The Bautin bifurcation or GH point separates the super-critical and sub-critical Hopf bifurcation in the parameter plane. On one side of this point, the system has two limit cycles, and on the other, there exists a unique limit cycle. At this point, the first Lyapunov coefficient is zero. What is the biological/ecological interpretation of this point?
A bifurcation is a change in the qualitative behavior of a dynamical system as a parameter is varied. A Hopf bifurcation is a type of bifurcation that occurs when a fixed point of a system loses stability and a limit cycle emerges near it. A limit cycle is a closed trajectory in the phase space that represents a periodic oscillation of the system.
A Bautin bifurcation or generalized Hopf (GH) bifurcation is a codimension-2 bifurcation that occurs when a Hopf bifurcation changes its type from supercritical to subcritical or vice versa. In a supercritical Hopf bifurcation, the limit cycle is stable and grows outwards from the fixed point as the parameter increases. In a subcritical Hopf bifurcation, the limit cycle is unstable and shrinks towards the fixed point as the parameter increases.
From an ecological perspective, a Bautin bifurcation can represent a transition between different types of oscillatory behavior in a population or an ecosystem. For example, ¹ studied a forest-grassland ecosystem model with human-environment interactions and found that Bautin bifurcations occur in the model as the level of human influence changes. They showed that for low levels of human influence, the system exhibits supercritical Hopf bifurcations and stable oscillations between forest and grassland states. For high levels of human influence, the system exhibits subcritical Hopf bifurcations and unstable oscillations that may lead to extinction of one or both states.
Another example is ² who investigated a prey-predator model with prey's fear cost and variable predator search speed. They showed that Bautin bifurcations occur in the model as the search speed changes. They showed that for low search speed, the system exhibits supercritical Hopf bifurcations and stable oscillations between high and low prey-predator densities. For high search speed, the system exhibits subcritical Hopf bifurcations and unstable oscillations that may lead to extinction of one or both species.
(1) Bautin bifurcations in a forest-grassland ecosystem with human ... - Nature. https://www.nature.com/articles/s41598-019-39296-x.
(2) Hopf bifurcation and Bautin bifurcation in a prey–predator model with .... https://www.sciencedirect.com/science/article/pii/S0378475422000398.
(3) On the complex dynamics of savanna landscapes | PNAS. https://www.pnas.org/doi/10.1073/pnas.1712356115.
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