First Passage and Machine Learning perspectives of RPS/RPSSL systems

Abstract

The evolution of RPS/RPSSL can be studied from multiple perspectives. Once the rate reactions pertaining to birth, death, predation, and other activities including migration and mutation are drawn for the system, the simulation of the system can be done through multiple stochastic techniques like the stochastic lattice simulation or the Gillespie simulation. In this thesis, we study system evolution using two techniques. In a small-scale system with mutation, we study the extinction of a system under the lens of First-Passage formulation. We formulate a First-Passage Problem to derive the exact analytical solutions for extinction state and extinction time. We then verify the agreement between our First-passage solutions, Gillespie solutions, and mean-field theory. The first-passage solutions provide us a deeper look towards species extinction compared to stochastic simulations. Our findings suggest first-extinction time and state distribution in a system with mutation follows intriguing behaviour which promotes coexistence. There also exists a depression in the state space post which mutation extends the first-extinction time. Moreover, a system devoid of mutation exhibits a discernible inclination towards probabilities that lean in the direction of an endangered state space. However, the formulation of a first-passage problem is computationally very expensive and stochastic simulations become redundant for multiple initial conditions with same system parameters. Hence, we move towards a more complex problem of understanding and forecasting a system’s evolution through Machine Learning. To provide the system with tractable information, we move to lattice-based simulation where we draw actions during each timestep using the Gillespie method. We also tackle a more intricate problem of migration, through which the spatio-temporal visualizations create spiral patterns in a system. Using Machine Learning we expedite the tasks of extinction prediction and system evolution by generating lattices using machine learning - which were computationally expensive using simulations and first-passage problems