We discuss theoretical and experimental approaches to three distinctdevelopmental systems that illustrate how theory can influence experimental workand vice-versa. The chosen systems – Drosophila melanogaster,bacterial pattern formation, and pigmentation patterns – illustrate thefundamental physical processes of signaling, growth and cell division, and cellmovement involved in pattern formation and development. These systems exemplifythe current state of theoretical and experimental understanding of how theseprocesses produce the observed patterns, and illustrate how theoretical andexperimental approaches can interact to lead to a better understanding ofdevelopment. As John Bonner said long ago
`We have arrived at the stage where models are useful to suggestexperiments, and the facts of the experiments in turn lead to new and improvedmodels that suggest new experiments. By this rocking back and forth between thereality of experimental facts and the dream world of hypotheses, we can moveslowly toward a satisfactory solution of the major problems of developmentalbiology.'

The problem of finite-dimensional asymptotics of infinite-dimensional dynamicsystems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do nottend to fixed positions, and the path covered tends to infinity as t $\to$ ∞. The drift equationsfor peak motion are obtained. Various types of distribution stability are studied: internalstability (stability with respect to perturbations that do not extend the support), externalstability or uninvadability (stability with respect to strongly small perturbations that extendthe support), and stable realizability (stability with respect to small shifts and extensions ofthe density peaks). Models of self-synchronization of cell division are studied, as an exampleof selection in systems with additional symmetry. Appropriate construction of the notionof typicalness in infinite-dimensional space is discussed, and the notion of “completely thin”sets is introduced.

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turinginstability and the interpretation refers to adaptive evolution. By analogy with other formalismsused in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sumof Dirac masses) will happen in the limit of small mutations. In the present work we study thisasymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation.We prove the convergence analytically and illustrate it numerically. We also illustrate numericallyhow the constraint is related to the concentration points. We investigate numerically some featuresof these concentration points such as their weights and their numbers. We show analytically howthe constrained Hamilton-Jacobi gives the so-called canonical equation relating their motion withthe selection gradient. We illustrate this point numerically.

Gliomas are highly invasive brain tumors that exhibit high and spatially heterogeneouscell proliferation and motility rates. The interplay of proliferation and migrationdynamics plays an important role in the invasion of these malignant tumors. We analyze theregulation of proliferation and migration processes with a lattice-gas cellular automaton(LGCA). We study and characterize the influence of the migration/proliferation dichotomy(also known as the “GO-or-Grow" mechanism) on avascular glioma invasion, in terms ofinvasion speed and width of the infiltration zone. We show that the invasive behavior ofthe (macroscopic) tumor colony is a highly complex phenomenon that cannot be extrapolatedby the sole knowledge of the (microscopic) individual cell phenotype.

In L 2(ℝd ;ℂn ), we consider a wide class of matrix elliptic secondorder differential operators $\mathcal{A}$ εwith rapidly oscillating coefficients (depending on x/ε).For a fixed τ > 0 and small ε > 0, we findapproximation of the operator exponential exp(− $\mathcal{A}$ ε τ) in the(L 2(ℝd ;ℂn ) →H 1(ℝd ;ℂn ))-operator norm with an error term of orderε. In this approximation, the corrector is taken into account. Theresults are applied to homogenization of a periodic parabolic Cauchy problem.

As one of the major directions in applied and computational harmonic analysis, theclassical theory of wavelets and framelets has been extensively investigated in thefunction setting, in particular, in the function spaceL2(ℝd). A discrete wavelettransform is often regarded as a byproduct in wavelet analysis by decomposing andreconstructing functions in L2(ℝd)via nested subspaces of L2(ℝd) ina multiresolution analysis. However, since the input/output data and all filters in adiscrete wavelet transform are of discrete nature, to understand better the performance ofwavelets and framelets in applications, it is more natural and fundamental to directlystudy a discrete framelet/wavelet transform and its key properties. The main topic of thispaper is to study various properties of a discrete framelet transform purely in thediscrete/digital setting without involving the function spaceL2(ℝd). We shall develop acomprehensive theory of discrete framelets and wavelets using an algorithmic approach bydirectly studying a discrete framelet transform. The connections between our algorithmicapproach and the classical theory of wavelets and framelets in the function setting willbe addressed. Using tensor product of univariate complex-valued tight framelets, we shallalso present an example of directional tight framelets in this paper.

We discuss how distributed delays arise in biological models and review theliterature on such models. We indicate why it is important to keep thedistributions in a model as general as possible. We then demonstrate, throughthe analysis of a particular example, what kind of information can be gainedwith only minimal information about the exact distribution of delays.In particular we show that a distribution independent stability region maybe obtained in a similar way that delay independent results are obtained forsystems with discrete delays. Further, we show how approximations to theboundary of the stability region of an equilibrium point may be obtained withknowledge of one, two or three moments of the distribution. We compare theapproximations with the true boundary for the case of uniform and gammadistributions and show that the approximations improve as more moments are used.

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. Themembrane is characterized by its Canham-Helfrich energy (Willmore energy with areaconstraint) and acts as a boundary force on the Navier-Stokes system modeling anincompressible fluid. We give a concise description of the model and of the associatednumerical scheme. We provide numerical simulations with emphasis on the comparisonsbetween different types of flow: the geometric model which does not take into account thebulk fluid and the biomembrane model for two different regimes of parameters.

Simple epidemiological models with information dependent vaccination functions cangenerate sustained oscillations via Hopf bifurcation of the endemic state. The onset of these oscillations depend on the shape of the vaccination function. A “global” approach is used to characterize the instability condition and identify classes of functions that always lead to stability/instability. The analysis allows the identification of an analytically determined “threshold vaccination function” having a simple interpretation: coverage functions lying always above the threshold always lead to oscillations, whereas coverage functions always below never lead to instability.

The spatio-temporal dynamics of a population present one of the most fascinating aspects and challenges for ecological modelling. In this article we review some simple mathematical models, based on one dimensional reaction-diffusion-advection equations, for the growth of a population on a heterogeneous habitat. Considering a number of models of increasing complexity we investigate the often contrary roles of advection and diffusion for the persistence of the population. When it is possible we demonstrate basic mathematical techniques and give the critical conditions providing the survival of a population in simple systems and in more complex resource-consumer models which describe the dynamics of phytoplankton in a water column.

Since cancer is a complex phenomenon that incorporates events occurring on differentlength and time scales, therefore multiscale models are needed if we hope to adequately addresscancer specific questions. In this paper we present three different multiscale individual-cell-basedmodels, each motivated by cancer-related problems emerging from each of the spatial scales: extracellular,cellular or subcellular, but also incorporating relevant information from other levels.We apply these hybrid models to investigate the influence of the microenvironement on tumourinvasion, cell-cell collaboration and competition leading to the initiation and growth of epithelialtumours, and to evolution of cell phenotypes/genotypes arising in tumours growing in differentoxygen concentrations. We also discuss how these models relate to one another and can be used tobridge biological scales relevant to cancer.

A conceptual minimal model demonstrating spatially heterogeneous wave regimesinterpreted as pursuit-evasion in predator-prey system is constructed and investigated. Themodel is based on the earlier proposed hypothesis that taxis accelerations of prey and predators are proportional to the density gradient of another population playing a role of taxis stimulus. Considering acceleration rather than immediate velocity allows obtaining realistic solutions even while ignoring variations of total abundances of both modelled populations. Linear analysis of the model shows that stationary homogeneous regime becomes oscillatory unstable with respect to small heterogeneous perturbations if either taxis activities or total population abundances are high enough. The ability for active directed movement of both prey and predators is the necessary condition for spatial self-organization. Numerical simulations illustrate analytical results. The relation between the proposed model and conventional two-component systems with cross-diffusion is discussed.

We study the mathematical properties of a general model of cell division structuredwith several internal variables. We begin with a simpler and specific model with two variables, wesolve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-timeconvergence. The main difficulty comes from natural degeneracy of birth terms that we overcomewith a regularization technique. We then extend the results to the case with several parameters andrecall the link between this simplified model and the one presented in [6]; an application to thenon-linear problem is also given, leading to robust subpolynomial growth of the total population.

In order to better understand the dynamics of acute leukemia, and in particular to findtheoretical conditions for the efficient delivery of drugs in acute myeloblastic leukemia,we investigate stability of a system modeling its cell dynamics.

The overall system is a cascade connection of sub-systems consisting of distributeddelays and static nonlinear feedbacks. Earlier results on local asymptotic stability areimproved by the analysis of the linearized system around the positive equilibrium. For thenonlinear system, we derive stability conditions by using Popov, circle and nonlinearsmall gain criteria. The results are illustrated with numerical examples andsimulations.

The immune system is able to protect the host from tumor onset, and immune deficienciesare accompanied by an increased risk of cancer. Immunology is one of the fields in biologywhere the role of computational and mathematical modeling and analysis were recognized theearliest, beginning from 60s of the last century. We introduce the two most common methodsin simulating the competition among the immune system, cancers and tumor immunologystrategies: differential equations and rule-based models. Several specific implementationsare presented, describing in details how they work and how they advance or contribute thefield of tumor immunology.

There is a growing interest in high-order finite and spectral/hp elementmethods using continuous and discontinuous Galerkin formulations. In this paper weinvestigate the effect of h- and p-type refinement onthe relationship between runtime performance and solution accuracy. The broad spectrum ofpossible domain discretisations makes establishing a performance-optimal selectionnon-trivial. Through comparing the runtime of different implementations for evaluatingoperators over the space of discretisations with a desired solution tolerance, wedemonstrate how the optimal discretisation and operator implementation may be selected fora specified problem. Furthermore, this demonstrates the need for codes to support bothlow- and high-order discretisations.

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviourthan for the usual Fisher equation. A striking numerical observation is that a travelingwave with minimal speed can connect a dynamically unstable steady state 0 to a Turingunstable steady state 1, see [12]. This is provedin [1, 6] inthe case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simpleanalytical traveling wave solutions that exhibit various behaviours. These travelingwaves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connectthese two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. Thelatter exist in a regime where time dynamics converges to another object observed in[3, 8]: awave that connects 0 to a pulsating wave around 1.

HIV infection is multi-faceted and a multi-step process. The virus-induced pathogenicmechanisms are manifold and mediated through a range of positive and negative feedbackregulations of immune and physiological processes engaged in virus-host interactions. Thefundamental questions towards understanding the pathogenesis of HIV infection are nowshifting to ‘dynamic’ categories: (i) why is the HIV-immune response equilibrium finallydisrupted? (ii) can one modify the dynamic equilibrium for host benefit? (iii) can onepredict the outcome of a system perturbation via antiviral drugs or drugs modulating thehost immune response dynamics? Answering these questions requires a majorinterdisciplinary effort, and in particular, the development of novel mathematicalapproaches for a coherent quantitative description and prediction of intra-patient HIVevolution, the immunological responses to HIV infection, and the systems level homeostaticregulation of specific effector and regulatory lymphocyte populations in correlation withdisease status. Here we summarized fundamental biological features of HIV infection andcurrent mathematical modelling attempts to understand HIV pathogenesis.

In this paper, we investigate the complex dynamics of a spatial plankton-fish system withHolling type III functional responses. We have carried out the analytical study for bothone and two dimensional system in details and found out a condition for diffusiveinstability of a locally stable equilibrium. Furthermore, we present a theoreticalanalysis of processes of pattern formation that involves organism distribution and theirinteraction of spatially distributed population with local diffusion. The results ofnumerical simulations reveal that, on increasing the value of the fish predation rates,the sequences spots → spot-stripe mixtures → stripes → hole-stripe mixtures holes → wavepattern is observed. Our study shows that the spatially extended model system has not onlymore complex dynamic patterns in the space, but also has spiral waves.

The discovery of nearly periodic vegetation patterns in arid and semi-arid regionsmotivated numerous model studies in the past decade. Most studies have focused onvegetation pattern formation, and on the response of vegetation patterns to gradients ofthe limiting water resource. The reciprocal question, what resource modifications areinduced by vegetation pattern formation, which is essential to the understanding ofdryland landscapes, has hardly been addressed. This paper is a synthetic review of modelstudies that address this question and the consequent implications for inter-specificplant interactions and species diversity. It focuses both on patch and landscape scales,highlighting bottom-up processes, where plant interactions at the patch scale give rise tospatial patterns at the landscape scale, and top-down processes, where pattern transitionsat the landscape scale affect inter-specific interactions at the patch scale.