Sharks, Mullet, and Mathematical Ecology Using ODEs to model predator-prey relationships Chandler Shaeffer

Lotka – Volterra Equations Imagine you have a closed system with a predator population and a prey population With the logistic map equation, you have a set number of rabbits for each year. However, this isn’t realistic! Predator and prey populations influence each other – how do you model this? Lotka – Volterra Equations dx / dt = αx – βxy dy / dt = -γy + δxy

Let’s say we have 100 lemon sharks and 400 mullet. dx / dt = αx – βxy dy / dt = -γy + δxy Where x represents the number of mullet y represents the number of sharks α represents the growth rate of the mullet population β represents the rate of predation upon the mullet γ represents the rate of loss of the predator population δ represents the growth rate of the predator population Use fourth-order Runge-Kutta method to solve ODE system (adapted code from labs and homework)

All models are wrong, but some are useful Lower γ Lower α Higher β

Refine model by introducing competition Model intraspecific competition w/ new parameter c*x^2 Model interspecific competition by adding new ODE