutations as changes to individuals' traits, not as changes to their interactions. That shift in thinking led to a general conclusion about mutualism, as we will see, which wouldn't make sense without it.
In the Lotka-Volterra model, we describe an ecological community by spelling out all the interaction terms in the community. These are numbers, each one describing how one kind of creatures is affected by one other. We typically describe them in terms of their signs, two at a time. For instance, the impact of worms on robins is positive, because encountering a worm helps a robin grow and reproduce, while the impact of robins on worms is negative, because it reduces their numbers. A positive-negative pair is the signature of a predator-prey interaction. A negative-negative pair is a competition, while a positive-positive pair is a mutualism — an ecologically cooperative relationship, like the one between flowering plants and their pollinators.
So suppose when there's a mutation, all the interaction terms are a little different from the existing ones — and because new and old features affect each other in unpredictable ways, the little changes are all independent and random.
That's all we need. The Lotka-Volterra model says how the new and old populations grow and shrink, and which go extinct. A sequence of occasional mutations, with older populations often going extinct as they are replaced by newer ones, produces long-term change in the makeup of the ecological community. We can visualize it by just plotting all the interaction terms on a single axis, as the matrix changes through time.
Here is the result. In the simulations, for simplicity, I don't start with a predator and prey, just a single species with a ‒1 term for competition with itself. The self-interaction terms are plotted in black, while the rest are red, so we can use the black points to track the number of populations through time.
- Figure:
- Interactions vs. time, in evolving Lotka-Volterra matrix model.
There is a very strong pattern here: the one initial type splits into two different but coexisting types of creatures, and the interactions between them (the diagonally rising red points) steadily improve, changing from negative to positive numbers.
The positive red numbers mean that each of these populations is good for the other, not bad for it. That is, instead of competing with each other, or preying on each other, they are now mutualists.
Check back for a followup post in which I do some math, and find a general reason why interactions like to become mutualistic.
Links
How Selection Acts on Interactions and Why It Favors Mutual Benefit
A working draft of a paper on the subject.
pdf
Slides from a 2008 talk, which became the above draft paper.
pdf
This model and results were published in chapter 5 of my Ph.D. dissertation:
Evolution, Constraint, Cooperation, and Community Structure in Simple Models.
Worden, L., 2003 (PhD Dissertation). Subject matter includes modeling self-organization processes in ecological communities; evolution in the context of ecological communities; transformations between competitive and cooperative interactions in ecological and game theory models; and ways scientific models function as rhetorical tools in social contexts, and how alternative models can be used to intervene in accepted economic and social discourses.
pdf (warning — 14MB download)
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