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Extinction of oscillating populations

Established populations often exhibit oscillations in their sizes. If a population is isolated, intrinsic stochasticity of elemental processes can ultimately bring it to extinction. Here we study extinction of oscillating populations in a stochastic version of the Rosenzweig-MacArthur predator-prey model. To this end we extend a WKB approximation (after Wentzel, Kramers and Brillouin) of solving the master equation to the case of extinction from a limit cycle in the space of population sizes. We evaluate the extinction rates and find the most probable paths to extinction by applying Floquet theory to the dynamics of an effective WKB Hamiltonian. We show that the entropic barriers to extinction change in a non-analytic way as the system passes through the Hopf bifurcation. We also study the subleading pre-exponential factors of the WKB approximation.