Origin of Cooperators and Defectors: Difference between revisions

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[[Image:Cost & benefit functions in continuous Snowdrift Game.png|thumb|300px|Quadratic cost (dotted) and benefit (dashed) functions, constrained to the trait interval [0,1], together with the monomorphic population payoff (solid). The two vertical lines highlight that the evolutionary dynamics is quite unrelated to maximizing the population payoff (dash-dotted) as opposed to the branching point (dashed). Also note that there is no easy way to predict the evolutionary outcome simply by looking at the shape of the cost and benefit functions. This allows to rule out some scenarios but in general it requires a detailed analysis of the slopes and curvature of the two functions. The parameter values correspond to the branching scenario.]]
[[Image:Cost & benefit functions in continuous Snowdrift Game.png|thumb|300px|Quadratic cost (dotted) and benefit (dashed) functions, constrained to the trait interval [0,1], together with the monomorphic population payoff (solid). The two vertical lines highlight that the evolutionary dynamics is quite unrelated to maximizing the population payoff (dash-dotted) as opposed to the branching point (dashed). Also note that there is no easy way to predict the evolutionary outcome simply by looking at the shape of the cost and benefit functions. This allows to rule out some scenarios but in general it requires a detailed analysis of the slopes and curvature of the two functions. The parameter values correspond to the branching scenario.]]


Consider a monomorphic population with strategy x. The growth rate of a rare mutant y in the resident population x is determined by fx(y)=P(y,x)−P(x,x)= B(x+y)−C(y)−(B(2x)−C(x)). The evolution of trait x is then given by \(dx/dt = D(x) = df_x/dy |_{y=x} = B^\prime(2x)-C^\prime(x)\), i.e. \(D(x)^{\prime\prime} = 4 b_2 x+b_1-2 c_2 x-c_1\). If an equilibrium x∗ with D(x∗)=0 exists, it is called a convergent stable strategy if \(dD/dx|_{x=x^*} = 2 B^{\prime\prime}(2 x^*)-C^{\prime\prime}(x^*)<0 (2b_2-c_2 <0)\) and x∗ is a repellor otherwise. Interestingly, the CSS condition does not imply evolutionary stability of x∗. Evolutionary stability requires that x∗ is a maximum of fx, i.e. \(d^2 f_x/dy^2|_{y=x^*}<0 (b_2-c_2<0)\). If this does not hold x∗ is a branching point (\(2 b_2 < c_2 < b_2 < 0\)) because mutants on both sides of x∗ can invade.
Consider a monomorphic population with strategy x. The growth rate of a rare mutant y in the resident population x is determined by fx(y)=P(y,x)−P(x,x)= B(x+y)−C(y)−(B(2x)−C(x)). The evolution of trait x is then given by  
 
\[\frac{dx}{dt} = D(x) = \frac{df_x}{dy}\bigg|_{y=x} = B^\prime(2x)-C^\prime(x),\]
i.e. D(x)=4b2x+b1−2c2x−c1. If an equilibrium x∗ with D(x∗)=0 exists, it is called a convergent stable strategy (CSS) if  
\[\frac{dD}{dx}\bigg|_{x=x^*} = 2 B^{\prime\prime}(2 x^*)-C^{\prime\prime}(x^*)<0,\]
which reduces to \(2b_2-c_2 <0\) for quadratic cost- and benefit functions. Otherwise, x∗ is a repellor. Interestingly, the CSS condition does not imply evolutionary stability of x∗. Instead, evolutionary stability requires that x∗ is a maximum of fx, i.e.  
\[\frac{d^2 f_x}{dy^2}\bigg|_{y=x^*}<0\]
or \(b_2-c_2<0\). If this does not hold x∗ is a branching point\((2 b_2 < c_2 < b_2 < 0)\) because mutants on both sides of x∗ can invade, which then results in spontaneous diversification.
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