Species can survive in both scenarios, but it is important to understand how they differ and which is best suited to an ecological system (prey-predator). Where both cases fit with an ecological subsystem, what is the biological importance of the Hopf bifurcation?
Thanks!
For population models, we observe two types of Hopf-bifurcation, namely sub-critical and super-critical. Importantly, we have to keep in mind that in nature we rarely observe the transition from steady-state coexistence to oscillatory coexistence through Hopf-bifurcation. The oscillatory coexistence of several species can be explained as a result of super-critical Hopf-bifurcation and the observed dynamics is the behaviour of the model under consideration for a particular set of parameter values. On the other hand, once we find the ecological scenario that the stable steady-state coexistence depends upon the initial population density - such kind of situation can be explained through the result of sub-critical Hopf bifurcation. In this situation, the unstable limit cycle surrounding the stable equilibrium point marks the basin of attraction of the coexistence steady-state.
Unstable equilibrium with stable limit cycle means (biologically) that the populations oscillate with a period determined by the cycle. On the other hand, an stable equilibrium point, means that the populations coexist at the density determined by the equilibrium point. That is, oscillation in the first case and coexistence in the second one.
an unstable equilibrium point with a stable limit cycle means populations of all species repeats within a particular range as time increases. On the other side, a stable equilibrium means population size of the species is very closed to the actual population size (stable equilibrium).
I think the stable limit cycle has an unstable equilibrium point (unstable spiral) as the trajectories go away from the equilibrium to the stable limit cycle's direction (from the inside).
The stable equilibrium point, means that the population coexists at the density determined by the equilibrium point. While it means unstable equilibrium with a (biologically) constant boundary cycle, it is the approach of population behavior to an unknown density around a stable point.
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