Routing functions and ecological models

Biological systems can undergo sudden transitions when parameters change, making stability analysis of steady states essential. We develop an algebraic framework for studying the parameter spaces of ecological ODE models with polynomial or rational rates. Using tools from computational algebra, we identify boundaries separating stability regimes, and using routing functions we compute the regions in their complement together with the stability behavior in each region. We demonstrate the method on the classical Levins–Culver competition–colonization model and a modern tripartite symbiosis model, where it uncovers rich and previously hidden stability structure.