1. Spatial Invasion Fitness: Interesting Research Avenues Régis Ferrière, Jean-Francois Le Galliard, and Ulf Dieckmann

2. n … in Metapopulations n … on Lattices n … on Graphs n … in Continuous Space Spatial Invasion Processes

3. n First question How can we define invasion fitness for spatially extended ecological systems? n Second ( related, and perhaps even more important ) question How can we derive invasion probabilities for spatially extended ecological systems? n Basic problem Even at the onset of the invasion process, mutants interact with themselves — while being globally rare they can already be locally common. Mutant invasion focus Resident background

4. The Typical Three Phases of Spatial Invasion log N mut  log N res Initial nonlinear phase Linear phase (might not even exist) Final nonlinear phase Time Invasion fitness?

5. Open Problems n (1) How can we best characterize the resident equilibrium, in a way that is maximally relevant for predicting the mutant's dynamics? n (2) How can we approximate the mutant's exponential growth rate in the linear phase? n (3) Can we really assume that there always is a linear phase? What are the prerequisites for its existence? In other words, under what ecological conditions is the mutant still globally rare after having gone through the initial nonlinear part of its invasion dynamics? How about invariably compact invasion foci? n (4) What is the relation between the exponential growth rate during the linear phase and the mutant's invasion probability?

6. Open Problems n (5) What deterministic effects operating during the initial nonlinear phase can significantly modify the invasion probability? (In fact, the invasion probability is expected to strongly depend on the mutant's dynamics when it is rarest: this gives it a large quantitative weight.) n (6) How do fluctuation corrections during the initial nonlinear phase influence the mutant's invasion probability? n (7) How do the various layered contributions formally affect the invasion probability? Always multiplicatively? How about interaction effects? n (8) How do these considerations relate to issues of multi-level selection and group selection? Are these concepts perhaps too crude when levels cannot be sharply defined?