title: Computational complexity of ecological and evolutionary spatial dynamics.
abstract: There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).
authors.affiliation: Institute of Science and Technology Austria, A-3400 Klosterneuburg, Austria; ribsen@ist.ac.at.
authors.forename: Rasmus
authors.initials: R
authors.lastname: Ibsen-Jensen
authors.affiliation: Institute of Science and Technology Austria, A-3400 Klosterneuburg, Austria;
authors.forename: Krishnendu
authors.initials: K
authors.lastname: Chatterjee
authors.affiliation: Program for Evolutionary Dynamics, Departments of Organismic and Evolutionary Biology and Mathematics, Harvard University, Cambridge, MA 02138.
authors.forename: Martin A
authors.initials: MA
authors.lastname: Nowak
mesh_terms: D000465:Algorithms; D005075:Biological Evolution; D019295:Computational Biology; D004463:Ecology; D017753:Ecosystem; D011336:Probability
keywords: complexity classes; evolutionary games; fixation probability