Rock-paper-scissors dynamics in space

Basic problem

A central problem in ecology is explaining how biodiversity is maintained within communities.  Here we will explore how the type of competitive interaction that occurs between species, combined with other ecological parameters such as population size (i.e. drift) and dispersal, influence the level of biodiversity that is maintained within communities. Intransitive interactions are illustrated by the rock-paper-scissors game, in which rock beats scissors, scissors beats paper and paper beats rock.

General approach

We will employ an approach using a cellular automaton, which allows simple implementation of changes in the types of competitive interactions, population size, and rate of dispersal.  First we will explore the trivial case in which fitness interactions between species are transitive; we will then quickly move to the more interesting case in which fitness interactions are intransitive.

What can be learned?

Concepts:

Fitness interactions
Interactive effects of drift and selection
Expected times to extinction
Paradox of enrichment

Methods:

Cellular automata

Starting point

Download the Downloadhandout (PDF, 164 KB) containing the description of the model system and the Downloadstarting script (R, 4 KB).

Interesting questions that you can investigate

How does the maintenance of diversity (i.e., expected time to extinction for all species except one) depend on:

population size
initial frequencies of the species
the strength of fitness interactions (i.e., the magnitude of the fitness advantage)
differences in the strength of fitness interaction between species (i.e., if A always beats B, but B sometimes beats C, and C sometimes beats A)
the distance over which organisms interact/disperse?

How does the magnitude of population fluctuations depend on the same set of factors?
How do the dynamics of the system change when there are greater numbers of species interacting?
What happens if only some species interact locally while others are capable of greater dispersal?
What is the effect of disturbance (e.g. local fires) on the maintenance of diversity?

Glossary

Intransitive fitness: a situation in which the fitness rankings of organisms is not linear (i.e. if W_A > W_B and W_B > W_C, one would expect that W_A > W_C. Intransitivity means that W_A < W_C).
Cellular automaton: a regular grid of cells in which the state of any cell at time t is a function of the states of a finite number of other cells (its neighbourhood) at time t-1.
Allelopathy: the chemical inhibition of one species by another.
Colicins: plasmid-encoded polypeptide toxins produced by and active against E. coli and related bacteria.
Periodic boundary conditions (a.k.a. infinite periodic tiling): when one goes off the top of the grid, one comes in at the corresponding position on the bottom, and when one goes off the left of the grid, one comes in on the right. This creates a situation in which no cells exist at the "edge" of the grid.

Literature & Weblinks

Sinervo B, Lively CM. (1996) external pageThe rock-paper-scissors game and the evolution of alternative male strategies. Nature 380: 240-243.
Durrett R, Levin S. (1997) external pageAllelopathy in spatially distributed populations. J.Theor.Biol. 185 (2): 165-171.
Frean M, Abraham ER (2001) external pageRock-scissors-paper and the survival of the weakest. Proc. Roy. Soc. 268 (1474): 1323-1327.
Kerr B, Riley MA, Feldman MW, et al. (2002) external pageLocal dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418 (6894): 171-174.
Nowak MA, Sigmund K. (2002) external pageBacterial game dynamics. Nature 418 (6894): 138-139.
Czaran TL, Hoekstra RF, Pagie L. (2002) external pageChemical warfare between microbes promotes biodiversity. PNAS 99 (2): 786-790.
Kirkup BC, Riley MA. (2004) external pageAntibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature 428 (6981): 412-414.
Laird RA, Schamp BS. (2006) external pageCompetitive intransitivity promotes species coexistence. Am. Nat. 168 (2): 182-193.
Karolyi G, Neufeld Z, Scheuring I. (2005) external pageRock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms. J. Theor. Biol. 236 (1): 12-20.
Reichenbach T, Mobilia M, Frey E. (2007) external pageMobility promotes and jeopardizes biodiversity in rock–paper–scissors games. Nature 448: 1046-9.

external pageExtensive description of the rock-paper-scissors game in Wikipedia