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 an instantaneous localized source as they propagate over and become trapped by elongated underwater banks or ridges. To find the solutions of the corresponding equations, we use an earlier-developed asymptotic approach based on a generalization of Maslov’s canonical operator, which provides a relatively simple and efficient analytic-numerical algorithm for the wave field computation. An analysis of simple examples (where the bank and source shapes are given by certain elementary functions) shows that the appearance and dynamics of trapped wave trains is closely related to a cascade of bifurcations of space-time caustics, the bifurcation parameter being the bank length-to-width ratio.

We derive the conserved vectors for the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption 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 upper atmosphere between the Ferrel and Polar cells. As a particualr example, the conserved densities are analyzed by visualizing the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.

The formation, persistence and movement of self-organised biological aggregations are mediated by signals (e.g., visual, acoustic or chemical) that organisms use to communicate with each other. To investigate the effect that communication has on the movement of biological aggregations, we use a class of nonlocal hyperbolic models that incorporate social interactions and different communication mechanisms between group members. We approximate the maximum speed for left-moving and right-moving groups, and show numerically that the travelling pulses exhibited by the nonlocal hyperbolic models actually travel at this maximum speed. Next, we use the formula for the speed of a travelling pulse to calculate the reversal time for the zigzagging behaviour, and show that the communication mechanisms have an effect on these reversal times. Moreover, we show that how animals communicate with each other affects also the density structure of the zigzags. These findings offer a new perspective on the complexity of the biological factors 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 of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a ‘highest’, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.

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

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, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(ℝd). A discrete wavelet transform is often regarded as a byproduct in wavelet analysis by decomposing and reconstructing functions in L2(ℝd) via nested subspaces of L2(ℝd) in a multiresolution analysis. However, since the input/output data and all filters in a discrete wavelet transform are of discrete nature, to understand better the performance of wavelets and framelets in applications, it is more natural and fundamental to directly study a discrete framelet/wavelet transform and its key properties. The main topic of this paper is to study various properties of a discrete framelet transform purely in the discrete/digital setting without involving the function space L2(ℝd). We shall develop a comprehensive theory of discrete framelets and wavelets using an algorithmic approach by directly studying a discrete framelet transform. The connections between our algorithmic approach and the classical theory of wavelets and framelets in the function setting will be addressed. Using tensor product of univariate complex-valued tight framelets, we shall also present an example of directional tight framelets in this paper.

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

We review recent results on stability of traveling waves in partly parabolic reaction-diffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently mathematical technicalities has led to theorems that allow one to convert spectral calculations, which are used in the sciences and engineering to study stability of a wave, into detailed, theoretically-based information about the behavior of perturbations of the wave.

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

We study a nonlinear diffusion equation ut = uxx + f(u) with Robin boundary condition at x = 0 and with a free boundary condition at x = h(t), where h(t) > 0 is a moving boundary representing the expanding front in ecology models. For any f ∈ C1 with f(0) = 0, we prove that every bounded positive solution of this problem converges to a stationary one. As applications, we use this convergence result to study diffusion equations with monostable and combustion types of nonlinearities. We obtain dichotomy results and sharp thresholds for the asymptotic behavior of the solutions.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved.

The release of chemicals following herbivore grazing on primary producers may provide feeding cues to carnivorous predators, thereby promoting multitrophic interactions. In particular, chemicals released following grazing on phytoplankton by microzooplankton herbivores have been shown to elicit a behavioural foraging response in carnivorous copepods, which may use this chemical information as a mechanism to locate and remain within biologically productive patches of the ocean. In this paper, we use a 1D spatial reaction-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 released by the primary producers forming the bottom trophic level. The effect of the infochemical-mediated predation is investigated by comparing the case where copepods forage randomly to the case where copepods adjust their vertical position to follow the distribution of grazing-induced chemicals. Results indicate that utilization of infochemicals for foraging provides fitness benefits to copepods and stabilizes the system at high nutrient load, whilst also forming a possible mechanism for phytoplankton bloom formation. We also investigate how the copepod efficiency to respond to infochemicals affects the results, and show that small increases (2%) in the ability of copepods to sense infochemicals can promote their persistence in the system. Finally we argue that effectively employing infochemicals for foraging can be an evolutionarily stable strategy for 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 behaviour than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing unstable steady state 1, see [12]. This is proved in [1, 6] in the 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 simple analytical traveling wave solutions that exhibit various behaviours. These traveling waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The latter exist in a regime where time dynamics converges to another object observed in [3, 8]: a wave that connects 0 to a pulsating wave around 1.

Habitat fragmentation is an important area of concern in species conservation. Habitat fragmentation can affect population distributions through reductions in suitable habitat, and through organism responses to different habitat types and the transitions between them. In earlier work, the effect of habitat fragmentation on cyclic populations was investigated in the context of populations that show no behavioural response to the interface between habitat types. In this paper, we extend the earlier work by adding edge-mediated behaviour to the models. That is, we investigate the dynamics that result when oscillatory predator and prey species also exhibit behavioural responses to habitat interfaces. Our results show generally that habitat loss decreases the amplitude and the average density of the prey and predator populations, but that most of the reponses observed in the two models exhibit marked differences. This work highlights the complexity of the interplay between population cycles, habitat fragmentation, and edge-mediated behaviour, and the need to study such systems in greater detail.