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.
Under this grant research was performed on spatial models in three settings: ecology; spread of diseases, opinions, and fads through social networks; models of cancer growth and progression. In the area of ecology, work with Ted Cox and Ed Perkins on "Voter Model Perturbations" developed a general method for understanding equilibria of spatial models, providing a rigorous proof of a crtierion for the persistence of altruism in a spatial population. This work also applies to recently developed spatail evolutionary games used to model cancer. Work with Yuan Zhang on Schelling's 1975 model provided one of the first rigorous results about the emergence of segregation in a spatial model as tolerance for neighbors with different opinions is decreased. Work of a group of eight authors begun during the 2010-2011 year on complex networks led to an understanding of the phase transition in a voter model where the individuals opinions and the network coevolve. Later work with a postdoc, David Sivakoff, and two students at nearby NC School for Science and Math found the equilibirium distribution for a model in which individuals prefer to connect with others with similar opinions or with those who have many connections. The results in the last two paragraphs give insights into how the spatail or social relationships of individuals shape the behavior of the system as a whole. However, more important are the contributions made to quanitfying heterogeniety in cancer, an important obstacle to treatment. These were done for a nonsptial branching process model and more recently for the spatial Moran model. Though that model is simple, it provides theoretical results to help understand the "field effect in cancer," the existence of premalignant transformation in the area surrounding a tumor. The research described above shows the intellectual merit of the work. The broader impacts come in two directions. (i) Mentoring postdocs and training graduate students to work in these areas, and exposing talented high school students to matheamtical research. (ii) The impact of contributions to understanding the mechanisms of cancer and ultimately to improving treatment. The last four words ae not an empty promise. Durrett andhis postdoc marc Ryser are working with two Doctors in the Duke medical center on applications to head and neck squamous cell carcinoma.
