Stationary moments, diffusion limits, and extinction times for logistic growth with random catastrophes

A variety of organisms, from arctic reindeer to gastrointestinal bacteria, face stochastic environmental fluctuations that result in population catastrophes. Understanding the statistics of growthcatastrophe processes is a particularly relevant problem at present, due for example to expected increases in environmental volatility as a consequence of climate change and to growing interest in applying ecological theory to the human microbiome. Progress has proven challenging, as the abrupt nature of real catastrophes fails to be captured by traditional Gaussian noise models and is better described by coupling growth to explicitly discontinuous stochastic processes. Statistical properties of these stochastic jump differential equations remain poorly characterized, in part due to a lack of analytically tractable approaches that enable generally applicable insights. To address this, I revisit a minimal model of logistic growth coupled to density-independent catastrophes and derive exact expressions for its stationary moments, neglecting the possibility of extinction. I then use these results to address two outstanding problems concerning random catastrophes and also to show how they are related: (1) the unexplained existence of an effective catastrophe parameter that largely controls low-order statistics, and (2) the quantitative comparison of extinction risks in random catastrophe models with their Gaussian noise counterparts. With regards to the latter, two distinct methods are proposed that both show significantly higher risk of extinction under random catastrophe dynamics over a wide range of parameters, a particularly important conclusion for species conservation efforts. Together, these findings enable succinct and predicative characterizations of an important class of stochastic dynamical systems.