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Nonnormality and Its Influence on the Stability and Behavior of Ecological Food Webs
 Vogel, Kaela
 Advisor(s): Hastings, Alan
Abstract
The historic approach to food web research in theoretical ecology is frequently computational with the focus on figuring out what sort food web attributes (e.g. topological structure, network weights, etc.) confer stability to food webs. Much of the theory derived from the computational approach for food web research relies on the linear stability of systems of ordinary differential equations; where the stability of the resulting Jacobian matrices that encapsulate all the species interaction structure and weights is used to determine whether we expect to see a community of that type (or not). This approach almost always depends on using the eigenvalues of the matrix to judge stability, which say something about the eventual asymptotic decay of a perturbation to that system. Asymptotic metrics, such as the rightmost eigenvalue, are a suboptimal metric for stability to use for several reasons. First, natural systems are in a constant state of experiencing disturbances. Second, the nonequilibrium dynamics during the transient phase of a system after a disturbance can be strikingly different from the asymptotic dynamics and may take a surprisingly long time to decay. Third, field observations frequently happen on much shorter timescales than the system dynamics, contributing to a mismatch between empirical observations and theoretical predictions.
This research aims to understand how food web structure and the network weights influence the transient finitetime behavior of food webs using a mixture of the old computation methods with new techniques yet to be fully explored in ecology. I use the generalized LotkaVolterra equations to parameterize eight common food web module structures and create a large data set of feasible systems based on random draws of the of original parameters (on the order of thousands to millions depending on the structure). Since the old methods focus on linearizations of nonlinear systems, I will focus on one source of odd transient dynamics that afflicts linear systems: Nonnormality of the Jacobian matrix leading to sensitivity to perturbations of its entries and transient amplification of perturbations to the equilibrium. One powerful technique used for nonnormal systems is pseudospectra, which uses the norm of the resolvent to understand the finitetime dynamical behavior of nonnormal systems. I will introduce nonnormality, pseudospectra and its relationship to old methods in ecology to recognize systems with transient growth in Chapter 2, such as reactivity and the Kreiss constant.
One weakness of both of using pseudospectra and the numerical abscissa (otherwise known as reactivity in ecological literature) is that the worst case behavior predicted by these metrics is dependent on the particular structure of the perturbation. To explore how perturbation structure may contribute transient growth in natural systems, I look at two common types of equilibrium perturbations. Resource pulse perturbations, where just the basal species is perturbed, and removal pulse perturbations where I put the additional condition on the vector sign structure that all entries must be negative. I found that resource pulse perturbations are unlikely to be amplified, but simultaneous removal of both the top predator and its prey is far more likely to cause transient amplification in model food webs than what one would predict given the probability of randomly drawing a vector with all negative entries.
I hypothesize that sensitivity to perturbations is common is due to unavoidable asymmetry in the predator prey interactions due to assimilation efficiency or predatorprey body size ratios. I test this by comparing a metric of nonnormality, Henrici's departure from normality to the ratio between predator and prey interactions (coupling symmetry), and find that there is a cutoff coupling symmetry to determine whether a system is forced to become reactive and this is strongly dependent on the number of trophic levels in the module. Reactivity is strongly linearly correlated with nonnormality for our simulated food webs, and this seems to be due in part to the most nonnormal systems also having eigenvalues near zero. Finally, to understand how long transient in response to perturbations of the equilibrium may relate to sensitivity to changes in the underlying model parameters, I also calculate a pseudospectral metric, real distance to instability, for a subset of our data. We find that the eigenvalue may also be an unreliable metric for how close a system is to mathematical instability as well as how a system responds to equilibrium perturbations.
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