Open Access
October 2018 A simple evolutionary game arising from the study of the role of IGF-II in pancreatic cancer
Ruibo Ma, Rick Durrett
Ann. Appl. Probab. 28(5): 2896-2921 (October 2018). DOI: 10.1214/17-AAP1378

Abstract

We study an evolutionary game in which a producer at x gives birth at rate 1 to an offspring sent to a randomly chosen point in x+Nc, while a cheater at x gives birth at rate λ>1 times the fraction of producers in x+Nd and sends its offspring to a randomly chosen point in x+Nc. We first study this game on the d-dimensional torus (ZmodL)d with Nd=(ZmodL)d and Nc = the 2d nearest neighbors. If we let L then t the fraction of producers converges to 1/λ. In d3 the limiting finite dimensional distributions converge as t to the voter model equilibrium with density 1/λ. We next reformulate the system as an evolutionary game with “birth-death” updating and take Nc=Nd=N. Using results for voter model perturbations we show that in d=3 with N= the six nearest neighbors, the density of producers converges to (2/λ)0.5 for 4/3<λ<4. Producers take over the system when λ<4/3 and die out when λ>4. In d=2 with N=[clogN,clogN]2 there are similar phase transitions, with coexistence occurring when (1+2θ)/(1+θ)<λ<(1+2θ)/θ where θ=(e3/(πc2)1)/2.

Citation

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Ruibo Ma. Rick Durrett. "A simple evolutionary game arising from the study of the role of IGF-II in pancreatic cancer." Ann. Appl. Probab. 28 (5) 2896 - 2921, October 2018. https://doi.org/10.1214/17-AAP1378

Information

Received: 1 March 2017; Revised: 1 December 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974768
MathSciNet: MR3847976
Digital Object Identifier: 10.1214/17-AAP1378

Subjects:
Primary: 60K35 , 92D15

Keywords: reaction-diffusion equation , Replicator equation , voter model perturbation , weak selection

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 5 • October 2018
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