Spatial evolutionary games with small selection coefficients

Spatial evolutionary games with small selection coefficients

Rick Durrett May 5, 2014

Abstract
Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturba- tions to study spatial evolutionary games on Zd, d ≥ 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games we consider have payoff matrices of the form 1 + wG where 1 is matrix of all 1’s and w is small and positive. Since our population size N = ∞, we call our selection small rather than weak which usually means w = O(1/N). We prove that the effect of space is equiv- alent to replacing the replicator ODE by a related PDE where the reaction term is the replicator equation for a game matrix with some of the entries changed. The first idea is well known in the theory of stochastic spatial processes [58, 16, 62, 63]. The second is inspired by work of Ohtsuki and Nowak [28] (for the pair approximation). A remarkable aspect of our result is that the modifications of the game matrix depend on the interaction kernel only through the values of two simple probabilities for an associated coalescing random walk. 

link: http://www.math.duke.edu/~rtd/evog/spaceg.pdf

Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth

Sébastien Benzekry, Clare Lamont, Afshin Beheshti, Amanda Tracz, John M.L. Ebos, Lynn Hlatky, Philip Hahnfeldt

(Submitted on 5 Jun 2014)

Despite internal complexity, tumor growth kinetics follow relatively simple macroscopic laws that have been quantified by mathematical models. To resolve this further, quantitative and discriminant analyses were performed for the purpose of comparing alternative models for their abilities to describe and predict tumor growth. For this we used two in vivo experimental systems, an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: to 1) determine a statistical model for description of the volume measurement error, 2) establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) assess the models ability to forecast future tumor growth.
Nine models were compared that included the exponential, power law, Gompertz and (generalized) logistic formalisms. The Gompertz and power law provided the most parsimonious and parametrically identifiable description of the lung data, whereas the breast data were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power for the breast tumor growth curves, with excellent prediction scores (greater than 80%) extending out as far as 12 days. In contrast, for the lung data, none of the models were able to achieve substantial prediction rates (greater than 70%) further than the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement of predictions.
These results not only have important implications for biological theories of tumor growth and the use of mathematical modeling in preclinical anti-cancer drug investigations, but also may assist in defining how mathematical models could serve as potential prognostic tools in the clinical setting.

link: http://arxiv.org/abs/1406.1446