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Gustav W. Delius

Publications and source records attributed to Gustav W. Delius.

At least 19 recordsLinked to original sources
arXiv2017

Plankton: the paradox and the power law

Two basic features of assemblages of unicellular plankton: (1) their high biodiversity and (2) the power-law structure of their abundance, can be explained by an allometric scaling of cell growth and mortality with respect to cell size. To show this, we describe a numerical study of a size-structured, multispecies, population-dynamic model; the model has a single resource, supporting an arbitrary number of phytoplankton and zooplankton species. If the number of plankton species is large enough, the death rate of prey and cell growth rate of predators have approximate allometric scalings with cell size. Together, these scalings give rise to an equilibrium distribution of abundance near the power law, on which many species can coexist. Scalings of this kind cannot be achieved if the number of species is small. This suggests that the conjunction of species-richness and power-law structures in plankton communities is more than a coincidence. Although the exact allometric scalings used here should not be expected in practice, exclusion of species should be relatively slow if they lie close to the power law. Thus the forces needed to achieve coexistence could be effective, even if they are relatively weak.

q-bio.PE↗
arXiv2016

Sheldon Spectrum and the Plankton Paradox: Two Sides of the Same Coin. A trait-based plankton size-spectrum model

The Sheldon spectrum describes a remarkable regularity in aquatic ecosystems: the biomass density as a function of logarithmic body mass is approximately constant over many orders of magnitude. While size-spectrum models have explained this phenomenon for assemblages of multicellular organisms, this paper introduces a species-resolved size-spectrum model to explain the phenomenon in unicellular plankton. A Sheldon spectrum spanning the cell-size range of unicellular plankton necessarily consists of a large number of coexisting species covering a wide range of characteristic sizes. The coexistence of many phytoplankton species feeding on a small number of resources is known as the Paradox of the Plankton. Our model resolves the paradox by showing that coexistence is facilitated by the allometric scaling of four physiological rates. Two of the allometries have empirical support, the remaining two emerge from predator-prey interactions exactly when the abundances follow a Sheldon spectrum. Our plankton model is a scale-invariant trait-based size-spectrum model: it describes the abundance of phyto- and zooplankton cells as a function of both size and species trait (the maximal size before cell division). It incorporates growth due to resource consumption and predation on smaller cells, death due to predation, and a flexible cell division process. We give analytic solutions at steady state for both the within-species size distributions and the relative abundances across species.

q-bio.PE↗
arXiv2012

Predation effects on mean time to extinction under demographic stochasticity

Methods for predicting the probability and timing of a species' extinction are typically based on a combination of theoretical models and empirical data, and focus on single species population dynamics. Of course, species also interact with each other, forming more or less complex networks of interactions. Models to assess extinction risk often lack explicit incorporation of these interspecific interactions. We study a birth and death process in which the death rate includes an effect from predation. This predation rate is included via a general nonlinear expression for the functional response of predation to prey density. We investigate the effects of the foraging parameters (e.g. attack rate and handling time) on the mean time to extinction. Mean time to extinction varies by orders of magnitude when we alter the foraging parameters, even when we exclude the effects of these parameters on the equilibrium population size. In particular we observe an exponential dependence of the mean time to extinction on handling time. These findings clearly show that accounting for the nature of interspecific interactions is likely to be critically important when estimating extinction risk.

q-bio.PE↗
arXiv2010

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a "jump-growth" equation, a first order approximation which is the widely used McKendrick-von Foerster equation, and a second order approximation which is the McKendrick-von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick-von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick-von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator : prey mass ratio, a large diet breadth and a high feeding efficiency.

q-bio.PE↗
arXiv2010

A scale-invariant model of marine population dynamics

A striking feature of the marine ecosystem is the regularity in its size spectrum: the abundance of organisms as a function of their weight approximately follows a power law over almost ten orders of magnitude. We interpret this as evidence that the population dynamics in the ocean is approximately scale-invariant. We use this invariance in the construction and solution of a size-structured dynamical population model. Starting from a Markov model encoding the basic processes of predation, reproduction, maintenance respiration and intrinsic mortality, we derive a partial integro-differential equation describing the dependence of abundance on weight and time. Our model represents an extension of the jump-growth model and hence also of earlier models based on the McKendrick--von Foerster equation. The model is scale-invariant provided the rate functions of the stochastic processes have certain scaling properties. We determine the steady-state power law solution, whose exponent is determined by the relative scaling between the rates of the density-dependent processes (predation) and the rates of the density-independent processes (reproduction, maintenance, mortality). We study the stability of the steady-state against small perturbations and find that inclusion of maintenance respiration and reproduction in the model has astrong stabilising effect. Furthermore, the steady state is unstable against a change in the overall population density unless the reproduction rate exceeds a certain threshold.

q-bio.PE↗
arXiv2010

A jump-growth model for predator-prey dynamics: derivation and application to marine ecosystems

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker-Planck equation for the stochastic fluctuations. The McKendrick--von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation and the McKendrick--von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.

q-bio.PE↗
arXiv2002

Quantum affine reflection algebras of type d_n^(1) and reflection matrices

Quantum affine reflection algebras are coideal subalgebras of quantum affine algebras that lead to trigonometric reflection matrices (solutions of the boundary Yang-Baxter equation). In this paper we use the quantum affine reflection algebras of type d_n^(1) to determine new n-parameter families of non-diagonal reflection matrices. These matrices describe the reflection of vector solitons off the boundary in d_n^(1) affine Toda field theory. They can also be used to construct new integrable vertex models and quantum spin chains with open boundary conditions.

math.QA↗
arXiv1999

Particle Reflection Amplitudes in a_n^(1) Toda Field Theories

We determine the exact quantum particle reflection amplitudes for all known vacua of a_n^(1) affine Toda theories on the half-line with integrable boundary conditions. (Real non-singular vacuum solutions are known for about half of all the classically integrable boundary conditions.) To be able to do this we use the fact that the particles can be identified with the analytically continued breather solutions, and that the real vacuum solutions are obtained by analytically continuing stationary soliton solutions. We thus obtain the particle reflection amplitudes from the corresponding breather reflection amplitudes. These in turn we calculate by bootstrapping from soliton reflection matrices which we obtained as solutions of the boundary Yang-Baxter equation (reflection equation). We study the pole structure of the particle reflection amplitudes and uncover an unexpectedly rich spectrum of excited boundary states, created by particles binding to the boundary. For a_2^(1) and a_4^(1) Toda theories we calculate the reflection amplitudes for particle reflection off all these excited boundary states. We are able to explain all physical strip poles in these reflection factors either in terms of boundary bound states or a generalisation of the Coleman-Thun mechanism.

hep-th↗
arXiv1998

Soliton-preserving boundary condition in affine Toda field theories

We give a new integrable boundary condition in affine Toda theory which is soliton-preserving in the sense that a soliton hitting the boundary is reflected as a soliton. All previously known integrable boundary conditions forced a soliton to be converted into an antisoliton upon reflection. We prove integrability of our boundary condition using a generalization of Sklyanin's formalism.

hep-th↗
arXiv1998

Restricting affine Toda theory to the half-line

We restrict affine Toda field theory to the half-line by imposing certain boundary conditions at $x=0$. The resulting theory possesses the same spectrum of solitons and breathers as affine Toda theory on the whole line. The classical solutions describing the reflection of these particles off the boundary are obtained from those on the whole line by a kind of method of mirror images. Depending on the boundary condition chosen, the mirror must be placed either at, in front, or behind the boundary. We observe that incoming solitons are converted into outgoing antisolitons during reflection. Neumann boundary conditions allow additional solutions which are interpreted as boundary excitations (boundary breathers). For $a_n^{(1)}$ and $c_n^{(1)}$ Toda theories, on which we concentrate mostly, the boundary conditions which we study are among the integrable boundary conditions classified by Corrigan et.al. As applications of our work we study the vacuum solutions of real coupling Toda theory on the half-line and we perform semiclassical calculations which support recent conjectures for the $a_2^{(1)}$ soliton reflection matrices by Gandenberger.

hep-th↗
arXiv1997

The structure of quantum Lie algebras for the classical series B_l, C_l and D_l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint ---> adjoint. We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B_l, C_l and D_l. In the quantum case also the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

q-alg↗
arXiv1996

Quantum Lie algebras; their existence, uniqueness and $q$-antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras. On them the quantum Lie bracket is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g_h independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra g_h. In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same g are isomorphic, 2) the quantum Lie bracket of any quantum Lie algebra is $q$-antisymmetric. We also describe a construction of quantum Lie algebras which establishes their existence.

q-alg↗
arXiv1996

The Problem of Differential Calculus on Quantum Groups

The bicovariant differential calculi on quantum groups of Woronowicz have the drawback that their dimensions do not agree with that of the corresponding classical calculus. In this paper we discuss the first-order differential calculus which arises from a simple quantum Lie algebra. This calculus has the correct dimension and is shown to be bicovariant and complete. But it does not satisfy the Leibniz rule. For sl_n this approach leads to a differential calculus which satisfies a simple generalization of the Leibniz rule.

q-alg↗
arXiv1996

Introduction to Quantum Lie Algebras

Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in $h$. They are derived from the quantized enveloping algebras $\uqg$. The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. In this paper the recent general results about quantum Lie algebras are introduced with the help of the explicit example of $(sl_2)_h$.

q-alg↗
arXiv1996

On Quantum Lie Algebras and Quantum Root Systems

As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum Lie bracket given by the quantum adjoint action. The structure constants of these algebras depend on the quantum deformation parameter $q$ and they go over into the usual Lie algebras when $q=1$. The notions of q-conjugation and q-linearity are introduced. q-linear analogues of the classical antipode and Cartan involution are defined and a generalised Killing form, q-linear in the first entry and linear in the second, is obtained. These structures allow the derivation of symmetries between the structure constants of quantum Lie algebras. The explicitly worked out examples of $g=sl_3$ and $so_5$ illustrate the results.

q-alg↗
arXiv1995

Quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$

Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on $\qlie{g}$ is induced by the quantum adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the quantum root systems, which are now functions of the quantum parameter $q$. They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.

q-alg↗
arXiv1995

Exact S-Matrices with Affine Quantum Group Symmetry

We show how to construct the exact factorized S-matrices of 1+1 dimensional quantum field theories whose symmetry charges generate a quantum affine algebra. Quantum affine Toda theories are examples of such theories. We take into account that the Lorentz spins of the symmetry charges determine the gradation of the quantum affine algebras. This gives the S-matrices a non-rigid pole structure. It depends on a kind of ``quantum'' dual Coxeter number which will therefore also determine the quantum mass ratios in these theories. As an example we explicitly construct S-matrices with $U_q(c_n^{(1)})$ symmetry.

hep-th↗