Computational complexity of ecological and evolutionary spatial dynamics

Significance An important question in evolution is: how does population structure affect the outcome of the evolutionary process? The theory of evolution in structured population has provided an impressive range of results, but an understanding of the computational complexity of even simple questions was missing. We prove that some fundamental problems in ecology and evolution can be precisely characterized by well-established computational complexity classes. This implies that the problems cannot be answered by simple equations. For example, there cannot be simple formulas for the fixation probability of a mutant given frequency-dependent selection in a structured population. We also show that, for example, calculating the molecular clock of neutral evolution in structured populations admit efficient algorithmic solutions. 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).