CAREER: Dynamics and harvesting of stochastic populations

Texas A&M University, College Station TX

Investigators

Abstract

Environmental fluctuations have been shown to drive populations extinct, facilitate persistence, reverse competitive exclusion, change genetic diversity, and modify the spread of infectious diseases. It is important to study the interplay between environmental fluctuations, both deterministic and random, and the persistence of interacting species. Developing a rigorous mathematical theory for coexistence, in conjunction with data-driven applications, will help theoretical ecologists pinpoint how harvesting and periodic or random environmental fluctuations affect the long term dynamics of ecological communities. The research project will provide much-needed theoretical underpinning for this fast-moving area. The application related to the harvesting of marine animals will be key for conservation and management of vulnerable or endangered species. Questions around optimal control of stochastic models are vital in today's world. Ecologists and evolutionary biologists invoke stochasticity as a key determinant of everything from population genetics to extinct. But the exposure that scientists from such disciplines actually get to the mathematical concepts underpinning stochastic processes is incomplete. An integral component of the educational objectives will be the organization of a summer school at the interface of biology and stochastics targeted to advanced undergraduate and graduate students from mathematics and biology. In order to have realistic models for the coexistence of species it is important to incorporate both periodic and random environmental fluctuations. Connecting ideas from dynamical systems and stochastic processes, it will be possible to show that the long-term dynamics is determined by the invasion rates (Lyapunov exponents) of the periodic measures living on the boundary of the state space. The developed ideas will then be used to look at non-stationary community theory where the long term behavior of the system can not be described by an equilibrium, an attractor, or a stationary distribution. An important question from conservation biology is how to harvest a given population in order to maximize the yield while not driving the population extinct. While there are a few results for single-species systems, little is known in the significantly more realistic setting of interacting species. By using a combination of novel approaches from stochastic control and Markov chain approximation methods one can analyze multi-species harvesting problems and then apply the results in order to gain insight for important real-life applications from fishery management. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.