Figure 1

Stability of equilibrium points: black dot represents a stable point, white dot represents an unstable point. (a) with \(\alpha <\alpha _1\), the \({\hat{x}}=1\) is unstable and \({\hat{y}}=1\) is asymptotically stable. (b) if \(\alpha _1<\alpha \le 1\), an unstable equilibrium \({\hat{x}}\) (denoted by \({\tilde{x}}\)) appears. The value of the equilibrium point is \({\hat{x}}={\tilde{x}}=(n-r)/r(n-1)\alpha\), changing the stability of \({\hat{x}}=1\) that becomes a stable equilibrium point. The possibility of obtaining full cooperation depends on the existence of the interior equilibrium point \({\tilde{x}}\). This equilibrium point appears when \(\alpha _1 < \alpha\), and the outcome: full cooperation or full defection will depend on the frequency of cooperators related to the equilibrium \({\tilde{x}}\) (see Fig.  1 b). For values of \(\alpha < \alpha _1\), the interior equilibrium point \({\tilde{x}}\) does not exists and consequently the outcome is full defection (see Fig.  1 a).