# Josef Hofbauer's Publications

A short proof of the Lagrange-Good formula. * Discrete Math.* **25** (1979) 135-139.
(with P. Schuster and Karl Sigmund)
A note on evolutionarily stable strategies and
game dynamics. * J. theor. Biol. * ** 81** (1979), 609-612.

(with Peter Schuster, Karl Sigmund and Robert Wolff)
Dynamical systems under constant organisation. Part 2: Homogeneous growth functions of degree 2. * SIAM J. Appl. Math. * **38 **(1980), 282-304.
(with Peter Schuster and Karl Sigmund) Competition and cooperation in catalytic selfreplication. * J. Math. Biol. * ** 11** (1981), 155-168.
On the occurrence of limit cycles in the Volterra-Lotka equation. * Nonlinear Analysis * ** 5 ** (1981) 1003-1007.
A general cooperation theorem for hypercycles.
* Monatshefte Math. * ** 91 **(1981) 233-240.
(with Peter Schuster, Karl Sigmund and Robert Wolff) Selfregulation of behaviour in animal societies. Part 1: Symmetric contests.
* Biol. Cybern.* ** 40 ** (1981), 1-8.
(with Peter Schuster, Karl Sigmund and Robert Wolff) Selfregulation of behaviour in animal societies. Part 2: Games between two populations without selfinteraction.
* Biol. Cybern.* ** 40 ** (1981), 9-15.
(with Peter Schuster, Karl Sigmund, R. Gottlieb and P. Merz)
Selfregulation of behaviour in
animal societies. Part 3: Games between two populations with
selfinteraction. * Biol. Cybern.* ** 40 ** (1981), 16-25.

Lagrange Inversion. * Actes du Seminaire Lotharingien de Combinatoire * **6 ** (1982).
(with Ethan Akin) Recurrence of the unfit. * Math. Biosciences * ** 61 ** (1982) 51-63.
(with Peter Schuster and Karl Sigmund) Game dynamics for Mendelian populations. * Biol. Cybern.* ** 43** (1982), 51-57.

(with Karl Sigmund) ** Evolutionstheorie und dynamische Systeme**. Parey, Hamburg-Berlin (1984). 200 pp.
A q-analog of the Lagrange expansion. * Archiv Math. * ** 42 ** (1984) 536-544.
(with Peter Schuster) Dynamics of linear and nonlinear autocatalysis and
competition. In: Stochastic Phenomena and Chaotic Behaviour in Complex
Systems (Proc. Conf. Flattnitz 1983). Ed. P. Schuster. Springer Series in
Synergetics 21, pp. 159-171. 1984.
(with Gerard Iooss)
A Hopf bifurcation theorem for difference equations approximating a differential equation. * Monatshefte Math. * **98 **(1984) 99-113.
A difference equation model for the hypercycle.
* SIAM J. Appl. Math. * ** 44** (1984) 762-772.

(with Erwin Amann) Permanence in Lotka-Volterra and replicator equations.
In: Lotka-Volterra Approach to Cooperation and Competition (Proc. Wartburg
1984). Ed. W. Ebeling and M. Peschel. pp. 23-34. Akademie-Verlag
Berlin 1985.
(with J. Fürlinger) q-Catalan numbers.
* J. Combinatorial Theory * ** A40 **(1985) 248-264.
Gradients versus cycling in genetic selection models. In: Dynamics of
Macrosystems (Proc. Laxenburg 1984). Ed. J. P. Aubin, D. Saari and
K. Sigmund. pp. 90-101. Springer Lecture Notes in Economics and
Mathematical Systems 257. 1985.
The selection mutation equation. * J. Math. Biol. * ** 23 ** (1985) 41-53.

(with K. Sigmund) Permanence for replicator equations.
In: Dynamical Systems. A. Kurzhansky and K. Sigmund (eds.). Springer Lecture Notes Econ. Math. Syst. 287 (1987), pp.70-91.
(with John Maynard Smith) The `battle of the sexes': A genetic model with limit cycle behavior. * Theor. Population Biology * **32** (1987) 1-14.
(with Vivian Hutson und Wolfgang Jansen) Coexistence for systems governed by difference equations of Lotka-Volterra type. * J. Math. Biol.* **25** (1987) 553-570.
Heteroclinic cycles on the simplex. Proc. Int. Conf. Nonlinear
Oscillations. Ed. M. Farkas, V. Kertesz, G.Stepan.
Budapest: Janos Bolyai Math. Society. 1987.

(with
Karl Sigmund ): ** The Theory of Evolution and Dynamical Systems. **
Cambridge University Press 1988.
Saturated equilibria, permanence and stability for ecological systems.
In: * Mathematical Ecology * (Proc. Conf. Trieste 1986).
Eds. L. J. Gross, T. G. Hallam and S. A. Levin.
Singapore-Hong Kong: World Scientific. 1988.

A unified approach to persistence. * Acta
Applicandae Math. * ** 14** (1989) 11 - 22.
(with Karl Sigmund): On the stabilizing effect of predators and competitors on ecological communities. * J. Math. Biology * ** 27** (1989) 537 - 548.
(with J. W.-H. So )
Uniform persistence and repellors for maps.
* Proc. Amer. Math. Soc.* **107** (1989) 1137-1142.

An index theorem for dissipative semiflows.
* Rocky Mountains J. Math. * ** 20** (1990) 1017-1031.
(with J. W.-H.
So)
Multiple limit cycles for predator-prey models.
* Math. Biosciences * **99 ** (1990) 71-75.
(mit Karl
Sigmund )
Adaptive dynamics and evolutionary stability.
* Appl. Math. Letters * **3** (1990) 75-79.

(with J. Mallet-Paret and
Hal Smith )
Stable periodic solutions for the hypercycle system.
* J. Dynamics and Diff. Equs. * **3** (1991) 423-436.
(with
Andrea
Gaunersdorfer
and Karl Sigmund
)
On the dynamics of asymmetric games.
* Theor. Pop. Biol. * ** 39** (1991) 345-357.

Heteroclinic cycles in ecological differential
equations.
* Tatra Mount. Math. Publ. *
**4** (1994), 105-116.
(with J. W.-H.
So)
Competition in the gradostat: the global stability problem.
* Nonlinear Analysis * ** 22** (1994), 1017-1031.
(with R.
Bürger) Mutation load and mutation-selection balance in quantitative genetic traits. * J. Math. Biol. * ** 32** (1994) 193-218.
(with J. W.-H. So ) Multiple limit cycles in three dimensional Lotka--Volterra equations. * Appl. Math. Letters * ** 7** (1994) 65-70.

(with
A. Gaunersdorfer ) Fictitious play, Shapley polygons, and the
replicator equation . * Games and Economic Behaviour *
**11** (1995) 279-303.
Rational versus Evolutionary Dynamics for Games.
In: * Dynamic Evolutionary Game Theory in Biology and Economics. *
August 1995. Waterloo and Guelph. Conference Report, ed.: R. Cressman.
(with Eva Jablonka, B. Oborny, I. Molnar, Éva Kisdi , T.
Czárán)
The adaptive advantage of phenotypic memory in changing environments.
* Phil. Trans. Royal Soc. London *
** B 350 **(1995) 133-141.

(with
J. W.-H. So and Yasuhiro Takeuchi)
Global stability of competition in a spatially heterogenous environment.
* Differential Equations Dynamical Systems *
** 4** (1996) 213-223.
Evolutionary dynamics for bimatrix games: A Hamiltonian system? * J. Math. Biol. * **34** (1996), 675-688.
Springer Link.
(with Ross Cressman, W.G.S. Hines) Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. * J. Math. Biol. * **34** (1996), 707-733.
Springer Link.
(with Jörgen W. Weibull
) Evolutionary selection against dominated strategies. * J. Economic Theory * ** 71 ** (1996), 558-573.

(with Barnabas Garay) Chain recurrence and discretization.* Bull. Austral. Math. Soc. * ** 55**, 63-71 (1997).
(with
Vivian Hutson ,
Glenn T. Vickers )
Travelling waves for games in economics and biology.
* Nonlinear Analysis, Theory, Methods & Applications *
** 30**, 1235-1244 (1997).
Equilibrium selection via travelling waves.
In: Werner Leinfellner and Eckehart Köhler (eds.):
Vienna Circle Institute Yearbook 5/1997 :
Game Theory, Experience, Rationality.
Foundations of Social Sciences, Economics and Ethics.
In Honor of John C. Harsanyi. Dordrecht-Boston-London: Kluwer.
pp. 245-259.

(with Karl
Sigmund )
**Evolutionary Games and Population Dynamics ** , Cambridge University Press 1998.
( contents)

(with Thomas Nagylaki and Pavol Brunovský)
Convergence of multilocus systems under weak epistasis or weak selection. * J. Math. Biology * ** 38** (1999) 103-133.
(with
Gerhard Sorger)
Perfect foresight and equilibrium selection in symmetric
potential games. * J. Economic Theory * ** 85**, 1-23
(1999).
The spatially dominant equilibrium of a game.
* Annals of Operations Research * ** 89 **
(1999) 233-251.
Leopold Gegenbauer (1849-1903) zum 150. Geburtstag.
* Jahresbericht des Piaristengymnasiums Krems an der Donau * 1999, pp.
39-41.

(with Joseph Wai-Hung
So)
Diagonal dominance and harmless off-diagonal delays.
*Proc. Amer. Math. Soc.* ** 128** (2000), 2675-2682.
(with
Karl Schlag):
Sophisticated imitation in cyclic games.
* J. Evolutionary Economics *** 10** (2000), 523-543.
From Nash and Brown to Maynard Smith: Equilibria, dynamics
and ESS.
* Selection * ** 1 **
(2000), 81-88.

(with Ross
Cressman and Garay József):
Evolutionary stability concepts for N-species
frequency-dependent interactions.
* J. theor. Biol.* ** 211** (2001) 1-10.
(with P. L. Simon):
An existence
theorem for parabolic equations on $\R^n$ with discontinuous nonlinearity.
* Electronic Journal of
Qualitative Theory of Differential Equations * (2001) No. 8, pp. 1-9.

Competitive exclusion of disjoint hypercycles.
* Z. Phys. Chemie * ** 216 ** (2002) 35-39.
A simple proof of 1 + 1/2^{2} + 1/3^{2} +
... = PI^{2}/6 and related identities.
* American Math. Monthly * ** 109 ** (Feb 2002) 196-200.
(with Gerhard Sorger): A differential game approach to evolutionary equilibrium selection. * Int. Game Theory Review * ** 4 ** (March 2002) 17-31.
(with Christoph
Hauert , Silvia DeMonte, Karl Sigmund ):
Volunteering as Red Queen Mechanism for Cooperation in Public Goods Game.
* Science * **296** (May 2002) 1129-1132.

(with Christoph
Hauert , Silvia DeMonte, Karl Sigmund ):
Replicator dynamics for optional public good games.
* J. Theor. Biol. * ** 218** (Sept 2002), 187-194.
(with Bill Sandholm )
On the global convergence of stochastic fictitious play.
* Econometrica * ** 70 ** (Nov 2002) 2265-2294.

(with Barnabas Garay):
Robust permanence for ecological differential equations, minimax, and
discretizations. * SIAM J. Math. Anal.* ** 34 ** (2003) 1007-1039.
(with K. Sigmund): Evolutionary Game Dynamics.
* Bull. Amer. Math. Soc.* ** 40 ** (2003), 479-519.
(with Y. Saito and R. Kon): Qualitative V-L stability and qualitative permanence for Lotka-Volterra equations.
Functional equations in mathematical models (in Japanese).
Surikaisekikenkyusho Kokyuroku 1309 (2003), 140--145.
%MR1998272
(with R. Kon): Structure of *n*-dimensional Lotka-Volterra systems for qualitative permanence.
Functional equations in mathematical models (in Japanese).
Surikaisekikenkyusho Kokyuroku 1309 (2003), 146--153.
%MR1998273

(with Franz Hofbauer, P. Raith, T. Steinberger):
Intermingled basins in a two species system. * J. Math. Biology * ** 49 **(Sept 2004), 293-309.
SpringerLink
(with S.J. Schreiber): To persist or not to persist. * Nonlinearity * ** 17** (2004), 1393-1406.

(with Ross Cressman): Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics. *Theor. Pop. Biol.* **67** (Feb 2005), 47-59.
(with Ed Hopkins)
Learning in perturbed asymmetric games. *Games and Economic Behavior * **52** (July 2005), 133-152.
(with M. Benaim and S. Sorin):
Stochastic approximation and differential inclusions.
* SIAM J. Control and Optimization* ** 44 (1) ** (August 2005), 328–348.

(with Sylvain Sorin ):
Best response dynamics for continuous zero-sum games. * Discrete and Continuous Dynamical Systems* ** B6 ** (Jan 2006), 215-224.
(with Ross Cressman and Frank Riedel):
Stability of the replicator equation for a single-species with a multi-dimensional continuous trait space. * J. Theor. Biol. * ** 239 ** (March 2006), 273-288.
(with Ulrich Berger):
Irrational behavior in the Brown-von Neumann-Nash dynamics. *Games and Economic Behavior * ** 56** (July 2006), 1-6.
(with M. Benaim and S. Sorin):
Stochastic approximation and differential inclusions II: Applications. * Math. Operations Research * ** 31 ** (Nov 2006), 649-672.
(with St. Müller, L. Endler, Ch. Flamm, P. Schuster)
A generalized model of the repressilator.
* J. Math. Biology* ** 53** (Dec 2006), 905-937.

(with W.H. Sandholm )
Evolution in games with randomly perturbed payoffs .
*J. Economic Theory*** 132** (Jan 2007), 47–69.
To persist or not to persist: Differential equations in ecology.
In: Trends in Dynamical Systems, F. Dumortier, D. Roose, A. Vanderbauwhede (eds.), Koninklijke Vlaamse Academie van Belgie voor Wetenschappen en Kunsten (pp. 19-32). 2007.
(with Simon Huttegger):
Selection-Mutation Dynamics of Signaling Games With Two Signals. ,
In: Proceedings of the
ESSLLI 2007 Workshop on Language, Games, and Evolution , pp. 25-32.
(with W. H. Sandholm): Stable Games.
Proceedings of the 46th IEEE Conference on Decision and Control (New Orleans, LA, USA, Dec. 12-14, 2007), pp. 3416-3421.

(with M. Benaim and W. H. Sandholm):
Robust permanence and impermanence for stochastic replicator dynamics. * J. Biological Dynamics* 2 (April 2008), 180 - 195.
(with D. Oyama, S. Takahashi):
Monotone methods for equilibrium selection under perfect foresight dynamics.
* Theor. Economics* 3 (June 2008), 155 - 192.
(with Simon Huttegger):
Feasibility of communication in binary signaling games. * J. Theor. Biol. * 254 (Oct 2008), 843-849.
(with R. Kon and Y. Saito):
Qualitative permanence of Lotka-Volterra equations.
* J. Math. Biology* 57 (Dec 2008), 863-881.

(with Joerg Oechssler and Frank Riedel):
Brown-von Neumann-Nash dynamics: The continuous strategy case. *Games and Econ. Behav. *** 65** (March 2009), 406-429.
(with Sylvain Sorin and Yannick Viossat):
Time average replicator and best reply dynamics.
* Math. Operations Research *** 34** (May 2009) 263-269.
(with W.H. Sandholm )
Stable games and their dynamics. * J. Economic Theory *** 144** (July 2009) 1665-1693.
(with M. Benaim and Ed Hopkins):
Learning in games with unstable equilibria. *J. Economic Theory *** 144** (July 2009) 1694-1709.
(with Lorens Imhof ):
Time averages, recurrence and transience in the stochastic replicator dynamics. *Annals of Applied Probability* ** 19** (Aug 2009), 1347-1368.

(with S. J. Schreiber):
Robust permanence for structured populations. *J. Diff. Equs. ***248** (April 2010), 1955-1971.

(with Daisuke Oyama and Satoru Takahashi):
Perfect foresight dynamics in binary supermodular games
International Journal of Economic Theory **7** (Sep 2011) 251-267.
(with W.H. Sandholm )
Survival of Dominated Strategies under Evolutionary Dynamics, * Theor. Economics* **6** (2011) 341-377.
Deterministic evolutionary game dynamics. In:
Evolutionary Game Dynamics. Edited by: Karl Sigmund.
Proceedings of Symposia in Applied Mathematics,
vol. 69, Amer. Math. Soc., Providence, RI, 2011, pp. 61-79.

(with M. Benaim and S. Sorin): Perturbations of set-valued dynamical systems, with applications to game theory. * Dynamic Games and Applications* **2** (2012) 195-205.

(with D. Balkenborg and C. Kuzmics): Refined best-response correspondence and dynamics.
* Theor. Economics* **8** (2013) 165-192.

(with Simon Huttegger): Selection-mutation dynamics in signaling games.
* Games* **6** (2015), 2-31.
(with D. Balkenborg and C. Kuzmics):
The refined best-response correspondence in normal form games.
* International Journal of Game Theory* **44** (Feb 2015), 165-193.
(with Boyu Zhang):
Equilibrium selection via replicator dynamics in 2x2 coordination games.
*International Journal of Game Theory* **44** (May 2015), 433-448.
(with Linlin Su ):
Global stability in diallelic migration-selection models. *J. Math. Anal. Appl. * **428** (2015), 677-695.
(with Stefan Müller):
Genetic recombination as a chemical reaction network.
Mathematical Modelling of Natural Phenomena (special issue on *Dynamics of Chemical Reaction Networks*)
**10** (5) (2015), 84-99. Link

(with Linlin Su ):
Global stability of spatially homogeneous equilibria in migration-selection models.
SIAM J Appl Math **76** (March 2016) 578-597.
(with Boyu Zhang):
Quantal response methods for equilibrium selection in 2 x 2 coordination games .
Games Econ. Behav. ** 97 ** (May 2016) 19-31.

(with Peter Bednarik):
Discretized Best-Response Dynamics for the Rock-Paper-Scissors Game.
J. Dynamics and Games **4** (Jan 2017) 75-86.
(with D. Balkenborg and C. Kuzmics):
The refined best-response correspondence and backward induction.
German Economic Review (2017)
(with Balázs Boros, Stefan Müller):
On global stability of the Lotka reactions with generalized mass-action kinetics. Acta Appl Math 151 (Oct 2017) 53-80

(with Martin Pontz and Reinhard Bürger):
Evolutionary dynamics in the two-locus two-allele model with weak selection.
J. Math. Biol. 76 (Jan 2018) 151-203.
(with Balázs Boros, Stefan Müller, Georg Regensburger):
The center problem for the Lotka reactions with generalized mass-action kinetics.
Qualitative Theory of Dynamical Systems, 17(2), 403-410 (2018).
( arXiv,
reprint)
Minmax via replicator dynamics.
Dynamic Games and Applications
(Sept 2018) 8 (3), 637–640.

(with Balázs Boros, Stefan Müller, Georg Regensburger):
Planar S-systems: Stability and the center problem. ( arXiv)
Discrete and Continuous Dynamical Systems-A, to appear
(mit Balázs Boros):
Planar S-systems: Permanence. ( arXiv)
J. Diff. Equations, to appear.

### Preprints

Copies of these papers are available upon request. Please send an email to
`Josef.Hofbauer univie.ac.at`

- Stability for the best response dynamics. Preprint Vienna 1994.
Revised Version Budapest 1995.
- Imitation dynamics for games. Preprint 1995.
- (with Jeroen Swinkels) A Universal Shapley Example. 1995
- (mit Ulrich Berger): The Nash dynamics. 1996.