The first model, based on [Silvertown, et al, 1992], uses empirical data about the invasion rates of 5 different species of grass to create a CA to study the effect of the initial pattern on the process of population spread. The second two models are based on [Durrett and Levin, 1998]'s response to Silvertown's model. They point out that while the process varies in Silvertown's model, the end state is very predictable. Their first model shows that dispersal range (or neighborhood size) and probability of invasion can form a trade-off that can shift the competitive dynamic one way or the other. Their second model involves cyclic competition, where for species 1, 2, and 3, 1<2<3<1 (< meaning "is invaded by"). Finally, I extend their final model to include different size neighborhoods for each species, and explore how this changes the population sizes.
Silvertowns model includes five species of grass with a table of invasion probabilities. Each column is a native species, and each row is an invader, and each value is the probability that the invader grass will invade an area occupied by the native grass. This data was empirically collected by [Thornhallsdottir, 1990]. The model is given an initial state where five bands of grass are placed adjacent to each other in a toroidal space. Alternating which bands are adjacent to which drastically changes how each species spreads at different times in the simulation. The probability that a given cell will be invaded is based on a neighborhood of the four closest neighbors, averaged together. The timesteps are synchronous.
In Durrett and Levin's first model, two species compete for space. One has a neighboorhood of the four closest members, which are invaded at each timestep with a probability of one; that is it will certainly invade these neighbors. The other species, however, has a larger neighborhood of 36 and a lower probability of invading the members of that neighborhood.
I wasn't able to perfectly copy their model from their description, but I did get one with the same behavior. One difference is that my model is synchronous and their's is asynchronous. Durett and Levin claim that with stochastic models like this one, this has no effect on the results. Another difference is that I believe their model chose a neighbor at random and invades it with a certain probability, whereas in my model, all neighbors are invaded, each with a certain probability, so my probabilities are much lower than theirs; about one tenth as high. But the behavior is identical to what they describe, which is that by varying this probability, competition can favor one species or the other. You can get very close to a balance (around .84 for them, around .085 for me). This simply shows that neighborhood size (or, in plant ecology terms, seed dispersal range), can impact the population dynamics. At this point, Durett and Levin have already diverged from using empirical data into theoretical populations.
Their main dissatisfaction with Silvertown's model, however, was the predictable end state, and we still haven't gotten away from that. For this, they look to nonhierarchical competition, in particular, cyclic competition where for species 1, 2, and 3, 1<2<3<1, where < means "is invaded by". This situation has been investigated in metapopulation studies using ODEs. In [May and Leonard, 1975], it is shown that there are some situations where cyclic competition does not lead to cyclic fluctuations in population sizes; rather the population dynamics diverge outward, getting ever closer to the case where one species completely dominates.
They point out that biologically, this is "nonsense". What's interesting about Durrett and Levin's paper is that this absurdity disappears in a spatially explicit model like a cellular automata.
The cells in this model are either empty (0) or occupied by one of three species (1, 2, 3). The have a neighborhood of 36. At each timestep, a neighbor is chosen at random and is "killed" with a certain probability, meaning the state of that cell changes to zero in the next timestep. Then that cell is "invaded" by the species at the current cell with another probability. I kept these probabilities constant for each of the three species for simplicity's sake.
Given a large enough area of cells, the populations in this model will fluctuate but never diverge to total dominance by one species. (See [Durrett and Levin, 1998] for a proof.) Also, the model has an interesting spatial structure; more interesting than model's A or B.
I decided to extend Model C to explore the effect of varying the neighboorhood sizes of the 3 species. I gave species 1 a neighborhood radius of 1.0, species 3 a neighborhood radius of 5.0, and species 2 a radius somewhere in between. I tried this with the situation where 1<2<3<1 and where 1>2>3>1. This gave each species very different-looking behaviors that I would like to explore more in the future. In this project, I just looked at how the neighborhood sizes effect the relative sizes of the populations.
I found that with the cyclic model 1<2<3<1 where neighborhood size is increasing from species 1 to 3, a mid-sized neighborhood leads to smaller population sizes. By varying the size of this middle neighborhood, I found that this gave no notable change to the population of species 2, although it did favor species one, with a small neighborhood, to increase the neighborhood size of species 2.
Reversing the competition cycle (1>2>3>1) so that the species with larger neighborhoods were dominated by those with smaller neighborhoods greatly favored species 2.
Results from Model D when 1<2<3<1, for various neighborhood radii for species 2
When I saw the population fluctuations that resembled sine waves, I wondered if an acoustic display would reveal some properties about these fluctuations. It seems that if they are at all periodic, they would create a tone. To try this out, I used the Maxlink library to send population data to a Max/MSP patch which stores it as sound data. The results were basically white noise, but with a slight bias to a certain frequency range. Increasing the size of the mid-sized neighborhood raises this bias a bit. They take a long time to generate, so here are 4 second recordings of the three species populations and the empty cells. I raised the size of the mid-sized neighborhood over the course of the recording, creating a slight change in overall pitch.4_species 1.aif
I find these sounds interesting because of how they were generated, but I don't claim that they are particularly interesting in themselves. They seem to have more spatial depth than simple noise, but not much. The main reason I've documented them here is because I think this approach to sound synthesis, using a chaotic system of distributed populations, or something along these lines, could be fruitful and is something I would like to explore more.
It would be interesting to be able to explore these models from the opposite end, specifying population sizes or other characteristics such as patch growth rate, population fluctuation periods, or rate of travel for patches, and have the details of probabilities and neighborhood sizes handled behind the scenes. For instance, can the population dynamics and behaviors be made to resemble predator/prey models (very few sharks, large patches of fish) without the need to represent the internal state of agents (age, hunger, etc.). To this end, more analysis of how the different settings effect the population behaviors would be needed. Also, it would be interesting to bring back in empirical data for more accuracy. It's not completely clear what sort of system can be represented with cyclic competition although Durrett and Levin propose that the succession of grass to shrubs to trees, and back to grass after a disruptive event, may qualify.
Durrett, R. and Levin, S. 1998. Spatial Aspects of Interspecific Competition, Theoretical Population Biology. 53, 30-43.
May, R. M. and Leonard, W.J. 1975. Nonlinear aspects of competition between species, SIAM Journal of Applied Mathematics. 29, 243-253
Silvertown, J., Holtier, S., Johnson, J., and Dale, P. 1992. Cellular automaton models of interspecific competition for space—the effect of pattern on process, Journal of Ecology. 80, 527-534.
Thornhallsdottir, T.E. 1990. The dynamics of five grasses and white clover in a simulation mosaic sward. Journal of Ecology, 78, 909-923.
While developing this project, all the models were kept in one sketch which has several features all activated with a confusing mess of key commands. Here are the source files:
Zip file of sketch folder:biosims_final.zip