Recently the effect of a quiescent phase (or dormant/resting phase in applications) onthe dynamics of a system of differential equations has been investigated, in particular with respectto stability properties of stationary points. It has been shown that there is a general phenomenonof stabilization against oscillations which can be cast in rigorous form. Here we investigate, forhomogeneous systems, the effect of a quiescent phase, and more generally, a phase with slowerdynamics. We show that each exponential solution of the original system produces two exponentialsolutions of the extended system whereby the stability properties can be controlled.

Two main approaches have been considered for modelling the dynamics of the SIS model oncomplex metapopulations, i.e, networks of populations connected by migratory flows whoseconfigurations are described in terms of the connectivity distribution of nodes (patches)and the conditional probabilities of connections among classes of nodes sharing the samedegree. In the first approach migration and transmission/recovery process alternatesequentially, and, in the second one, both processes occur simultaneously. Here we followthe second approach and give a necessary and sufficient condition for the instability ofthe disease-free equilibrium in generic networks under the assumption of limited (orfrequency-dependent) transmission. Moreover, for uncorrelated networks and under theassumption of non-limited (or density-dependent) transmission, we give a bounding intervalfor the dominant eigenvalue of the Jacobian matrix of the model equations around thedisease-free equilibrium. Finally, for this latter case, we study numerically theprevalence of the infection across the metapopulation as a function of the patchconnectivity.

Blood rheology is completely determined by its major corpuscles which are erythrocytes,or red blood cells (RBCs). That is why understanding and correct mathematical descriptionof RBCs behavior in blood is a critical step in modelling the blood dynamics. Variousphenomena provided by RBCs such as aggregation, deformation, shear-induced diffusion andnon-uniform radial distribution affect the passage of blood through the vessels. Hence,they have to be taken into account while modelling the blood dynamics. Other importantblood corpuscles are platelets, which are crucial for blood clotting. RBCs strongly affectthe platelet transport in blood expelling them to the vessel walls and increasing theirdispersion, which has to be considered in models of clotting. In this article we give abrief review of basic modern approaches in mathematical description of these phenomena,discuss their applicability to real flow conditions and propose further pathways fordeveloping the theory of blood flow.

We study pressure-driven, two-layer flow in inclined channels with high density andviscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and theKarman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics.Two distinguished limits are considered: where the viscosity ratio is small with densityratios of order unity, and where both density and viscosity ratios are small. The evolution equationsaccount for the presence of inertia, gravity, capillarity and viscous retardation; attention isrestricted to situations in which the flow is laminar. The results of our linear stability analysis andnumerical simulations indicate that the flow is destabilised by positive channel inclination in thestably stratified case. The dependence of the nonlinear wave dynamics on system parameters isalso explored.

We present the fiber-spring elastic model of the arterial wall with atheroscleroticplaque composed of a lipid pool and a fibrous cap. This model allows us to reproducepressure to cross-sectional area relationship along the diseased vessel which is used inthe network model of global blood circulation. Atherosclerosis attacks a region ofsystemic arterial network. Our approach allows us to examine the impact of the diseasedregion onto global haemodynamics.

In this paper we give a survey on modeling efforts concerning the CVRS. The material wediscuss is organized in accordance with modeling goals and stresses control and transportissues. We also address basic modeling approaches and discuss some of the challenges formathematical modeling methodologies in the context of parameter estimation and modelvalidation.

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

Thrombus formation in flowing blood is a complex time- and space-dependent process ofcell adhesion and fibrin gel formation controlled by huge intricate networks ofbiochemical reactions. This combination of complex biochemistry, non-Newtonianhydrodynamics, and transport processes makes thrombosis difficult to understand. That iswhy numerous attempts to use mathematical modeling for this purpose were undertaken duringthe last decade. In particular, recent years witnessed something of a transition from the“systems biology” to the “systems pharmacology/systems medicine” stage: computationalmodeling is being increasingly applied to practical problems such as drug development,investigation of particular events underlying disease, analysis of the mechanism(s) ofdrug’s action, determining an optimal dosing protocols, etc. Here we review recentadvances and challenges in our understanding of thrombus formation.

The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to ageneral class of reaction-diffusion (RD) system with and without sub-diffusion isanalyzed. For both the case of regular and sub-diffusion, the method of matched asymptoticexpansions is used to derive an ODE characterizing the spike locations in the absence ofany 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fastinstabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem(NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibriumsolution. For a particular sub-class of the reaction kinetics, it is shown that thediscrete spectrum of this NLEP is determined by the roots of some simple transcendentalequations. From a rigorous analysis of these transcendental equations, explicit sufficientconditions are given to predict the occurrence of either Hopf bifurcations or competitioninstabilities of the two-spike quasi-equilibrium solution. The theory is illustrated forseveral specific choices of the reaction kinetics.

We present a dynamic geometric model of phyllotaxis based on two postulates, primordiaformation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugateare all variations of the same unifying phenomenon and that the difference lies in thechanges in position of initial primordia. We explore the set of all initial positions andcolor-code its points depending on the phyllotactic pattern that arises.

Many models in biology and ecology can be described by reaction-diffusionequations wit time delay. One of important solutions for these type of equationsis the traveling wave solution that shows the phenomenon of wave propagation.The existence of traveling wave fronts has been proved for large class ofequations, in particular, the monotone systems, such as the cooperativesystems and some competition systems. However, the problem on the uniqueness oftraveling wave (for a fixed wave speed) remains unsolved. In this paper, we showthat, for a class of monotone diffusion systems with time delayed reaction term,the mono-stable traveling wave font is unique whenever it exists.

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, wherethe prey population is structured according to a certain life history trait. The traitdistribution within the prey population is the result of interplay between geneticinheritance and mutation, as well as selectivity in the consumption of prey by thepredator. The evolutionary processes are considered to take place on the same time scaleas ecological dynamics, i.e. we consider the evolution to be rapid. Previously publishedresults show that population structuring and rapid evolution in such predator-prey systemcan stabilize an otherwise globally unstable dynamics even with an unlimited carryingcapacity of prey. However, those findings were only based on direct numerical simulationof equations and obtained for particular parameterizations of model functions, whichobviously calls into question the correctness and generality of the previous results. Themain objective of the current study is to treat the model analytically and considervarious parameterizations of predator selectivity and inheritance kernel. We investigatethe existence of a coexistence stationary state in the model and carry out stabilityanalysis of this state. We derive expressions for the Hopf bifurcation curve which can beused for constructing bifurcation diagrams in the parameter space without the need for adirect numerical simulation of the underlying integro-differential equations. Weanalytically show the possibility of stabilization of a globally unstable predator-preysystem with prey structuring. We prove that the coexistence stationary state is stablewhen the saturation in the predation term is low. Finally, for a class of kernelsdescribing genetic inheritance and mutation we show that stability of the predator-preyinteraction will require a selectivity of predation according to the life trait.

Recently, adult stem cells have become a focus of intensive biomedical research, but thecomplex regulation that allows a small population of stem cells to replenish depleted tissues is stillunknown. It has been suggested that specific tissue structures delimit the spaces where stem cellsundergo unlimited proliferation (stem cell niche). In contrast, mathematical analysis suggests thata feedback control of stem cells on their own proliferation and differentiation (denoted QuorumSensing) suffices for preserving homeostasis in developing tissues. In the present work we examinethis discrepancy by analyzing a mathematical model of signalling pathways in a mammary stemcell. Our work shows that the model has a unique equilibrium depending on the environmentalconditions, which is stable for appropriate parameter values. We show how this dependence onthe micro-environment can control the cell decision to either proliferate or differentiate. Our workfurther suggests that this description is sufficient for the creation of niche conditions, with no needof additional structures. The analysis also pinpoints the pathways that may serve as the QuorumSensing mechanism.

We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

We study the structure of the spectrum of the infinite XXZ quantum spin chain, ananisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves thenumber of down spins (or particle number), allowing to represent it as a direct sum ofN-particleinteracting discrete Schrödinger-type operators restricted to the fermionic subspace. Inthe Ising phase of the model we use this representation to give a detailed determinationof the band and gap structure of the spectrum at low energy. In particular, we show thatat sufficiently strong anisotropy the so-called droplet bands are separated from higherspectral bands uniformly in the particle number. Our presentation of all necessarybackground is self-contained and can serve as an introduction to the mathematical theoryof the Heisenberg and XXZ quantum spin chains.

We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutionsof delay equations apply directly to a resource-consumer model with age-structured consumerpopulation.

This contribution reviews the nonlinearstochastic properties of turbulent velocity and passive scalarintermittent fluctuations in Eulerian and Lagrangian turbulence.These properties are illustrated with original data sets of (i)velocity fluctuations collected in the field and in thelaboratory, and (ii) temperature, salinity and in vivofluorescence (a proxy of phytoplankton biomass, i.e. unicelledvegetals passively advected by turbulence) sampled from highlyturbulent coastal waters. The strength of three of the mostpopular models describing intermittent fluctuations (thelognormal, log-Lévy and log-Poisson models) to fit thedistribution of in vivo fluorescence has subsequently beencritically assessed. A theoretical formulation for the stochasticproperties of biologically active scalars is also provided andvalidated. Finally, the potential effect of the intermittentproperties of turbulent velocity fluctuations on processesrelevant to the life of plankton organisms are theoreticallyinvestigated. It is shown that the intermittent nature ofmicroscale turbulence may result in (i) a decrease in the rate ofnutrient fluxes towards non-motile phytoplankton cells (6-62 %),(ii) a decrease in the physical coagulation of phytoplankton cells(25-48 %) and in the subsequent phytoplankton aggregate volumes(22-41 %), and (iii) a decrease of the turbulence contribution topredator-prey encounter rates (25-50 %).

One interesting example of a discrete mathematical model used in biology is a food web.The first biology courses in high school and in college present the fundamental nature ofa food web, one that is understandable by students at all levels. But food webs as part ofa larger system are often not addressed. This paper presents materials that can be used inundergraduate classes in biology (and mathematics) and provides students with theopportunity to explore mathematical models of predator-prey relationships, determinetrophic levels, dominant species, stability of the ecosystem, competition graphs, intervalgraphs, and even confront problems that would appear to have logical answers that are asyet unsolved.

In this paper, we study the numerical approximation of a size-structured population modelwhose dependency on the environment is managed by the evolution of a vital resource. Weshow that this is a difficult task: some numerical methods are not suitable for along-time integration. We analyze the reasons for the failure.