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.

View Article: PubMed Central - PubMed

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

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With very disparate growth rates, essentially only the fast-growing strategy changes its abundance until carrying capacity is reached. At carrying capacity, the difference in growth rates becomes ineffective, such that the structure of the payoff matrix, A, determines the dynamics. (a) If all payoffs are the same, , then the dotted line  is a continuum of equilibria. Thus, starting with an initial population composition, x1 remains more or less constant and x2 adjusts such that carrying capacity is reached—given that x1 is not too low and x2 not too high, initially. (b) If the complete symmetry in the payoffs is broken,  and , all trajectories move to a slow manifold (bold line) close to  relatively quickly. Trajectories are nearly horizontal since x1 grows much faster than x2. At this manifold, they slowly converge to the global attractor (0,10), because the payoff configuration favors strategy 2.
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f0015: With very disparate growth rates, essentially only the fast-growing strategy changes its abundance until carrying capacity is reached. At carrying capacity, the difference in growth rates becomes ineffective, such that the structure of the payoff matrix, A, determines the dynamics. (a) If all payoffs are the same, , then the dotted line is a continuum of equilibria. Thus, starting with an initial population composition, x1 remains more or less constant and x2 adjusts such that carrying capacity is reached—given that x1 is not too low and x2 not too high, initially. (b) If the complete symmetry in the payoffs is broken, and , all trajectories move to a slow manifold (bold line) close to relatively quickly. Trajectories are nearly horizontal since x1 grows much faster than x2. At this manifold, they slowly converge to the global attractor (0,10), because the payoff configuration favors strategy 2.

Mentions: As an example, consider a game with uniform payoffs, . Then, every point on is an equilibrium, no matter how growth rates are chosen. For a=10, , and , the corresponding phase portrait is depicted in Fig. 3a. It shows that even with very disparate growth rates the stability properties of the pure equilibria cannot be changed in the degenerate case . Thus, the effect of a large discrepancy in growth rates is neutral with respect to equilibria, but leads to nearly horizontal trajectories in strategy density space, such that effectively only the fast-growing strategy changes its abundance when the dynamics converges to a continuum of equilibria. However, the slightest change in payoffs breaks the symmetry, such that the curve of equilibria collapses and the equilibrium with the higher payoff is approached, see Fig. 3b. Trajectories move towards a slow manifold in a short initial phase, during which strategy 2 hardly changes in abundance. When population size is saturated, the difference in growth rates becomes effective, such that strategy 1 is able to out-compete strategy 2 (compare the concepts of r- and K-selection, MacArthur and Wilson, 1967). Thus, even a highly increased growth rate cannot make up for a slightly worse payoff in the long run.


Density games.

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

With very disparate growth rates, essentially only the fast-growing strategy changes its abundance until carrying capacity is reached. At carrying capacity, the difference in growth rates becomes ineffective, such that the structure of the payoff matrix, A, determines the dynamics. (a) If all payoffs are the same, , then the dotted line  is a continuum of equilibria. Thus, starting with an initial population composition, x1 remains more or less constant and x2 adjusts such that carrying capacity is reached—given that x1 is not too low and x2 not too high, initially. (b) If the complete symmetry in the payoffs is broken,  and , all trajectories move to a slow manifold (bold line) close to  relatively quickly. Trajectories are nearly horizontal since x1 grows much faster than x2. At this manifold, they slowly converge to the global attractor (0,10), because the payoff configuration favors strategy 2.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3753514&req=5

f0015: With very disparate growth rates, essentially only the fast-growing strategy changes its abundance until carrying capacity is reached. At carrying capacity, the difference in growth rates becomes ineffective, such that the structure of the payoff matrix, A, determines the dynamics. (a) If all payoffs are the same, , then the dotted line is a continuum of equilibria. Thus, starting with an initial population composition, x1 remains more or less constant and x2 adjusts such that carrying capacity is reached—given that x1 is not too low and x2 not too high, initially. (b) If the complete symmetry in the payoffs is broken, and , all trajectories move to a slow manifold (bold line) close to relatively quickly. Trajectories are nearly horizontal since x1 grows much faster than x2. At this manifold, they slowly converge to the global attractor (0,10), because the payoff configuration favors strategy 2.
Mentions: As an example, consider a game with uniform payoffs, . Then, every point on is an equilibrium, no matter how growth rates are chosen. For a=10, , and , the corresponding phase portrait is depicted in Fig. 3a. It shows that even with very disparate growth rates the stability properties of the pure equilibria cannot be changed in the degenerate case . Thus, the effect of a large discrepancy in growth rates is neutral with respect to equilibria, but leads to nearly horizontal trajectories in strategy density space, such that effectively only the fast-growing strategy changes its abundance when the dynamics converges to a continuum of equilibria. However, the slightest change in payoffs breaks the symmetry, such that the curve of equilibria collapses and the equilibrium with the higher payoff is approached, see Fig. 3b. Trajectories move towards a slow manifold in a short initial phase, during which strategy 2 hardly changes in abundance. When population size is saturated, the difference in growth rates becomes effective, such that strategy 1 is able to out-compete strategy 2 (compare the concepts of r- and K-selection, MacArthur and Wilson, 1967). Thus, even a highly increased growth rate cannot make up for a slightly worse payoff in the long run.

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