# Takashi Nagatani's research while affiliated with Shizuoka University and other places

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## Publications (333)

We investigate the traffic dynamics of route choice using real-time information in the case that there is an onramp in the two-route network. We propose the macroscopic two-route network model with an onramp for the route choice. The traffic behavior in two-route network changes by introducing the onramp. We study the effect of the onramp on the tr...

We present a dynamic model for major traffic jams in urban-scale networks. We investigate the traffic breakdown in the directed network consisted of cycle graphs when an intersection is closed to vehicles. We show how the traffic breakdown propagates from cycle graph to cycle graph. The deadlock (jam) of vehicles is induced by the closed intersecti...

We study the metapopulation dynamics of such mediated infectious disease as mosquito-borne infection in a patchy environment. The action range and habitat of mediators (e.g. mosquito) are different from those of humans. We focus on the effect of the difference on epidemic spreading in a patchy environment. The patchy environment and migration paths...

Ecosystems on earth are strongly affected by human life. We pay attention to pest control in a patchy environment. To date, many authors have reported the indeterminacy in pest control. Most of these works have been studied in single-habitat systems. In the present article, however, we consider a food chain model (prey, predator and top predator) o...

We study the urban-scale macroscopic traffic flow in city networks. Star graph is considered as traffic network. Star graphs with controlled traffic flow are transformed to various cell-transmission graphs by using the cell transmission method. The dynamic equations of vehicular densities on all nodes (roads) are presented on cell-transmission grap...

We present a metapopulation dynamic model for the diffusively-coupled rock-paper-scissors (RPS) game with mutation in scale-free hierarchical networks. We investigate how the RPS game changes by mutation in scale-free networks. Only the mutation from rock to scissors (R-to-S) occurs with rate μ. In the network, a node represents a patch where the R...

We study the urban-scale macroscopic traffic flow in fractal network which represents a percolation backbone. We show where and how the traffic densities and currents vary with increasing mean density in the fractal network. The fractal network is transformed to the cell-transmission graph where a node corresponds to a road. The dynamic equations o...

We study the urban-scale macroscopic traffic flow in city networks. Star graph is considered as traffic network. Star graphs with controlled traffic flow are transformed to various cell-transmission graphs by using the cell transmission method. The dynamic equations of vehicular densities on all nodes (roads) are presented on cell-transmission grap...

When the migration rate of predators is definitely different from that of prey, the metapopulation dynamics on heterogeneous graphs change greatly with migration rate. We study the effect of the migration difference on the metapopulation dynamics in the diffusively coupled prey–predator system on homogeneous and heterogeneous graphs. We present a m...

We study the urban-scale traffic flow on city networks when there is a bottleneck on a road. Directed cycle graphs with and without bypasses are considered for the city traffic network. The dynamic equations of vehicular densities on all roads are presented on the networks. When vehicles flow into the connected road, the drivers are matched their s...

We combine the rock–paper–scissors (RPS) game with SIS epidemic model. We study the effect of infection on population dynamics of RPS games with epidemic by using the metapopulation dynamic model on graphs. Via infection, an individual R changes to I with infection rate β. An infected individual I returns to R with recovery rate γ. All agents move...

When the action range of infected individuals is restricted in a patchy environment, epidemic spreading changes due to the restricted migration. We focus on the effect of the restricted migration of infected individuals to epidemic spreading in a patchy environment. Each individual is either susceptible (S) or infected (I). The restricted migration...

We study how the Allee effect with diffusion changes depending on the graph topology. We present the metapopulation dynamic model on heterogeneous and homogeneous graphs for the Allee effect of mobile individuals. We consider the small graphs with three and four nodes and the star and complete graphs with N nodes. A subpopulation (patch) is represe...

The behavior of animals during day and night can be demonstrated as time-varying migration. We study the effect of the temporal migration in a diffusively coupled Allee model on heterogeneous graphs. We present the metapopulation dynamic model for temporally heterogeneous graphs using the Allee effect of mobile individuals. We consider the small gr...

We study the effect of the network structure on the dynamic stability in the diffusively coupled Lotka–Volterra system. Here we present a metapopulation model for the Lotka–Volterra system on various graphs. The total population is assumed to consist of several subpopulations (nodes). Each individual migrates by random walk; the destination of migr...

Much literature exists on temporal networks for epidemic spreading. We present a metapopulation dynamic model on a temporal network; a subpopulation (patch) is represented by a node on a graph, and a link represents a migration path between patches. We consider the alternating graph as a temporal network. It alternates between star and complete gra...

Migration paths of animals are rarely the same. The paths may change according to seasonal and circadian rhythms. We study the effect of temporal migration on population dynamics of rock-paper-scissors (RPS) games with mutation by using the metapopulation dynamic model with two patches. Via mutation, an individual R changes to S with rate μ. All ag...

Recently, metapopulation models for rock-paper-scissors games have been presented. Each subpopulation is represented by a node on a graph. An individual is either rock (R), scissors (S) or paper (P); it randomly migrates among subpopulations. In the present paper, we assume victory rates differ in different subpopulations. To investigate the dynami...

Various theoretical models have been proposed to understand the basic nature of epidemics. Recent studies focus on the effects of mobility to epidemic process. However, uncorrelated random walk is typically assumed as the type of movement. In our daily life, the movement of people sometimes tends to be limited to a certain direction, which can be d...

Understanding mechanisms of biodiversity has been a central question in ecology. The coexistence of three species in rock-paper-scissors (RPS) systems are discussed by many authors; however, the relation between coexistence and network structure is rarely discussed. Here we present a metapopulation model for RPS game. The total population is assume...

Predator-prey systems have been studied intensively for over a hundred years. These studies have demonstrated that the dynamics of Lotka-Volterra (LV) systems are not stable, that is, exhibiting either cyclic oscillation or divergent extinction of one species. Stochastic versions of the deterministic cyclic oscillations also exhibit divergent extin...

The rock-paper-scissors (RPS) game is known as one of the simplest cyclic dominance models. This game is key to understanding biodiversity. Three species, rock (R), paper (P) and scissors (S), can coexist in nature. In the present paper, we first present a metapopulation model for RPS game with mutation. Only mutation from R to S is allowed. The to...

We present the metapopulation dynamic model for epidemic spreading of random walkers between subpopulations. A subpopulation is represented by a node on a graph. Each agent or individual is either susceptible (S) or infected (I). All agents move by random walk on the graph; namely, each agent randomly determines the destination of migration. The re...

We study the dynamic motion of a vehicle moving through the series of traffic signals on a roadway with bypasses. A vehicle moves on either main roadway or bypass. There are all signals on the main roadway and signals are controlled by both cycle time and phase difference. The dynamic model of the vehicular motion is described in terms of the nonli...

We study the effect of contingent movement on the persistence of cooperation on complex networks with empty nodes. Each agent plays Prisoner's Dilemma game with its neighbors and then it either updates the strategy depending on the payoff difference with neighbors or it moves to another empty node if not satisfied with its own payoff. If no neighbo...

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equa...

It is necessary and important to operate buses and trams on time. The bus schedule is closely related to the dynamic motion of buses. In this part, we introduce the nonlinear maps for describing the dynamics of shuttle buses in the transportation system. The complex motion of the buses is explained by the nonlinear-map models. The transportation sy...

We present an epidemic model combined with a traffic cellular automaton. Each agent or individual is either susceptible (S) or infected (I). An agent with a certain density moves to a fixed direction on one-dimensional lattice. Simulations for SIS model show that the epidemic spreads via migration. We find a dynamical phase transition between infec...

Spatial and temporal behaviors of the rock–paper–scissors (RPS) game is key to understanding not only biodiversity but also a variety of cyclic systems. It has been demonstrated that, in the stochastic cellular automaton of RPS game, three species cannot survive on one-dimensional (1-d) lattice; only a single species survives. Previous studies have...

When a cluster of vehicles with various speeds moves through the series of signals, the cluster breaks down by stopping at signals and results in smaller groups of vehicles. We present the nonlinear-map model of the motion of vehicles controlled by the signals. We study the breakup of a cluster of vehicles through the series of signals. The cluster...

Migration is observed across many species. Several authors have studied ecological migration by applying cellular automaton (CA). In this paper, we present a directional migration model with desert on a one-dimensional lattice where a traffic CA model and a lattice Lotka-Volterra system are connected. Here predators correspond to locomotive animals...

In most cases, physicists have studied the migration of biospecies by the use of random walk. In the present article, we apply cellular automaton of traffic model. For simplicity, we deal with an ecosystem contains a prey and predator, and use one-dimensional lattice with two layers. Preys stay on the first layer, but predators uni-directionally mo...

Stochastic cellular automata for rock-paper-scissors games are related to Lotka-Volterra model. Simulations are usually performed by two methods local and global interactions. It is well known that the population dynamics with local interaction is stable, where all species coexist. In contrast, global interaction leads to extinction. So far, theori...

We study the dynamic motion of a vehicle moving through the series of traffic signals controlled by the position-dependent phase of power law. All signals are controlled by both cycle time and position-dependent phase. The dynamic model of the vehicular motion is described in terms of the nonlinear map. The vehicular motion varies in a complex mann...

We investigate the dynamic behavior of a shuttle bus controlled the speed when passengers come periodically at the origin. We propose the nonlinear-map model for the dynamics of the speed-controlled bus with the periodic inflow. The bus schedule is closely connected to the motion. The motion of the speed-controlled bus is affected by the periodic i...

We study the dynamic motion of a bus moving through the series of traffic signals where the bus stops at bus stops during a time. The dynamic state of the bus depends highly on both stoppage time at the bus stop and cycle time of the signal. It is found that the bus motion has two kinds of dynamic states: the one is the normal state and the other i...

We present the dynamic model for the multiple-vehicle collisions to take into account the velocity-dependent friction force. We study the effect of the velocity-dependent friction on the chain-reaction crash on a road. In the traffic situation, drivers brake according to taillights of the forward vehicle and the friction force depends highly on the...

We study the bus schedule in the shuttle bus transportation system controlled by speedup. The bus schedule is closely related to the dynamic motion of the bus. The motion of a shuttle bus depends on the inflow rate of passengers and the delayed speedup control. The delayed speedup control has an important effect on the dynamic motion of the bus. We...

We study the chain-reaction crash (multiple-vehicle collision) in high-visibility condition on a highway. In the traffic situation, drivers control their vehicles by both gear-changing and braking. Drivers change the gears according to the headway and brake according to taillights of the forward vehicle. We investigate whether or not the first coll...

We study the traffic behavior of a group of vehicles moving through a sequence of signals with irregular splits on a roadway. We present the stochastic model of vehicular traffic controlled by signals. The dynamic behavior of vehicular traffic is clarified by analyzing traffic pattern and travel time numerically. The group of vehicles breaks up mor...

We investigate the dynamic behavior of a shuttle bus serving repeatedly between two terminals when passengers come periodically at two terminals A and B. The period and phase of the periodic inflow of passengers at terminal A are different from those at terminal B. The dynamic behavior of the bus is highly affected by the difference between two per...

We present the dynamic model of the chain-reaction crash to take account of the vehicular size. Drivers brake according to taillights of the forward vehicle. We investigate the effect of the vehicular size on the chain-reaction crash (multiple-vehicle collision) in the traffic flow controlled by taillights. In the multiple-vehicle collision, the fi...

We study the dynamic motion of two competing elevators when elevators stop over at some floors. We present the dynamic model of elevators to take into account the stopover effect. The dynamics of the elevator traffic system is described by a pair of deterministic nonlinear maps. The motion of two elevators is determined by the five parameters: the...

We present the stochastic model for the jam formation at the tollgates of which the number is adjusted by synchronizing with the jam's length. We study the jam formation and its fluctuation in front of the adjustable tollgates on a highway. Controlling the number of tollgates has an important effect on the jam formation. The jams are classified int...

We study the dynamic motion of two elevators by an elevator-choice strategy of passengers. An elevator is correlated with another elevator by the elevator choice. The dynamics of the elevator traffic system is described by a pair of deterministic nonlinear maps. The motion of two elevators is determined by the three parameters: the passenger’s pref...

We present the dynamic model of the chain-reaction crash to take into account the irregularity of the perception–reaction time. When a driver brakes according to taillights of the forward vehicle, the perception–reaction time varies from driver to driver. We study the effect of the perception irregularity on the chain-reaction crash (multiple-vehic...

We study the traffic behavior in the asymmetric two-route traffic system with real-time information. In the asymmetric two-route system, the length on route A is different from that on route B and there exists a bottleneck on route A. We extend the symmetric two-route dynamic model to the asymmetric case. We investigate the asymmetric effects of th...

We study the chain-reaction crash (multiple-vehicle collision) in low-visibility condition on a road. In the traffic situation, drivers brake according to taillights of the forward vehicle. The first crash may induce more collisions. We investigate whether or not the first collision induces the chain-reaction crash, numerically and analytically. Th...

We present the stochastic model for queueing at two tollgates with line changing of vehicles to take into account fluctuating inflow and outflow. We study the jam formation and its transition in front of two tollgates on a highway. The line changing between two queues of vehicles has an important effect on the jam formation. The jams are classified...

We study the dynamics of two-route bus traffic system with two buses using a bus choice scenario. The two-route bus traffic system with real-time information is not consistent with the two-route vehicular traffic system but is similar to the vehicular system. The two-route bus traffic system is mimicked by the physically dynamic model. The dynamics...

We study the vehicular traffic controlled by two kinds of signals which are positioned with a periodic configuration. We propose a microscopic model to explore the driving behavior in the traffic system with two kinds of signals. The control method of traffic flow by the combination of two kinds of signals is proposed. The dynamic model is describe...

We study the multiple-vehicle collision when a vehicle changes from the first lane to the second lane on a two-lane highway. If a vehicle with high (low) speed on the first lane enters into the second lane, it may crash into the forward (rear) vehicle on the second lane and the crash may induce more collisions. We investigate whether or not the inc...

We study the traffic behavior in the case that there exists a bottleneck
on a route in the two-route traffic system with real-time information.
We introduce the bottleneck into the two-route dynamic model proposed by
Wahle et al. When there is a bottleneck on route A, a traffic jam
occurs behind the bottleneck on route A. The drivers try to avoid t...

We study the dynamics and control of an airplane in air transportation
between two airports. The dynamic models are presented for the airplane
schedule. The dynamics of an airplane is described by the piecewise map
and delayed map models. The characteristics of the nonlinear maps are
studied analytically and numerically. The airplane displays the c...

We study the dynamics of traffic system with two elevators using a elevator choice scenario. The two-elevator traffic system with real-time information is similar to the two-route vehicular traffic system. The dynamics of two-elevator traffic system is described by the two-dimensional nonlinear map. An elevator runs a neck-and-neck race with anothe...

We introduce the preference parameter into the two-route dynamic model
proposed by Wahle et al. The parameter represents the driver's
preference for the route choice. When the driver prefers a route, the
traffic flow on route A does not balance with that on route B. We study
the signal control for the unbalanced two-route traffic flow at the
tour-t...

We study the dynamic behavior in the elevator traffic controlled by
capacity when the inflow rate of passengers into elevators varies
periodically with time. The dynamics of elevators is described by the
piecewise map model combined with the circle map. The motion of the
elevators depends on the inflow rate, its period, and the number of
elevators....

We investigate the dynamic behavior of shuttle buses when passengers switch to another bus B on route B from bus A on route A. By switching from bus A to bus B, the outflow of passengers from route A (inflow of passengers into route B) changes to the periodic inflow of a square wave. The dynamics of the shuttle buses with the change is described by...

We study the effect of restart at signals on the vehicular traffic controlled by a series of signals. The Nagel–Schreckenberg model (NS model) and Fukui–Ishibashi model (FI model) are applied to the vehicular motion. In the FI model, the step-by-step acceleration is not taken into account but the acceleration effect is included in the NS model. It...

We study the multiple-vehicle collision when a vehicle decelerates suddenly in a single-lane traffic flow. The extended optimal velocity model is used for the vehicular motion to take into account the relative velocity. If a vehicle slows down suddenly and the following vehicle does not decelerate successfully, it crashes into the vehicle ahead wit...

We study the schedule of shuttle buses in the transportation system controlled by capacity. The bus schedule is closely related to the dynamic motion of buses. We present the nonlinear-map model for the dynamics of shuttle buses. The motion of shuttle buses depends on the inflow rate. The dependence of the fixed points on the inflow is derived. The...

We study the dynamical behavior of vehicular traffic through a series of traffic signals. The vehicular traffic is controlled with the use of the cycle time generated by a logistic map. Each signal changes periodically with a cycle time, and the cycle time varies from signal to signal. The nonlinear dynamic model of the vehicular motion is presente...

We study the traffic states and fundamental diagram of vehicular traffic controlled by a series of traffic lights using a deterministic cellular automaton (CA) model. The CA model is not described by a set of rules but is given by a difference equation. The vehicular traffic varies highly with both signal’s characteristics and vehicular density. Th...

We study the effect of signals on the vehicular traffic in the two-route system at the tour-time feedback strategy where the vehicles move ahead through a series of signals. The Nagel–Schreckenberg model is applied to the vehicular motion. The traffic signals are controlled by both cycle time and split. The tour times on two routes fluctuate period...

We study the dynamical behavior of counter traffic flow through a sequence of signals (traffic lights) controlled by a phase shift. There are two lanes for the counter traffic flow: the first lane is for east-bound vehicles and the second lane is for west-bound vehicles. The green-wave strategy is studied in the counter traffic flow where the phase...

We study the dynamic behavior of an elevator induced by a periodic inflow of passengers. An elevator schedule is closely related to the dynamics. We present the modified circle map model for the dynamics of the elevator traffic. The motion of the elevator depends on both loading parameter and inflow period. The elevator displays the periodic and ir...

We study a bus schedule in a shuttle bus transportation system controlled by capacity. The motion of shuttle buses depends on the inflow rate of passengers, the number of buses, and the delayed increase of buses. The bus schedule is closely related to the dynamic motion of buses. The delayed increase of buses has an important effect on the arrival...

We study the dynamic process of the multiple-vehicle collision when a vehicle stops suddenly in a traffic flow. We apply the optimal-velocity model to the vehicular motion. If a vehicle does not decelerate successfully, it crashes into the vehicle ahead with a residual speed. The collision criterion is presented by vi(t)/Δxi(t)→∞vi(t)/Δxi(t)→∞ if Δ...

We study the traffic behavior when a vehicle changes from the first lane to the second lane on a two-lane highway. The incoming vehicle decelerates or accelerates by interacting with the vehicle ahead or behind on the second lane. We apply the extended optimal velocity model to the vehicular motion to take into account the velocity difference. We i...

A model for the facing pedestrian traffic on a passage with a partition line at rush hour is developed. The model is described by a bi-directional cellular automaton (CA) model with four species. The CA model is not stochastic but deterministic. If the passage is congested and the local density is superior to the threshold, walkers to the east and...

We study the dynamical behavior and transitions of shuttle buses in a transportation system reducing energy consumption. We present the nonlinear-map model for the dynamics of M buses. The motion of shuttle buses depends on the loading parameter and the number. The dependence of the fixed points on the loading parameter is derived. The dynamical tr...

We study the dynamic behavior of vehicular traffic in a two-route system with a series of signals (traffic lights) at low density where the number of signals on route A is different from that on route B. We investigate the dependence of the tour time on the route for some strategies of signal control. The nonlinear dynamic model of a two-route traf...

We have studied the dynamic behavior of a bus in the shuttle bus transportation with a periodic inflow. A bus schedule is closely related to the dynamics. We present the modified circle map model for the dynamics of the shuttle bus. The motion of the shuttle bus depends on the loading parameter and the inflow period. The shuttle bus displays the pe...

We study the control and regularization of irregular motion of a vehicle moving through the series of traffic signals positioned at disordered intervals. All signals are controlled by both cycle time and phase shift. The nonlinear dynamic model of the vehicular motion controlled by signals is described in terms of the stochastic nonlinear map. The...

We have studied the dynamic behavior and dynamic transitions of elevators in a system for reducing energy consumption. We present a nonlinear-map model for the dynamics of M elevators. The motion of elevators depends on the loading parameter and their number M. The dependence of the fixed points on the loading parameter is derived. The dynamic tran...

We study the dynamic behavior of vehicular traffic through the series of traffic lights controlled by phase shift in two-dimensional (2D) city traffic network. The nonlinear-map model is presented for the vehicular traffic. The city traffic network is made of one-way perpendicular streets arranged in a square lattice with traffic signals where vert...

We study the traffic behavior when a vehicle changes from the first lane to the second lane on a two-lane highway. We apply the optimal velocity model to the vehicular motion. If the incoming vehicle does not decelerate successfully, it crashes into the vehicle ahead. On the other hand, if the headway between the incoming vehicle and the vehicle be...

When a vehicle moves through a series of green lights, avoiding red signals in a two-dimensional (2d) city traffic network, the vehicle describes a characteristic trajectory (green-light path) and the travel time has a minimal value. The green-light path depends on the cycle time, split, signal-control strategy, and fluctuations of vehicular speed....

We study the regularization of irregular motion of a vehicle moving through the sequence of traffic signals with a disordered configuration. Each traffic signal is controlled by both cycle time and phase shift. The cycle time is the same for all signals, while the phase shift varies from signal to signal by synchronizing with intervals between a si...

We study the traffic states and queuing occurring in traffic flow on a toll highway with multi-lane tollgates. The traffic states change with increasing density and varying number of tollgates. When the manual-collection vehicles sort themselves into the tollgates, the queues occur just in front of the tollgates if the vehicular density is higher t...

We study the traffic behavior in the facing and crossing traffic of pedestrians numerically and analytically. There are four kinds of walkers, those moving to east, to west, to north, and to south. We present the mean-field approximation (MFA) model for the four-directional traffic. The model is described in terms of four nonlinear difference equat...

We present a bi-directional cellular automaton (CA) model for facing traffic of pedestrians on a footway with a soft boundary. Walkers are allowed to go across a soft boundary to avoid congestion. The soft boundary effect is taken into account in addition to the excluded-volume effect and bi-directionality. The CA model is not stochastic but determ...

We study the traffic states and jams occurring in traffic flow on a two-lane toll highway with electronic and manual (traditional) tollgates. The electronic and manual collection vehicles sort themselves into their respective lanes at low density, while they mix at each tollgate at high density. We derive the fundamental diagrams (flow–density diag...

We study the freezing transition in the counter flow of pedestrians within the channel numerically and analytically. We present the mean-field approximation (MFA) model for the pedestrian counter flow. The model is described in terms of a couple of nonlinear difference equations. The excluded-volume effect and bi-directionality are taken into accou...

We present the model for the vehicular traffic in a city traffic network controlled by traffic lights. The vehicular motion on a selected path is described in terms of the nonlinear map. We study the dynamic behavior of vehicular traffic through the series of traffic lights on selected paths in two-dimensional (2d) traffic network. The vehicle move...

We present a bi-directional cellular automaton (CA) model for facing traffic of pedestrians on a wide passage. The excluded-volume effect and bi-directionality of facing traffic are taken into account. The CA model is not stochastic but deterministic. We study the jamming and freezing transitions when pedestrian density increases. We show that the...

A two-dimensional square lattice system, on which flexible, chainlike, self-driven objects move randomly but are drifted to a same direction, causing a unidirectional net flow, is investigated by numerical simulations. It is shown that the objects exhibit a freezing transition from a smoothly flowing state to a completely jammed state, in which the...

We study the dynamic behavior of vehicular traffic through a series of traffic lights on selected paths in a two-dimensional (2d) traffic network. The city traffic network is made of one-way perpendicular streets arranged in a square lattice with traffic signals where vertical streets are oriented upwards and horizontal streets are oriented rightwa...

## Citations

... Similarly, recovered individual (R) is divided into R N and R Q to distinguish the total and apparent infections. Model (2) resembles SIQR model 24,43,44 . In the latter case, infected agent (I) always transitions to Q with a constant probability (per unit time). ...

... We can model dynamic processes arising in information systems, such as traffic networks [57], by performing diffusion on the associated graph structures described in the previous section. Let Φ and T ij represent the matter to be diffused and the velocity at which matter travels between nodes i and j. ...

... The dynamics in such patchy environment have been studied by metapopulation and network models (Allesina and Levine, 2011;Barabási and Pósfai, 2016;Kivelä et al., 2014;Szczesny et al., 2014). Some authors have applied RPS game to metapopulation model (Czárán and Hoekstra, 2003;Nagatani et al., 2018a;Nagatani and Ichinose, 2020;Voit and Meyer-Ortmanns, 2019). A distinct point of the present paper is to set a lattice as a patch. ...

... (1)-(4) to hierarchical small-world networks. [27][28][29] Figure 1(a) shows the schematic of the first generation for the network. The boundaries are periodic. ...

... One of the very first and most celebrated example was to identify that heterogeneous population could be a cooperator supporting environment [24]. The heterogeneity may originate from an irregular interaction networks where some players have significantly more neighbors than for others hence they can collect higher payoff [25,26,27]. Diversity may also originate from different individual skills, like strategy teaching capacity or other social status, which could also result in similar effect [28,29,30,31,32]. ...

... [38][39][40] Recently, urban-scale traffic system in city networks has drawn great attention. [41][42][43][44][45][46] Urban-scale macroscopic fundamental diagrams have been studied by the conventional density equations. The macroscopic fundamental diagrams have not been derived successfully at extended densities except for low density because of numerical instability. ...

... In the present situation, all streets are essentially equivalent since our simulation is highly symmetric, and so each street is topologically equivalent to any other. In more complex city topologies, 25 we may expect situations in which streets become jammed for long times, requiring routing strategies that are based on much longer historical traffic data. We will analyze these effects in a future manuscript. ...

... That is why every country in the world is experiencing a rapid development of motorized transport. However, the development of road infrastructure has not kept pace, which implies several bottlenecks [23]. This phenomenon has a direct impact on the quality of life, especially in urban areas, which are still characterized by almost permanent traffic jams and high levels of air pollution [24]. ...

... The mathematical meta-model of diffusively coupled Lotka-Volterra systems on heterogenous graphs in presented in [21]. When the number of systems is limited to two, the model of diffusively coupled predator-prey systems reads [21]: ...

... The research is performed by implementing stochastic simulations based on the May-Leonard implementation of the rock-paper-scissors model, where the total number of individuals is not conserved [25][26][27][28][29][30][31]. The disease spreading happens between two immediate neighbours in the lattice, with a sick individual transmitting the virus to a healthy organism, irrespective of the species [24,[32][33][34]. Sick organisms act as viral vectors, with a probability of being cured or dying, according to the disease virulence; no organism or species is immune, even after being cured of the disease. ...