Classification of parameter spaces for a reaction-diffusion model on stationary domains

This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary rectangular domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories of the phase-plane around the uniform steady state. In the presence of diffusion, the main findings are the quantitative analysis relating the domain-size with the reaction and diffusion rates and their corresponding influence on the dynamics of the reaction-diffusion system when perturbed in the neighbourhood of the uniform steady state. Theoretical predictions are supported by numerical simulations both in the presence as well as in the absence of diffusion. Conditions on the domain size with respect to the diffusion and reaction rates are related to the types of diffusion-driven instabilities namely Turing, Hopf and Transcritical types of bifurcations. The first condition is a lower bound on the area of a rectangular domain in terms of the diffusion and reaction rates, which is necessary for Hopf and Transcritical bifurcation to occur. The second condition is an upper bound on the area of domain in terms of reaction-diffusion rates that restricts the diffusion-driven instability to Turing type behaviour, whilst forbidding the existence of Hopf and Transcritical bifurcation. Theoretical findings are verified by the finite element solution of the coupled system on a two dimensional rectangular domain

Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains

Reaction-diffusion systems have been widely studied in developmental biology, chemistry and more recently in financial mathematics. Most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions. It therefore becomes convenient to look for numerical solutions: finite difference, finite element, finite volume and spectral methods are typical examples of the numerical methods used. Most of these methods are locally based schemes. We examine the implications of mesh structure on numerically computed solutions of a well-studied reaction-diffusion model system on two-dimensional fixed and growing domains. The incorporation of domain growth creates an additional parameter – the grid-point velocity – and this greatly influences the selection of certain symmetric solutions for the ADI finite difference scheme when a uniform square mesh structure is used. Domain growth coupled with grid-point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure. We compare our results to those obtained by use of finite elements on unstructured triangular elements

Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution

The internationalisation of the war on terrorism and making of a modern threat to the ethic of political liberalism : a conceptualisation of the current threat to global peace and security.

Doctor of Philosophy in Ethics. University of KwaZulu-Natal, Pietermaritzburg, 2018.The terrorist attacks in the United States of America (USA) on 11 September 2001 unquestionably caused anguish for the nation. Instead of seeking justice, the USA went on a retribution mission which led it to lose self-control as the terrorists lured it to behave like a rogue state. The stage was thus set for a cycle of violence between the protagonists, one represented by the self-centred USA, and the other by militant modern terrorists who do not value life, to lock horns in the international arena. This thesis demonstrates that the USA’s desire for vengeance led to the internationalisation of the war on terrorism, whose actions have, on numerous occasions, constituted an affront to the ethic of political liberalism which, being centred on liberty and the respect of the individual, demand justice and fairness, equality, tolerance, respect for the rule of law, and various individual rights such as freedom of conscience and non-discrimination. While there is no agreed upon definition of terrorism, this study showed that terrorism is an illegal form of warfare that thrives on the use of violence and intimidation which is targeted mainly at civilians to achieve political objectives. This study demonstrated that the USA has taken advantage of the illegality of terrorism to persuade and coerce other nations to join it in the War on Terror which it has used, to a great extent, to pursue its strategic interests all over the world. This study shows how, in pursuit of its foreign policy objectives, the USA has adopted a rapacious foreign policy that disregards international law and multilateral institutions. The superpower has not hesitated to use force where it has felt that its interests are under threat. It has lost morality as it embraces various tyrants around the world while punishing those despots who are not on its side in the War on Terror. While exercising its right to hunt down terrorists and bring them to justice, it has failed to differentiate combatants from non-combatants. The extensive abuse of suspects in secret detention camps by its security forces, which has been characterised by a gross violation of individual rights, constitutes an insult to the just war principle of jus in bello. In the war, the USA has failed to strike a balance between national security and the requirement for the respect of individual rights. This study demonstrates how it has supplanted the rule of law by the ‘rule of men’ as Arabs and other minority groups have been profiled and detained arbitrarily as public officials have denied them their freedom of conscience and the right to equality. Liberal provisions which give suspects the right to legal representation have been unfairly and unjustly dispensed with as the criminal justice system has been replaced by military tribunals. This study shows how the government, which has exhibited a lack of tolerance for minority groups, has denied individuals their liberty as it has moved them illegally from one country to another where they have been subjected to torture. This study concludes that the USA’s disrespect for individual rights and national sovereignty has made the War on Terror unjust, given its association with lawlessness, immorality and impunity. The USA’s actions confirm the thesis that the War on Terror constitutes a threat to the ethic of political liberalism and is indeed a threat to global peace and security

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth

Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics

Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell

The aim of this article is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. Actin is a polymer that can exist either in fil- amentous form (F-actin) or in monometric form (G-actin) (Chen et al. 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrish- nan et al. 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its bio-chemical dy- namics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al. 2003). Furthermore, by assuming slow deforma- tions of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the math- ematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contrac- tion, cell translation and cell relocation as well as cell protrusions. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter