The inverse stable subordinator provides a probability model for time-fractional differential equations, and leads to explicit solution formulae. This paper reviews properties of the inverse stable subordinator, and applications to a variety of problems in mathematics and physics. Several different governing equations for the inverse stable subordinator have been proposed in the literature. This paper also shows how these equations can be reconciled.

We study the behavior of linear nonstationary shallow water waves generated by aninstantaneous localized source as they propagate over and become trapped by elongatedunderwater banks or ridges. To find the solutions of the corresponding equations, we usean earlier-developed asymptotic approach based on a generalization of Maslov’s canonicaloperator, which provides a relatively simple and efficient analytic-numerical algorithmfor the wave field computation. An analysis of simple examples (where the bank and sourceshapes are given by certain elementary functions) shows that the appearance and dynamicsof trapped wave trains is closely related to a cascade of bifurcations of space-timecaustics, the bifurcation parameter being the bank length-to-width ratio.

We derive the conserved vectors for the nonlinear two-dimensional Euler equationsdescribing nonviscous incompressible fluid flows on a three-dimensional rotating sphericalsurface superimposed by a particular stationary latitude dependent flow. Under theassumption of no friction and a distribution of temperature dependent only upon latitude,the equations in question can be used to model zonal west-to-east flows in the upperatmosphere between the Ferrel and Polar cells. As a particualr example, the conserveddensities are analyzed by visualizing the exact invariant solutions associated with thegiven model for the particular form of finite disturbances for which the invariantsolutions are also exact solutions of Navier-Stokes equations.

The paper is devoted to a reaction-diffusion equation in an infinite two-dimensionalstrip with nonlinear boundary conditions. The existence of travelling waves is proved inthe bistable case by the Leray-Schauder method. It is based on a topological degree forelliptic problems in unbounded domains and on a priori estimates of solutions.

The formation, persistence and movement of self-organised biological aggregations aremediated by signals (e.g., visual, acoustic or chemical) that organisms use to communicatewith each other. To investigate the effect that communication has on the movement ofbiological aggregations, we use a class of nonlocal hyperbolic models that incorporatesocial interactions and different communication mechanisms between group members. Weapproximate the maximum speed for left-moving and right-moving groups, and shownumerically that the travelling pulses exhibited by the nonlocal hyperbolic modelsactually travel at this maximum speed. Next, we use the formula for the speed of atravelling pulse to calculate the reversal time for the zigzagging behaviour, and showthat the communication mechanisms have an effect on these reversal times. Moreover, weshow that how animals communicate with each other affects also the density structure ofthe zigzags. These findings offer a new perspective on the complexity of the biologicalfactors behind the formation and movement of various aggregations.

We prove the existence of a global bifurcation branch of 2π-periodic,smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset ofsolutions in the global branch contains a sequence which converges uniformly to somesolution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along theglobal bifurcation branch. In addition, a spectral scheme for computing approximations tothose waves is put forward, and several numerical results along the global bifurcationbranch are presented, including the presence of a turning point and a ‘highest’, cuspedwave. Both analytic and numerical results are compared to traveling-wave solutions of theKdV equation.

One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

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.

During the last two decades, molecular genetic studies and the completion of thesequencing of the Arabidopsis thaliana genome have increased knowledge ofhormonal regulation in plants. These signal transduction pathways act in concert throughgene regulatory and signalling networks whose main components have begun to be elucidated.Our understanding of the resulting cellular processes is hindered by the complex, andsometimes counter-intuitive, dynamics of the networks, which may be interconnected throughfeedback controls and cross-regulation. Mathematical modelling provides a valuable tool toinvestigate such dynamics and to perform in silico experiments that may not be easilycarried out in a laboratory. In this article, we firstly review general methods formodelling gene and signalling networks and their application in plants. We then describespecific models of hormonal perception and cross-talk in plants. This mathematicalanalysis of sub-cellular molecular mechanisms paves the way for more comprehensivemodelling studies of hormonal transport and signalling in a multi-scale setting.

Pollen tubes are tip growing plant cells that display oscillatory growth behavior. It hasbeen demonstrated experimentally that the reduction of the average pollen tube growth ratethrough elevated extracellular calcium or borate concentrations coincides with a greateramplitude of the growth rate oscillation and a lower oscillation frequency. We present asimple numerical model of pollen tube growth that reproduces these results, as well asanalytical calculations that suggest an underlying mechanism. These data show that thepollen tube oscillator is non-isochronous, and is different from harmonic oscillation.

We review recent results on stability of traveling waves in partly parabolicreaction-diffusion systems with stable or marginally stable equilibria. We explain howattention to what are apparently mathematical technicalities has led to theorems thatallow one to convert spectral calculations, which are used in the sciences and engineeringto study stability of a wave, into detailed, theoretically-based information about thebehavior of perturbations of the wave.

We study a nonlinear diffusion equationut = uxx + f(u)with Robin boundary condition at x = 0 and with a free boundary conditionat x = h(t), whereh(t) > 0 is a moving boundary representing theexpanding front in ecology models. For anyf ∈ C1 with f(0) = 0, weprove that every bounded positive solution of this problem converges to a stationary one.As applications, we use this convergence result to study diffusion equations withmonostable and combustion types of nonlinearities. We obtain dichotomy results and sharpthresholds for the asymptotic behavior of the solutions.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We brieflyreview properties of pn that have been usedto derive upper and lower bounds for the largest and smallest zero ofpn. Bounds for theextreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained usingdifferent approaches are numerically compared and new bounds for extreme zeros ofq-Laguerre and little q-Jacobi polynomials are proved.

The release of chemicals following herbivore grazing on primary producers may providefeeding cues to carnivorous predators, thereby promoting multitrophic interactions. Inparticular, chemicals released following grazing on phytoplankton by microzooplanktonherbivores have been shown to elicit a behavioural foraging response in carnivorouscopepods, which may use this chemical information as a mechanism to locate and remainwithin biologically productive patches of the ocean. In this paper, we use a 1D spatialreaction-diffusion model to simulate a tri-trophic planktonic system in the water column,where predation at the top trophic level (copepods) is affected by infochemicals releasedby the primary producers forming the bottom trophic level. The effect of theinfochemical-mediated predation is investigated by comparing the case where copepodsforage randomly to the case where copepods adjust their vertical position to follow thedistribution of grazing-induced chemicals. Results indicate that utilization ofinfochemicals for foraging provides fitness benefits to copepods and stabilizes the systemat high nutrient load, whilst also forming a possible mechanism for phytoplankton bloomformation. We also investigate how the copepod efficiency to respond to infochemicalsaffects the results, and show that small increases (2%) in the ability of copepods tosense infochemicals can promote their persistence in the system. Finally we argue thateffectively employing infochemicals for foraging can be an evolutionarily stable strategyfor copepods.

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.

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.

Habitat fragmentation is an important area of concern in species conservation. Habitatfragmentation can affect population distributions through reductions in suitable habitat,and through organism responses to different habitat types and the transitions betweenthem. In earlier work, the effect of habitat fragmentation on cyclic populations wasinvestigated in the context of populations that show no behavioural response to theinterface between habitat types. In this paper, we extend the earlier work by addingedge-mediated behaviour to the models. That is, we investigate the dynamics that resultwhen oscillatory predator and prey species also exhibit behavioural responses to habitatinterfaces. Our results show generally that habitat loss decreases the amplitude and theaverage density of the prey and predator populations, but that most of the reponsesobserved in the two models exhibit marked differences. This work highlights the complexityof the interplay between population cycles, habitat fragmentation, and edge-mediatedbehaviour, and the need to study such systems in greater detail.