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On the fractional-order games with non-uniform interaction rate and asymmetric games. (English) Zbl 1488.34030

Summary: The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only in Mathematics, but also in Physics, Engineering and Mathematical Biology (see E. Ahmed et al. [Int. J. Mod. Phys. C 16, No. 7, 1017–1025 (2005; Zbl 1080.65536)] E. Ahmed et al. [J. Math. Anal. Appl. 325, No. 1, 542–553 (2007; Zbl 1105.65122); Phys. Lett., A 358, No. 1, 1–4 (2006; Zbl 1142.30303)] J. P. C. dos Santos et al. [Adv. Difference Equ. 2011, Article ID 642013, 15 p. (2011; Zbl 1216.45003)] A. M. A. El-Sayed [J. Fractional Calc. 7, 89–100 (1995; Zbl 0839.34069); Int. J. Theor. Phys. 35, No. 2, 311–322 (1996; Zbl 0846.35001)] A. M. A. El-Sayed and F. M. Gaafar [Appl. Math. Comput. 144, No. 1, 117–126 (2003; Zbl 1049.35002)] A. M. A. El-Sayed [J. Fractional Calc. 10, 101–107 (1996; Zbl 0888.34060); Nonlinear Anal., Theory Methods Appl. 33, No. 2, 181–186 (1998; Zbl 0934.34055)] A. M. A. El-Sayed and F. M. Gaafar [PU.M.A., Pure Math. Appl. 12, No. 3, 297–310 (2002; Zbl 1006.34008)] A. M. A. El-Sayed et al. [Comput. Appl. Math. 23, No. 1, 33–54 (2004; Zbl 1213.34025)] A. M. A. El-Sayed et al. [Math. Sci. Res. J. 8, No. 11, 336–348 (2004; Zbl 1068.45008)] A. M. A. El-Sayed et al. [Appl. Math. Lett. 20, No. 7, 817–823 (2007; Zbl 1140.34302)] P. √Črdi [Complexity explained. Berlin: Springer (2008; Zbl 1162.37001)] R. Gorenflo and F. Mainardi [CISM Courses Lect. 378, 223–276 (1997; Zbl 1438.26010)] for example).
In this paper we are concerned with the fractional-order differential equations describing games with non-uniform interaction rate C. Taylor and M. A. Nowak [Theor. Popul. Biol. 69, No. 3, 243–252 (2006; Zbl 1109.92037)] and asymmetric games J. Hofbauer and K. Sigmund [Evolutionary games and population dynamics. Cambridge: Cambridge University Press (1998; Zbl 0914.90287)]. Existence, uniqueness and stability of the solutions of these systems are studied.

MSC:

34A08 Fractional ordinary differential equations
91A40 Other game-theoretic models
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