Research will be carried out on three topics: stochastic spatial models, processes taking place on random graphs, and questions related to the evolution of biological systems. Three of the proposed questions in the first topic concern "When can species coexist", while a fourth concerns the possibility that the quadratic contact process in two dimensions can have two phase transitions one for the existence of stationary distributions and a larger one for survival from a finite set. In the second topic one interesting mathematical problem concerns "explosive percolation" conjectured to have a discontinuous transition, while the more biologically important question concerns how the outcomes of epidemics and ecological competitions change when they take place on random graphs, which arguably provide better models of the real social networks. The third topic concern situations in which the characteristics of individuals in ecological competitions are also not static but evolve in response to their environment. My first steps in the area "adaptive dynamics" were taken in a study of predator-prey systems with John Mayberry. Here, we propose to study more complex examples that lead to evolutionary cycling and a second problem on the evolution of virulence, which leads to consideration of the role of spatial structure in increasing the virulence of diseases.
Many interesting mathematical questions arise from biology. Here we address some questions that arise from ecology and evolution. Three examples should illustrate the nature of our work. (1) At the turn of the century, observations of social networks revealed that we live in a small world in which everyone on the planet is separated by six degrees of separation. Now we need to understand how this geometry of social networks effects the spread of epidemics and the other biological and social processes. (2) The world shows much more biodiversity than mathematical models predict, so it is important to understand the mechanisms which allow for species coexistence. More generally, we will also be interested in how spatial structure changes the outcome of ecological competition. (3) In most situations the characteristics of individuals involved in competition with other species or with infectious agents are not static but evolve in time. For example, in most cases diseases evolve to be less virulent, but in a spatially structured population the opposite may occur. Co-evolution of hosts and parasites can lead to interesting evolutionary cycling, sometimes called ?Red Queen Dynamics after the character in Alice in Wonderland who has to keep running to stay in the same place.
