Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.
Phys Rev E. 2016 Mar;93(3):032109. doi: 10.1103/PhysRevE.93.032109. Epub 2016 Mar 7.
Established populations often exhibit oscillations in their sizes that, in the deterministic theory, correspond to a limit cycle in the space of population sizes. If a population is isolated, the 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 develop a WKB (Wentzel, Kramers and Brillouin) approximation to the master equation, employing the characteristic population size as the large parameter. Similar WKB theories have been developed previously in the context of population extinction from an attracting multipopulation fixed point. We evaluate the extinction rates and find the most probable paths to extinction from the limit cycle by applying Floquet theory to the dynamics of an effective four-dimensional WKB Hamiltonian. We show that the entropic barriers to extinction change in a nonanalytic way as the system passes through the Hopf bifurcation. We also study the subleading pre-exponential factors of the WKB approximation.
已建立的种群通常会表现出其规模的波动，在确定性理论中，这些波动对应于种群规模空间中的极限环。如果一个种群是孤立的，基本过程的内在随机性最终可能导致其灭绝。在这里，我们在 Rosenzweig-MacArthur 捕食者-被捕食者模型的随机版本中研究了振荡种群的灭绝。为此，我们针对主方程发展了 WKB（Wentzel、Kramers 和 Brillouin）近似，将特征种群规模作为大参数。在从吸引多种群固定点灭绝的背景下，以前已经开发出了类似的 WKB 理论。我们通过将 Floquet 理论应用于有效四维度 WKB 哈密顿量的动力学来评估灭绝率，并找到从极限环灭绝的最可能路径。我们表明，当系统通过 Hopf 分岔时，灭绝的熵障碍以非解析的方式发生变化。我们还研究了 WKB 近似的次幂前因子。
