Search results

title=Stable focus ''Q'' - multiple stable and unstable limit cycles (Bautin bifurcation)| ...\) as the second parameter. Increasing \(N\) turns the super-critical Hopf bifurcation into a subcritical one. ...

...surrounded by a stable limit cycle - the hallmark of a super-critical Hopf bifurcation. Cooperators and defectors co-exist in never ending periodic oscillations. ...ation of the replicator equation. The Hopf-bifurcation occurs at \(r_\text{Hopf} = 2.7672\).}} ...

title=Stable focus ''Q'' - unstable limit cycle (sub-critical Hopf bifurcation)| doc=For non-zero birthrates simple sub-critical Hopf bifurcations are observed. The fixed point \(Q\) is stable and surrounded b ...

doc=Increasing \(r\) further leads to a Hopf bifurcation, the interior fixed point \(Q\) becomes a stable focus and the limit cycle ...

...nstable and in (d), (e) it is stable. Between scenarios (c) and (d) a Hopf bifurcation occurs and over a very narrow range of \(r\), stable and unstable limit cyc ...able and unstable limit cycles that collide and disappear in a saddle-node bifurcation of periodic orbits. ...

...px|Stable limit cycles are observed in the vicinity of a sub-critical Hopf-bifurcation, which gives rise to ever lasting periodic oscillations of cooperators and ...Y. & Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, ''Theor. Pop. Biol.'' '''73''', 257-263 [http://dx.doi.org/10.1016/j.tpb.2 ...

...sing from a Hopf-bifurcation and spatial instability arising from a Turing-bifurcation. Moreover, a detailed analysis of the highly irregular dynamics through pow ...mes and Population Dynamics|Ecological public goods games: Cooperation and bifurcation]]=== ...