A moving average transform in the plane with a variable size and shape window depending on the position and the ’time’ is studied. The main objective is to select the window parameters in such a way that the new transform converges smoothly to the identity transform at the boundary of a prescribed bounded plane region. A new approximation of solitary waves arising from Korteweg-de Vries equation is obtained based on results in the paper. Numerical implementation and examples are included.

\n The problem of finding the distribution of functional of a trajectory of a particle executing a random walk in a disordered medium containing both traps and obstacles is considered. As a model of disordered medium, the Schirmacher model, a combination of random barriers model and multiple-trapping model, is used. Forward and backward Feynman-Kac equations with the boundary conditions at discontinuity points are formulated. As an example, the distribution of the residence time in a half-space is obtained. It is shown that the anomalous subdiffusion due to traps and that due to obstacles give very different distributions.\n

In this paper, we present a review about mathematical models of population and community dynamics incorporating individual behaviour. The considered models consist in coupling a game theoretic model, going on at a fast time scale and that describes the change of tactics among individuals in a population or in a community to a model going on at a slow time scale describing the effects in the long term of individual tactics on reproduction and mortality of the populations. As application, an extension of a stage-structured population with intra-individual behavioural variability is presented. Individuals can adopt three different tactics: hawk, dove and bully. We consider a stage-structured population with immature and mature individuals. A mature individual experiences the different tactics in his life and it is assumed that the game is played at a fast time scale in comparison to population dynamics. This fast part of the model is coupled to a slow part, which describes the growth of each sub-population. By use of aggregation of variables methods, we obtain a reduced system of two differential equations governing immature and total mature sub-populations at the slow time scale. The analysis of the model shows that when the cost of fightings is smaller, the proportion of hawks is larger. The total population size equilibrium shows a maximum with respect to the gain of the game.

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyse the probability distribution functions and we derive the mean squared displacement in $x$ and $y$ directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both $x$ and $y$ directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

A short review on pharmacokinetics-pharmacodynamics (PK-PD) presented below aims to show the evolution of some concepts and ideas in this field. Some of them are developed in more detail in the papers of this issue. The key question for a practical application of PK-PD models is the ability to estimate the model parameters using patients data. In [1] a novel approach to an accurate quantification of the uncertainty in parameter estimates attributed to inter-individual variability is proposed. The analyzed PK-PD model is formulated as a compartmental ODE system. The methodology of recognizing and capturing the uncertainty in predicted quantities of interest due to inter-individual variability when the individual is not available for repeated measurements may prove to be invaluable in the risk assessment of future experiments and drug applications. Anticancer molecular PK-PD in a cell population dynamics model with drug delivery optimisation is discussed in [2]. The works [3] and [4] deal with various aspects of hemostasis modelling, and [5] with metabolic aspects in a whole-body setting. These studies represent the diversity of aspects of PK-PD modelling nowadays: theoretical about parameter estimation in general versus applied to medical questions in particular, localised cell population versus whole-body settings, cell population and whole-body versus patient population settings.

We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal’s intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.

Soft-bottomed mussel beds provide an important example of ecosystem-scale self-organisation. Field data from some intertidal regions shows banded patterns of mussels, running parallel to the shore. This paper demonstrates the use of numerical bifurcation methods to investigate in detail the predictions made by mathematical models concerning these patterns. The paper focusses on the “sediment accumulation model” proposed by Liu et al (Proc. R. Soc. Lond. B 14 (2012), 20120157). The author calculates the parameter region in which patterns exist, and the sub-region in which these patterns are stable as solutions of the original model. He then shows how his results can be used to explain numerical observations of history-dependent wavelength selection as parameters are varied slowly.

We consider an intraspecific resource competition model with asymmetric nonlocal dispersal and interaction. Both interaction and dispersal are modeled using convolution integrals. We introduce two parameters for each convolution kernel describing the range of nonlocality and the extent of asymmetry. It is shown that the spatially homogenous equilibrium of this model becomes unstable for sufficiently small diffusion rates by performing a linear stability analysis. Our numerical observations indicate that traveling and stationary wave type patterns arise near the stability boundary. Away from the stability boundary, solutions to the model exhibit colony formation i.e. one can observe formation of a new colony island just ahead of a vanishing stationary wave island. We further analyze the behavior of solutions to the model near the stability boundary using the techniques of weakly nonlinear analysis. We obtain a Stuart-Landau type equation and give its parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities and extents of asymmetries of two kernel functions. We show that both continuous and discontinuous transitions from disordered behavior to ordered one are possible. We also verify these results numerically.

\n This paper explores by means of extensive numerical simulation how unidirectional and\n bidirectional symmetric dispersal can affect the mean total population size and the\n fluctuation range in a two-patch population model described by coupled difference\n equations. The obtained results show that the response to dispersal varies not only with\n the type of connection between subpopulations, but also with the intrinsic dynamics in\n each subpopulation. We find that the mean total population size increases monotonically\n with unidirectional dispersal from a region with local complicated dynamics to a region\n with an attracting equilibrium, whereas in the other studied scenarios the response is\n generically unimodal. Constancy and persistence are considered by relating them to the\n fluctuation range. Our results show that dispersal is capable of enhancing constancy and\n persistence only in certain situations. Additionally, we show that multistability affects\n the behaviour of the mean total population size and its fluctuation range. Hence it can be\n concluded that, in contrast to other control strategies, only if there is a good knowledge\n of the local population dynamics, then a modification of the natural dispersal rate\n between regions might be used as a way to control the size and stability of the\n population. \n

Mathematical modelling in ecology, epidemiology and eco-epidemiology is a vast and constantly growing research field. This is perhaps unsurprising since mathematical models can provide a wide-ranging exploration of the biological problem without a need for experiments which are usually expensive and can be potentially dangerous to ecosystems. The current Special Issue of MMNP presents recent findings and developments in the following areas of theoretical ecology and epidemiology: (i) revealing the biological processes behind observed complex patterns; (iii) modelling the role of dispersal and spatial heterogeneity in species persistence; (ii) applications of bifurcation theory in population dynamics and epidemiology; (iv) the role of selection in self-replicating biological and ecological systems; (v) sensitivity, predictability and control of biological models in a noisy environment. It is important to emphasize that many of the contributions to this issue are not limited to one of the above topics, but rather lie at the interface of several of them. The work of D. Ahmed and S. Petrovskii [1] studies the movement of insects whose spatial dispersal is a random variable and is described by a certain probability distribution. Understanding the mechanisms of insects movement behaviour is a highly relevant problem from the practical point of view, since it can help us to correctly interpret the results of insect trapping, which is the basis of many current pest monitoring programs [17]. There is a lively ongoing debate in the literature regarding the possibility of Levy-type movement behaviour by animals, as opposed to the Brownian motion, and various mechanisms have been proposed to explain the observed patterns [3, 16, 24]. Ahmed and Petrovskii provide an alternative description of Levy-type movement by introducing time-dependent diffusivity. In particular, the authors claim that the trap counts of insects whose dynamics is given by an example of a Levy-type distribution -a Cauchy type random walk can be successfully modelled via a different movement pattern: classical Brownian motion with a timedependent diffusion coefficient. Thus, the authors provide an excellent example of how two completely different mechanisms can describe the same pattern of field observations. Among the central topics in theoretical ecology is the understanding of the role of space in population dynamics and species persistence across different time scales [6,7,11,12]. In this issue, three contributions [2, 4, 20] directly focus on the role of space in population dynamics. Modelling the mechanisms of biological aggregation and patchiness is a pertinent topic in theoretical ecology, with various models suggested [6, 14, 19]. In their work, R. Eftimie and A. Coulier [4] explore the effects of communication between organisms on the formation and structure of large biological ag-