This project investigates the interplay of nonlinear dynamics and random abiotic fluctuations on distribution and abundance of interacting populations. To investigate this interplay, the theory of random dynamical systems which combines dynamical systems techniques with probabilistic machinery are applied to a class of random ecological maps sufficiently general to account for spatial structure, stage structure, and multispecies interactions. New mathematical methods for verifying stochastic dissipativeness, persistence, and extinction will be developed. The application of these methods to populations dispersing in random environments, the storage effect and the Allee effect in stage-structured populations, random replicator dynamics, and antagonistic interactions of ideal free populations in source-sink environments will be pursued. For models of structured populations, new methods will be developed for understanding monotonicity and convexity properties of the dominant Lyapunov exponent for parameterized families of random products of non-negative matrices. Since these dominant Lyapunov exponents determine the asymptotic growth rate of populations at low densities, they are critical for understanding persistence.
The interaction between environmental fluctuations, for example those due to anthropogenic disturbances or weather, and biological processes can dramatically affect the outcomes of species interactions. Consequently, the applications of the mathematical techniques developed in this project may be of practical value to the conservation or the restoration of ecological communities, the prevention of biological invasions, the management of natural resources, and biological control of agricultural pests.
