Limits...
Density games.

Novak S, Chatterjee K, Nowak MA - J. Theor. Biol. (2013)

Bottom Line: Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation.Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks.For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.

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Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. sebastian.novak@ist.ac.at

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Eigenvalues of the internal equilibrium for a=0.8, b=10, c=1, d=9, , and . The real parts of the eigenvalues are depicted by the solid curves, their imaginary parts by the dashed curves. Eigenvalues turn complex at , the Hopf bifurcation occurs at .
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f0025: Eigenvalues of the internal equilibrium for a=0.8, b=10, c=1, d=9, , and . The real parts of the eigenvalues are depicted by the solid curves, their imaginary parts by the dashed curves. Eigenvalues turn complex at , the Hopf bifurcation occurs at .

Mentions: An example of an attracting limit cycle is illustrated in Fig. 4. Fig. 5 shows the real parts (solid) and imaginary parts (dashed) of the eigenvalues along a path . First, they collide on the negative real axis and become complex, thereby transforming the internal equilibrium into an oscillatory attractor. Then, they cross the imaginary axis, turning the fixed point into a repellor and creating a limit cycle. This example also shows that indeed both scenarios, D1 and D2, are feasible: For instance, with a=0.8, b=10, c=1, d=9, and , the internal equilibrium is asymptotically stable, whereas with , it is repelling (see Fig. 4).


Density games.

Novak S, Chatterjee K, Nowak MA - J. Theor. Biol. (2013)

Eigenvalues of the internal equilibrium for a=0.8, b=10, c=1, d=9, , and . The real parts of the eigenvalues are depicted by the solid curves, their imaginary parts by the dashed curves. Eigenvalues turn complex at , the Hopf bifurcation occurs at .
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3753514&req=5

f0025: Eigenvalues of the internal equilibrium for a=0.8, b=10, c=1, d=9, , and . The real parts of the eigenvalues are depicted by the solid curves, their imaginary parts by the dashed curves. Eigenvalues turn complex at , the Hopf bifurcation occurs at .
Mentions: An example of an attracting limit cycle is illustrated in Fig. 4. Fig. 5 shows the real parts (solid) and imaginary parts (dashed) of the eigenvalues along a path . First, they collide on the negative real axis and become complex, thereby transforming the internal equilibrium into an oscillatory attractor. Then, they cross the imaginary axis, turning the fixed point into a repellor and creating a limit cycle. This example also shows that indeed both scenarios, D1 and D2, are feasible: For instance, with a=0.8, b=10, c=1, d=9, and , the internal equilibrium is asymptotically stable, whereas with , it is repelling (see Fig. 4).

Bottom Line: Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation.Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks.For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.

View Article: PubMed Central - PubMed

Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. sebastian.novak@ist.ac.at

Show MeSH
Related in: MedlinePlus