Extinction times and moment closure in the stochastic logistic process

• Published in: Theoretical population biology Vol. 65; no. 2; pp. 115 - 126
• Main Authors: Newman, T.J., Ferdy, Jean-Baptiste, Quince, C.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.03.2004
• Subjects: Animals; Biological Evolution; Conservation; Demographic stochasticity; Environment; Fluctuations; Logistic Models; Models, Biological; Moment closure approximation; Mortality; Population Density; Population Dynamics; Quasi-stationary distribution; Stochastic Processes; Time Factors; Time to extinction; Demographic stochasticity; Quasi-stationary distribution; Fluctuations; Time to extinction; Moment closure approximation; Conservation
• ISSN: 0040-5809; 1096-0325
• DOI: 10.1016/j.tpb.2003.10.003

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that V max is of order K 1/4/ M 1/2, which is much larger than unity so long as M is small compared to K 1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.