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Royal Society Open Science RSS feed -- recent Mathematics articles2054-5703Royal Society Open Science<![CDATA[Spatial correlated games]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/11/171361?rss=1
This article studies correlated two-person games constructed from games with independent players as proposed in Iqbal et al. (2016 R. Soc. open sci.3, 150477. (doi:10.1098/rsos.150477)). The games are played in a collective manner, both in a two-dimensional lattice where the players interact with their neighbours, and with players interacting at random. Four game types are scrutinized in iterated games where the players are allowed to change their strategies, adopting that of their best paid mate neighbour. Particular attention is paid in the study to the effect of a variable degree of correlation on Nash equilibrium strategy pairs.
]]>2017-11-15T00:06:00-08:00info:doi/10.1098/rsos.171361hwp:master-id:royopensci;rsos.1713612017-11-15Mathematics411171361171361<![CDATA[Decision landscapes: visualizing mouse-tracking data]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/11/170482?rss=1
Computerized paradigms have enabled gathering rich data on human behaviour, including information on motor execution of a decision, e.g. by tracking mouse cursor trajectories. These trajectories can reveal novel information about ongoing decision processes. As the number and complexity of mouse-tracking studies increase, more sophisticated methods are needed to analyse the decision trajectories. Here, we present a new computational approach to generating decision landscape visualizations based on mouse-tracking data. A decision landscape is an analogue of an energy potential field mathematically derived from the velocity of mouse movement during a decision. Visualized as a three-dimensional surface, it provides a comprehensive overview of decision dynamics. Employing the dynamical systems theory framework, we develop a new method for generating decision landscapes based on arbitrary number of trajectories. This approach not only generates three-dimensional illustration of decision landscapes, but also describes mouse trajectories by a number of interpretable parameters. These parameters characterize dynamics of decisions in more detail compared with conventional measures, and can be compared across experimental conditions, and even across individuals. The decision landscape visualization approach is a novel tool for analysing mouse trajectories during decision execution, which can provide new insights into individual differences in the dynamics of decision making.
]]>2017-11-08T02:01:34-08:00info:doi/10.1098/rsos.170482hwp:master-id:royopensci;rsos.1704822017-11-08Mathematics411170482170482<![CDATA[Multiple steady states and the form of response functions to antigen in a model for the initiation of T-cell activation]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/11/170821?rss=1
The aim of this paper is to study the qualitative behaviour predicted by a mathematical model for the initial stage of T-cell activation. The state variables in the model are the concentrations of phosphorylation states of the T-cell receptor (TCR) complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero and that the model possesses more than one positive steady state for certain values of the parameters. It can also exhibit damped oscillations. It is proved that the chemical concentration which represents the degree of activation of the cell, that of the maximally phosphorylated form of the TCR complex, is, in general, a non-monotone function of the activating signal. In particular, there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces a maximal activation of the T cell. This suggests that mechanisms taking place in the first few minutes after activation and included in the model studied in this paper suffice to explain the optimal dissociation time seen in experiments. In this way, the results of certain simulations in the literature have been confirmed rigorously and some important features which had not previously been seen have been discovered.
]]>2017-11-08T00:05:45-08:00info:doi/10.1098/rsos.170821hwp:master-id:royopensci;rsos.1708212017-11-08Mathematics411170821170821<![CDATA[Modelling the role of correctional services on gangs: insights through a mathematical model]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/10/170511?rss=1
Research has shown that gang membership increases the chances of offending, antisocial behaviour and drug use. Gang membership should be acknowledged as part of crime prevention and policy designs, and when developing interventions and preventative programmes. Correctional services are designed to rehabilitate convicted offenders. We formulate a deterministic mathematical model using nonlinear ordinary differential equations to investigate the role of correctional services on the dynamics of gangs. The recruitment into gang membership is assumed to happen through an imitation process. An epidemic threshold value, Rg, termed the gang reproduction number, is proposed and defined herein in the gangs’ context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that gangs may persist in the population even if Rg is less than one. Sensitivity analysis of Rg was performed to determine the relative importance of different parameters in gang initiation. The critical efficacy * is evaluated and the implications of having functional correctional services are discussed.
]]>2017-10-11T01:46:25-07:00info:doi/10.1098/rsos.170511hwp:master-id:royopensci;rsos.1705112017-10-11Mathematics410170511170511<![CDATA[A geometric method for eigenvalue problems with low-rank perturbations]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/9/170390?rss=1
We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci.22, 239–254. (doi:10.1007/s10827-006-0010-x)), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math.48, 249–264. (doi:10.1093/imamat/48.3.249)).
]]>2017-09-27T00:10:21-07:00info:doi/10.1098/rsos.170390hwp:master-id:royopensci;rsos.1703902017-09-27Mathematics49170390170390<![CDATA[Pedestrian motion modelled by Fokker-Planck Nash games]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/9/170648?rss=1
A new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.
]]>2017-09-13T00:09:20-07:00info:doi/10.1098/rsos.170648hwp:master-id:royopensci;rsos.1706482017-09-13Mathematics49170648170648<![CDATA[Huygens clocks revisited]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/9/170777?rss=1
In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.
]]>2017-09-06T01:49:28-07:00info:doi/10.1098/rsos.170777hwp:master-id:royopensci;rsos.1707772017-09-06Mathematics49170777170777<![CDATA[Modelling radicalization: how small violent fringe sects develop into large indoctrinated societies]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170678?rss=1
We model radicalization in a society consisting of two competing religious, ethnic or political groups. Each of the ‘sects’ is divided into moderate and radical factions, with intra-group transitions occurring either spontaneously or through indoctrination. We also include the possibility of one group violently attacking the other. The intra-group transition rates of one group are modelled to explicitly depend on the actions and characteristics of the other, including violent episodes, effectively coupling the dynamics of the two sects. We use a game theoretic framework and assume that radical factions may tune ‘strategic’ parameters to optimize given utility functions aimed at maximizing their ranks while minimizing the damage inflicted by their rivals. Constraints include limited overall resources that must be optimally allocated between indoctrination and external attacks on the other group. Various scenarios are considered, from symmetric sects whose behaviours mirror each other, to totally asymmetric ones where one sect may have a larger population or a superior resource availability. We discuss under what conditions sects preferentially employ indoctrination or violence, and how allowing sects to readjust their strategies allows for small, violent sects to grow into large, indoctrinated communities.
]]>2017-08-30T00:09:44-07:00info:doi/10.1098/rsos.170678hwp:master-id:royopensci;rsos.1706782017-08-30Mathematics48170678170678<![CDATA[The impact of weather and storm water management ponds on the transmission of West Nile virus]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170017?rss=1
West Nile virus (WNV) is the most widely distributed arbovirus in the world and the spread is influenced by complex factors including weather conditions and urban environmental settings like storm water management ponds (SWMP). The purpose of this work was to develop an ordinary differential equation model to explore the impacts of SWMP, temperature and precipitation on WNV vector abundance and the transmission of WNV between mosquito and bird populations. The model was used to analyse how weather conditions and SWMP can influence the basic reproduction number. The results found that an excess of precipitation and fiercer intraspecific competition will reduce vector population and the peak value of infectious vectors and birds. This information can be used to identify measures that would be useful to control larval abundance in SWMP and the transmission of WNV.
]]>2017-08-16T00:08:56-07:00info:doi/10.1098/rsos.170017hwp:master-id:royopensci;rsos.1700172017-08-16Mathematics48170017170017<![CDATA[Stochastic phase segregation on surfaces]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170472?rss=1
Phase separation and coarsening is a phenomenon commonly seen in binary physical and chemical systems that occur in nature. Often, thermal fluctuations, modelled as stochastic noise, are present in the system and the phase segregation process occurs on a surface. In this work, the segregation process is modelled via the Cahn–Hilliard–Cook model, which is a fourth-order parabolic stochastic system. Coarsening is analysed on two sample surfaces: a unit sphere and a dumbbell. On both surfaces, a statistical analysis of the growth rate is performed, and the influence of noise level and mobility is also investigated. For the spherical interface, it is also shown that a lognormal distribution fits the growth rate well.
]]>2017-08-16T00:08:56-07:00info:doi/10.1098/rsos.170472hwp:master-id:royopensci;rsos.1704722017-08-16Mathematics48170472170472