Some nonlinear problems in plankton ecology

Abstract

In marine ecology, the variability of the physical environment is often considered a main determinant of biological pattern. A common approach to identifying key environmental forcings is to match scales of variability: fluctuations of a biological variable at a particular frequency are attributed to forcing by the physical environment at a similar frequency. In nonlinear systems, however, different scales of variability interact and forcing at one frequency can produce variability at a different frequency. The general theme of this dissertation regards the interplay of scales in nonlinear ecological systems, with an emphasis on the mismatch of scales between biological variables and environmental forcings in the plankton. The approach is theoretical: I use simple models to identify conditions leading to such a mismatch. The models are motivated by planktonic systems and focus on one ubiquitous nonlinear ecological interaction, that between a consumer and a resource. This work is organized in three main parts as follows. In the first part, I consider the interaction between a phytoplankton population and a limiting nutrient resource. Most models for this interaction consider all cells as equal and group them under a single variable, the total biomass or cell density. They do not take into account any population heterogeneity resulting from the life histories of individual cells. However, single cells do have life histories: each cell progresses through a determinate sequence of events preceding cell division and the population is distributed in stages of the cell cycle. I incorporate this distribution (i.e. population structure) , as well as observations on resource control of cell cycle progression, into chemostat models for the phytoplankton-nutrient interaction. Simulation results demonstrate that the· population structure can generate oscillatory dynamics under a constant nutrient supply, and that such oscillations are important to population dynamics under a variable nutrient supply. Specifically, for a periodic resource supply, the population displays an aperiodic response with frequencies different from that of the forcing. I then show that a chemostat model without population structure (the Droop equations) does not exhibit this transfer of variability: a periodic nutrient supply produces a periodic population response of exactly the same frequency. In the second part, I consider a predator and a prey that interact and diffuse along an environmental gradient. The model is a reaction-diffusion equation, a type of model used in biological oceanography for planktonic interactions in turbulent flows. I demonstrate that weak diffusion along a spatial gradient may drive an otherwise periodic system into complex temporal dynamics that include chaotic behavior. I provide evidence for a quasiperiodic route to chaos as the diffusion rate decreases. Then, I focus on the spatial properties of the gradient and their consequences for the spatia-temporal dynamics of the system. In particular, I ask: how do the spatial patterns of the populations compare to the underlying environmental gradient in the different dynamic regimes (periodicity, quasiperiodicity, and chaos)? I show that the spatial patterns of predator and prey can differ strongly from the environmental gradient. In the route to chaos, as diffusion becomes weaker, this difference is magnified and the populations display smaller spatial scales. In the work summarized so far, nonlinearity leads to variability in biological variables at scales not present in the environmental forcings. In the third part of this work, I consider another consequence of the transfer of variability in nonlinear systems: the lack of a dominant scale. Patterns that lack a dominant scale but exhibit scale similarity are known as fractals. The characterization of numerical quantities that vary intermittently has motivated a generalization of fractals known as multifractals. Here, I give a first application of multifractals to biological oceanography. I analyze an acoustic data set on zooplankton biomass to describe the distribution in time of the total variability in the data. This distribution is highly intermittent: extreme localized contributions account for a large proportion of total variability. I show that multifractals provide a good characterization of such variability.