When
Where
Speaker: Alvin Bayliss, Northwestern University
Title: Back to Childhood - The Rock-Paper-Scissors Games for Mathematicians and Ecologists
Abstract: Consider an ecological community consisting of N ≥ 3 species and call them ui, i = 1, . . . , N . The community is said to experience cyclic competition if ui+1 has a competitive advantage over ui with uN having a competitive advantage over u1 (thereby closing the cycle). When N = 3 the competition scheme is analogous to the classical children’s game of Rock-PaperScissors (RP S). Species u2 has an advantage over species u1 (rock crushes scissors), u3 has an advantage over u2 (paper covers rock) and u3 has a competitive advantage over u1 (scissors cuts paper). Communities exhibiting RP S competition extend over a wide range of scales of life, including, as examples, bacterial communities, reptilian communities and plant communities. For strong interspecies competition such communities are dynamically unstable. No one species can stably emerge as the winner, because there will always be a disruptive species to displace it. This behavior can be manifested by heteroclinic cycles between the three single-species states (limit cycles are also possible). In contrast with N = 4, the community would be dynamically stable - for strong interspecies competition there are stable alliances of species which do not directly compete (u1 − u3 and u2 − u4) and the community is bistable with the ultimate outcome depending on initial condition. These results generalize for larger N depending on the parity (odd N - dynamically unstable, even N - stable alliances). In reality, competition schemes are never as clean as described above. As an example, if there is even parity (e.g., N = 4), there can be internal competition and predation in one or both of the stable alliances, i.e., an alliance can be fractured due to internal competitive effects. In this talk we consider such modified competition schemes and show that they can bridge the gap between even and odd parity communities. In particular, even though the parity is even there is an embedded RP S system which leads to dynamical behavior. We also consider a coupled network of such communities and describe synchronization as well as desynchronization of the communities within the network. We show, in particular, that coupling two dynamically unstable communities can lead to a dynamically stable community - an ecological manifestation of the phenomenon known as “oscillation death” observed by Lord Rayleigh for coupled acoustic oscillators (organ pipes). These results are obtained via a combination of analysis of the resulting system of differential equations as well as numerical computation.