Mathematics Colloquia and Seminars

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It now well known that both the network weights and the topological structure of links are important factors in structuring ecological food webs. Due to biological constraints, such as bioenergetics and body size, the weights between predator and prey of these bidigraph networks may be quite asymmetric. This asymmetry has important implications for the eigenvalue-stability of the Jacobian matrices representing food webs; due to the nonnormal nature of the eigenbasis, these systems may both be highly sensitive to parameter uncertainty and have long transient growth in response to perturbations that defies the conclusion one would get from looking at the eigenvalues. This talk will cover the counterintuitive behavior of nonnormal matrices and introduce pseudospectra, a method that uses the norm of the resolvent to better characterize the behavior of nonnormal operators. Then, using different ecological network structures parameterized by the generalized Lotka-Volterra equations, I explore how structure and interaction asymmetry may contribute to nonnormality and the consequences of this nonnormality on then tendency for the dynamics to amplify perturbations to the equilibrium. I also clarify the relationship between sensitivity to the underlying parameters (perturbations to the Jacobian) and sensitivity to perturbations of the equilibrium resulting in transient growth. Finally, one weakness of the pseudospectral approach is that it tells you something about sensitivity to perturbations but does not specify what structure of the perturbations bring about the "bad behavior," so I study how perturbations which only remove individuals of a species may result in odd transient dynamics and the consequences of this for managing multi-species fisheries and reserves.