Appl Math Optim 56:211–241 (2007)
2007 Springer Science+Business Media, Inc.
Nash Equilibria in Noncooperative Predator–Prey Games
Angel Manuel Ramos
and Tom´aˇs Roub´ıˇcek
Departamento de Matem´atica Aplicada, Universidad Complutense de Madrid,
Plaza de Ciencias 3, 28040, Madrid, Spain
Mathematical Institute, Charles University,
Sokolovsk´a 83, CZ-186 75 Praha 8, Czech Republic
Institute of Information Theory and Automation, Academy of Sciences,
Pod vod´arenskou vˇeˇz´ı 4, CZ-182 08 Praha 8, Czech Republic
Communicated by I. Lasiecka
Abstract. A noncooperative game governed by a distributed-parameter predator–
prey system is considered, assuming that two players control initial conditions for
predator and prey, respectively. Existence of a Nash equilibrium is shown under
the condition that the desired population proﬁles and the environmental carrying
capacity for the prey are sufﬁciently small. A conceptual approximation algorithm
is proposed and analyzed. Finally, numerical simulations are performed, too.
Key Words. Reaction–diffusion system, Lotka–Volterra system, Nash equilib-
rium, Second-order analysis, Existence, Numerical simulations.
AMS Classiﬁcation. 49N70, 91A10, 35Q80.
Mathematical models in ecological systems represent a long-time scrutinized area. In
a distributed-parameter version, they are essentially reaction–diffusion systems with
A.M. Ramos was supported by the “Plan Nacional de I+D+I (2000-2003)” of the MCyT (Spain),
through the AGL2003-06862-C02-02 project, and by the “Direcci´on General de Universidades e Investigaci´on
de la Consejer´ıa de Educaci´on de la Comunidad de Madrid y de la Universidad Complutense de Madrid”,
through the Ref. 910480 project. T. Roub´ıˇcek was partly supported by the Grants 201/03/0934 (GA
MSM 0021620839 (M