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Generalized Lotka-Volterra equations on graphs

Generalized Lotka-Volterra equations on graphs

We investigate the stability of generalized Lotka-Volterra equations in network topologies, such as trees and complete graphs. In particular, we have proved results on the stability of solutions where all species in the community are nonzero, namely all species persist. Our analytical findings are corroborated by numerical simulations and supplement published studies. We give a short proof of the result that tree networks with amensalistic, commensalistic and antagonistic interactions are stable regardless of the interaction strength, while tree networks with amensalistic, commensalistic, mutualistic and competitive interactions can be made unstable by choosing any of the interaction strengths large enough. We also present findings on the types of networks and interactions that are characterized by the largest real and imaginary parts of the eigenvalues of their corresponding Jacobian matrices. This is joint work with Lee DeVille and Shinhae Park.

## Department of Mathematics and Statistics