Feasibility conditions of ecological systems: Unfolding the links between model parameters

Mohammad AlAdwani, Civil and Environmental Engineering, MIT, Cambridge, MA and Serguei Saavedra, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA

Background/Question/Methods
Over more than 100 years, ecological research has been striving to derive internal and external conditions compatible with the coexistence of a given group of interacting species. Many efforts have been centered on developing phenomenological and mechanistic models to represent the dynamics of ecological systems and predict their behavior. To address this challenge, numerous studies have focused on the necessary conditions for species coexistence, namely feasibility. Feasibility corresponds to the existence of at least one equilibrium point whose components are all real and positive. Traditionally, feasibility conditions have been attained by finding the isocline equations and solving for the location of equilibrium roots before imposing positivity of at least one root. Due to mathematical limitations, currently there is no general methodology that can provide us with a full analytical understanding about feasibility (and coexistence conditions) for any given model. Even at the 2 species level, it is impossible to solve for the location of the roots of a generic model analytically if the isocline equations have five or more roots, making this traditional approach unsuitable for finding the necessary conditions for coexistence. Here, we propose a general formalism to obtain analytical conditions of feasibility (and non-feasibility) for any population dynamics model of any dimensions without the need to solve for the equilibrium locations.
Results/Conclusions
In general, our methodology can identify and separate feasibility conditions that guarantee exactly k feasible equilibrium points for any value k of a model system. We show that these feasibility conditions are always represented by polynomial inequalities in species abundances. We demonstrate that feasibility and non-feasibility conditions are represented by identical polynomial expressions, whose signs determine the number of feasible equilibrium points in the system. We show how to compact the derived feasibility conditions into the smallest minimal mathematical expressions to easily analyze them. We illustrate the power of our methodology by showing how it is possible to derive mathematical relationships between model parameters while maintaining feasibility in modified Lotka-Volterra models with functional responses and higher-order interactions (model systems with at least five equilibrium points)---a task that is impossible to do with simulations.