UK Nonlinear News, February 1998

# Recent Theses

## Fixed Point Indices and Existence Theorems for Semilinear Equations in Cones

### Department of Mathematics, Glasgow University

#### Supervisor: Professor J. Webb

The purpose of this thesis is to develop fixed point indices for A-proper semilinear operators defined on cones in Banach spaces and use the results to obtain existence theorems to semilinear equations. Semilinear equations of the form Lx=Nx are considered defined on cones, where L,N act between two Banach spaces, L is a linear Fredholm operator of index zero, N a nonlinear operator such that L-N is A-proper.

Source: Professor J.R.L. Webb jrlw@maths.gla.ac.uk).

## A Rational Dynamical-Systems Approach to Plankton Population Modelling

### Department of Applied Mathematical Studies, University of Leeds

#### Supervisor: Professor John Brindley

Abstract

Understanding the dynamics of plankton populations is of major importance since plankton form the basis of marine food webs throughout the world's oceans and play a significant role in the global carbon cycle. In this thesis we examine the dynamical behaviour of plankton models, exploring sensitivities to the number of variables explicitly modelled, to the functional forms used to describe interactions, and to the parameter values chosen. The practical difficulties involved in data collection lead to uncertainties in each of these aspects of model formulation.

The first model we investigate consists of three coupled ordinary differential equations, which measure changes in the concentrations of nutrient, phytoplankton and zooplankton. Nutrient fuels the growth of the phytoplankton, which are in turn grazed by the zooplankton. The recycling of excretion adds feedback loops to the system. In contrast to a previous hypothesis, the three variables can undergo oscillations when a quadratic function for zooplankton mortality is used. The oscillations arise from Hopf bifurcations, which we track numerically as parameters are varied. The resulting bifurcation diagrams show that the oscillations persist over a wide region of parameter space, and illustrate to which parameters such behaviour is most sensitive. The oscillations have a period of about one month, in agreement with some observational data and with output of larger seven-component models. The model also exhibits fold bifurcations, three-way transcritical bifurcations and Bogdanov-Takens bifurcations, resulting in homoclinic connections and hysteresis.

Under different ecological assumptions, zooplankton mortality is expressed by a linear function, rather than the quadratic one. Using the linear function does not greatly affect the nature of the Hopf bifurcations and oscillations, although it does eliminate the homoclinicity and hysteresis. We re-examine the influential paper by Steele and Henderson (1992), in which they considered the linear and quadratic mortality functions. We correct an anomalous normalisation, and then use our bifurcation diagrams to interpret their findings.

A fourth variable, explicitly modelling detritus (non-living organic matter), is then added to our original system, giving four coupled ordinary differential equations. The dynamics of the new model are remarkably similar to those of the original model, as demonstrated by the persistence of the oscillations and the similarity of the bifurcation diagrams. A second four-component model is constructed, for which zooplankton can graze on detritus in addition to phytoplankton. The oscillatory behaviour is retained, but with a longer period. Finally, seasonal forcing is introduced to all of the models, demonstrating how our dynamical systems approach aids understanding of model behaviour and can assist with model formulation.

Source: Andy Edwards ( andy@whoi.edu).

## Bifurcations and instabilities in rotating two-layer fluids

### Wolfson College, University of Oxford

#### Supervisor: Dr Peter L. Read

Abstract

This thesis describes a combined theoretical and experimental investigation of baroclinic instability in a rotating two-layer fluid. The first part of this thesis investigates baroclinic instability in an idealised theoretical model of a two-layer fluid, while the second part examines baroclinic instability in an experimental analogue of the two-layer model.

In the theoretical investigation, a set of spectral amplitude equations describing a two-layer fluid in a rectangular channel are examined for bifurcations as certain model parameters are varied. The bifurcation parameters investigated are: the dissipation parameter, r; the rotational Froude number, F; and $\beta$, a measure of the planetary vorticity gradient. Previous studies have yielded a number of multiple-scales'' approximations describing weakly-nonlinear instability in a two-layer channel flow. These multiple-scales approximations place certain restrictions on the size of r, $\beta$ and F and so are only valid in limited areas of the [$r,\beta,F$] parameter space. This thesis uses numerical continuation methods to show that the spectral amplitude equations, which are valid over a larger region of the [$r,\beta,F$] space, reproduce the behaviour observed in the areas of the parameter space in which the various multiple-scales approximations are valid. Furthermore, the manner in which various multiple-scales approximations, describing adjacent regions of the [$r,\beta,F$] parameter space, merge together is explained in terms of a single degenerate bifurcation on the inviscid axis. In addition, the effect of $\beta$ on the behaviour of a two-layer model is examined in more detail than has been possible previously. It is shown that the presence of $\beta$ alters the symmetry of the spectral amplitude equations, and thereby fundamentally alters the observed bifurcation structure, causing the $\beta$-plane spectral amplitude equations to give rise to circle-map dynamics and frequency-locking. Finally, the discrepancies between purely inviscid instability and instability in the inviscid limit of viscous theory, reported in previous studies, are examined. It is shown that instability in purely inviscid theory relies on a bifurcation mechanism found only in Hamiltonian systems, whereas instability in the inviscid limit of viscous theory is caused by a Hopf bifurcation. As a result the two theories cannot be compared directly.

In the experimental part of this thesis, a novel method of visualising flow in a rotating two-layer annulus is implemented. A new method of measuring two-layer instability is then developed and tested. These two complimentary techniques are used to examine the experimental [r,F] plane, and the principal results are summarised and compared with the results of the theoretical investigation. The steady wave flows and the flows undergoing amplitude vacillation that have been recorded in previous two-layer experiments, are observed in this study. It is shown, however, that these baroclinic waves do not travel at the speed of the mean-flow as has been suggested previously. New results concerning the generation of small-scale gravity waves by large-scale baroclinic disturbances are then presented. Observations indicate that there is no interaction between these two types of waves. This result has relevance to numerical weather forecasting in which filtered'' sets of equations, from which the gravity-waves have been removed'', are used. Observations of some complicated flows, characterised by competition between two unstable waves and by resonant interactions between three waves, which have not previously been reported in connection with two-layer experiments, are then given. These flows are similar to those that have been observed in thermal annulus experiments, indicating that they are probably a generic feature of baroclinic instability.