Solutions with a bounded support promote permanence of a distributed replicator equation

References

• Hofbauer J, Sigmund K. Evolutionary games and population dynamics. Cambridge: Cambridge University Press; 1998.
   Google Scholar
• Smith HL, Thieme HR. Dynamical systems and population persistence. Vol. 118, Graduate studies in mathematics. Providence (RI): American Mathematical Society; 2011.
   Google Scholar
• Lewontin RC. The meaning of stability. Brookhaven symposia in biology. 1969;22:13–24.
   PubMedGoogle Scholar
• Hofbauer J, Sigmund K. Evolutionary game dynamics. Bull. Am. Math. Soc. 2003;40:479–519.
   Web of Science ®Google Scholar
• Schuster P, Sigmund K. Replicator dynamics. J. Theor. Biol. 1983;100:533–538.
   Web of Science ®Google Scholar
• Cantrell RS, Cosner C. Spatial ecology via reaction–diffusion equations. New York, NY: Wiley; 2003.
   Google Scholar
• Dieckmann U, Law R, Metz JAJ. The geometry of ecological interactions: simplifying spatial complexity. Cambridge: Cambridge University Press; 2000.
   Google Scholar
• Lieberman E, Hauert C, Nowak MA. Evolutionary dynamics on graphs. Nature. 2005;433:312–316.
   PubMed Web of Science ®Google Scholar
• Schreiber SJ, Killingback TP. Spatial heterogeneity promotes coexistence of rock-paper-scissors metacommunities. Theor. Population Biol. 2013;86:1–11.
   PubMed Web of Science ®Google Scholar
• Perthame B. Parabolic equations in biology. Berlin: Springer; 2015.
   Google Scholar
• Novozhilov AS, Posvyanskii VP, Bratus AS. On the reaction--diffusion replicator systems: spatial patterns and asymptotic behaviour. Russ. J. Numer. Anal. Math. Model. 2012;26:555–564.
   Web of Science ®Google Scholar
• Bratus AS, Posvyanskii VP. Stationary solutions in a closed distributed Eigen-Schuster evolution system. Differ. Equ. 2006;42:1762–1774.
   Web of Science ®Google Scholar
• Bratus AS, Posvyanskii VP, Novozhilov AS. A note on the replicator equation with explicit space and global regulation. Math. Biosci. Eng. 2011;8:659–676.
   PubMed Web of Science ®Google Scholar
• Bratus AS, Posvyanskii VP, Novozhilov AS. Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates. Nonlinear Anal.: Real World Appl. 2010;11:1897–1917.
   PubMed Web of Science ®Google Scholar
• Cressman R, Dash AT. Density dependence and evolutionary stable strategies. J. Theor. Biol. 1987;126:393–406.
   Web of Science ®Google Scholar
• Cressman R, Vickers GT. Spatial and density effects in evolutionary game theory. J. Theor. Biol. 1997;184:359–369.
   PubMed Web of Science ®Google Scholar
• Hutson VCL, Vickers GT. The spatial struggle of tit-for-tat and defect. Philos. Trans. Roy. Soc. Ser. B: Biol. Sci. 1995;348:393–404.
   PubMed Web of Science ®Google Scholar
• Hutson VCL, Vickers GT. Travelling waves and dominance of ESS’s. J. Math. Biol. 1992;30:457–471.
   Web of Science ®Google Scholar
• Vickers GT. Spatial patterns and travelling waves in population genetics. J. Theor. Biol. 1991;150:329–337.
   PubMed Web of Science ®Google Scholar
• Vickers GT. Spatial patterns and ESS’s. J. Theor. Biol. 1989;140:129–35.
   PubMed Web of Science ®Google Scholar
• Weinberger ED. Spatial stability analysis of Eigen’s quasispecies model and the less than five membered hypercycle under global population regulation. Bull. Math. Biol. 1991;53:623–638.
   Web of Science ®Google Scholar
• Bratus AS, Hu C-K, Safro MV, et al. On diffusive stability of eigens quasispecies model. J. Dyn. Control Syst. 2016;22:1–14.
   Web of Science ®Google Scholar
• Vaidya N, Manapat ML, Chen IA, et al. Spontaneous network formation among cooperative RNA replicators. Nature. 2012;491:72–77.
   PubMed Web of Science ®Google Scholar
• Bratus AS, Posvyanskii VP, Novozhilov AS. Replicator equations and space. Math. Model. Nat. Phenom. 2014;9:47–67.
   Web of Science ®Google Scholar
• Barenblatt GI, Entov VM, Ryzhik VM. Theory of fluid flows through natural rocks. Springer-Verlag: Kluwer Academic Publishers; 1989.
   Google Scholar
• Knabner P, Angerman L. Numerical methods for elliptic and parabolic partial differential equations. Vol. 44. New York: Springer; 2003.
   Google Scholar
• Postnikov EB, Sokolov IM. Continuum description of a contact infection spread in a sir model. Math. Biosci. 2007;208:205–215.
   PubMed Web of Science ®Google Scholar
• Aronson DG. The porous medium equation. In: Fasano A, Primicerio M, editors. Nonlinear diffusion problems. Springer; 1986. p. 1–46.
   Google Scholar
• Bertsch M, Dal Passo R, Ughi M. Discontinuous viscosity solutions of a degenerate parabolic equation. Trans. Am. Math. Soc. 1990;320:779–798.
   Web of Science ®Google Scholar
• Dal Passo R, Luckhaus S. A degenerate diffusion problem not in divergence form. J. Differ. Equ. 1987;69:1–14.
   Web of Science ®Google Scholar
• Evans LC. Partial differential equations. 2nd ed. Providence, RI: American Mathematical Society; 2010.
   Google Scholar
• Horn RA, Johnson CR. Matrix analysis. Cambridge: Cambridge University Press; 2012.
   Google Scholar