Leiden University Student Repository

Busse, Julius (2024)

Extinction properties of population models with non locally Lipschitz noise coefficients

Master thesis | Mathematics (MSc)

This thesis explores the extinction properties of population models with non locally Lipschitz noise coefficients. We examine stochastic differential equations (SDEs) of the form dy(t) = f (y(t), t) dt + y(t)p dW (t), where the drift is positive around zero. The aim is to understand the extinction dynamics. We identify a critical threshold at p = 12 , in the sense that extinction occurs with a nonzero probability for p < 12 and extinction does not occur for p > 12 . After a logarithmic transformation of the equation, our analysis uses Feller’s test for explosions and combines it with two novel comparison theorems. We also provide numerical bounds for the extinction probabilities. Furthermore, we extend the analysis to stochastic delay differential equations (SDDEs) and show that their extinction properties can be related to those of non-delayed models. Finally, the Rosenzweig–MacArthur model serves as a proof of concept for applicability of our results to a multi-dimensional setting.

Stochastic Differential Equations, Stochastic Delay Differential Equations

Mathematical Biology

Applied Mathematics

Population Modelling

Author

Busse, Julius

Faculty

Faculty of Science

Specialisation

Applied Mathematics

Supervisors

Gaans, O. van

ECTS Credits

40

Language

en