The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations


  • Volker Grimm,

  • Christian Wissel

V. Grimm and C. Wissel, UFZ Umweltforschungszentrum Leipzig-Halle, Dept Ökologische Systemanalyse, Postfach 500 136, DE-04301 Leipzig, Germany (


Analysing the persistence and viability of small populations is a key issue in extinction theory and population viability analysis. However, there is still no consensus on how to quantify persistence and viability. We present an approach to evaluate any simulation model concerned with extinction. The approach is devised from general Markov models of stochastic population dynamics. From these models, we distil insights into the general mathematical structure of the risk of extinction by time t, P0(t). From this mathematical structure, we devise a simple but effective protocol – the ln(1−P0)-plot – which is applicable for situations including environmental noise or catastrophes. This plot delivers two quantities which are fundamental to the assessment of persistence and viability: the intrinsic mean time to extinction, Tm, and the probability c1 of the population reaching the established phase. The established phase is characterized by typical fluctuations of the population's state variable which can be described by quasi-stationary probability distributions. The risk of extinction in the established phase is constant and given by 1/Tm. We show that Tm is the basic currency for the assessment of persistence and viability because Tm is independent of initial conditions and allows the risk of extinction to be calculated for any time horizon. For situations where initial conditions are important, additionally c1 has to be considered.