We consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin... Show moreWe consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin coating. This coating admits regularly spaced gaps at the so-called nodes of Ranvier. Since the signal travels much faster through the coated regions, it appears to hop between the nodes of Ranvier. However, many mathematical models that describe this propagation do not take into account the discrete structure directly.More recently, a discrete version of the famous FitzHugh-Nagumo model has been proposed to capture this discrete behaviour. In this thesis, we consider several extensions to and generalisations of this discrete FitzHugh-Nagumo model. In particular, we study infinite-range interactions, periodic behaviour and spatial-temporal discretization. Our general aim is to establish the existence and, sometimes, non-linear stability of travelling wave solutions. Our main tools in this analysis are the spectral convergence method and exponential dichotomies. In addition, we extend some general mathematical theory to systems with infinite-range interactions. Show less
Jaïbi O., Doelman A., Chirilus-Bruckner M., Meron E. 2020
In this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad... Show moreIn this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad et al., 2004). The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction–diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that ‘jump’ between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction–diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings. Show less
Drug-discovery has become a complex disci- pline in which the amount of knowledge about human biology, physiology, and biochemistry have increased. In order to harness this complex body of... Show moreDrug-discovery has become a complex disci- pline in which the amount of knowledge about human biology, physiology, and biochemistry have increased. In order to harness this complex body of knowledge mathe- matics can play a critical role, and has actually already been doing so. We demonstrate through four case studies, taken from previously published data and analyses, what we can gain from mathematical/analytical techniques when nonlinear concentration-time courses have to be trans- formed into their equilibrium concentration-response (tar- get or complex) relationships and new structures of drug potency have to be deciphered; when pattern recognition needs to be carried out for an unconventional response- time dataset; when what-if? predictions beyond the obser- vational concentration-time range need to be made; or when the behaviour of a semi-mechanistic model needs to be elucidated or challenged. These four examples are typical situations when standard approaches known to the general community of pharmacokineticists prove to be inadequate. Show less
The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which... Show moreThe presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria. Show less
Benson, N.; Graaf, P.H. van der; Peletier, L.A. 2016
A recently developed system-pharmacological model for the dynamics of receptor tyrosine kinases is used to compare di erent targets for drug action: one aiming at binding endogenous ligand needed... Show moreA recently developed system-pharmacological model for the dynamics of receptor tyrosine kinases is used to compare di erent targets for drug action: one aiming at binding endogenous ligand needed for phosphorylation and another binding re- ceptors at their kinase domains and thus preventing them to generate phosphory- lation. We obtain quantitative estimates for the e ectivity of the inhibitor which demonstrate the influence of drug-properties such as dose, a nity to target and drug-elimination rates. Show less
In this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in... Show moreIn this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in very general types of systems emerge when geometrical techniques are applied. Show less
