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A central problem in population ecology is understanding the consequences ofstochastic fluctuations. Analytically tractable models with Gaussian drivingnoise have led to important, general insights, but they fail to capture rare,catastrophic events, which are increasingly observed at scales ranging fromglobal fisheries to intestinal microbiota. Due to mathematical challenges,growth processes with random catastrophes are less well characterized and itremains unclear how their consequences differ from those of Gaussian processes.In the face of a changing climate and predicted increases in ecologicalcatastrophes, as well as increased interest in harnessing microbes fortherapeutics, these processes have never been more relevant. To betterunderstand them, I revisit here a differential equation model of logisticgrowth coupled to density-independent catastrophes that arrive as a Poissonprocess, and derive new analytic results that reveal its statistical structure.First, I derive exact expressions for the model's stationary moments, revealinga single effective catastrophe parameter that largely controls low orderstatistics. Then, I use weak convergence theorems to construct its Gaussiananalog in a limit of frequent, small catastrophes, keeping the stationarypopulation mean constant for normalization. Numerically computing statisticsalong this limit shows how they transform as the dynamics shifts fromcatastrophes to diffusions, enabling quantitative comparisons. For example, themean time to extinction increases monotonically by orders of magnitude,demonstrating significantly higher extinction risk under catastrophes thanunder diffusions. Together, these results provide insight into a wide range ofstochastic dynamical systems important for ecology and conservation.

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