Narrow Results

Coupled map lattices are a popular and computationally simpler model of pattern formation in nonlinear systems. In this work, we investigate three-site interactions with linear multiplicative coupling in one-dimensional coupled logistic maps that cannot be decomposed into pairwise interactions. We observe the transition to synchronization and the transition to long-range order in space. We coarse-grain the phase space in regions and denote them by spin values. We use two quantifiers the flip rate $F(t)$ that quantify departure from expected band-periodicity as an order parameter. We also study a non-Markovian quantity, known as persistence $P(t)$ to study dynamic phase transitions. Following transitions are observed. (a) Transition to two band attractor state: At this transition $F(t)$ as well as $P(t)$ shows a power-law decay in the range of coupling parameters. Here all sites reach one of the bands. The $F(t)$ as well as $P(t)$ decays as power-law with the decay exponent ${\delta }_1=0.46$ and ${\eta }_1=0.28$, respectively. (b) The transition from a fluctuating chaotic state to a homogeneous synchronized fixed point: Here both the quantifiers $F(t)$ and $P(t)$ show power-law decay with decay exponent ${\delta }_2=1$ and ${\eta }_2=0.11$, respectively. We compare the transitions with the case, where pairwise interactions are also present. The spatiotemporal evolution is analyzed as the coupling parameter is varied.

This research rigorously investigates a stochastic Susceptible-Infectious-Susceptible (SIS) epidemic model incorporating a generalized nonlinear incidence rate. The study further expands the model scope by introducing white and Lévy noise components. Including jumps and noises enhances the likelihood of disease extinction within the population. Initially, we utilized a comprehensive mathematical analysis to establish the global stability of the equilibrium state devoid of the disease. Our findings demonstrate that the system maintains stability even in the presence of the disease. This serves as a fundamental insight into the dynamics of disease transmission. Moreover, we define a collection of adequate criteria that contribute to the continuous presence of the ailment, thereby guaranteeing its non-complete eradication. By identifying the critical factors that contribute to the sustained presence of the disease, we enhance our understanding of the long-term behavior of the epidemic. To validate our theoretical findings, we conduct numerical simulations. These simulations effectively demonstrate the accuracy and reliability of the presented results. By comparing the simulated data with our analytical predictions, we can confidently confirm the validity of our model and its applicability in practical scenarios.

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

This paper investigates dynamical behaviors of a stochastic plant–pollinator model with degenerate diffusion. We derive a sufficient and almost necessary condition for the permanence and extinction for plant–pollinator model, which characterizes the long-time behavior of the dynamic systems. Examples and numerical results are provided to illustrate our findings.

Predator–prey interactions are considered to be key components in the structural framework of an ecosystem. Giving rise to rich dynamics, these engagements have always drawn attention of budding researchers. Biological processes like Allee effect have been investigated by several researchers in the field of eco-epidemiology. Regulation of chaotic dynamics under the influence of refugia cannot be neglected either. Hence, in our present investigation, we have demonstrated our model under the cumulative impact of both Allee effect and refugia. We shall also observe the role of nutritional value in regulating ailment propagation. The stability conditions for the biologically feasible equilibrium points are constructed using local stability analysis. Conditions for existence of Hopf bifurcation along with uniform persistence and global asymptotic stability have been established. Rigorous numerical investigation has been performed so as to scrutinize the impact of force of infection and nutritional value on the dynamics of the proposed eco-epidemic model.

In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

In this paper, we study notions of persistent homotopy groups of compact metric spaces. We pay particular attention to the case of fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii–Plaut and Barcelo et al. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure which encodes more information than its persistent homology counterpart. We also consider the rationalization of the persistent homotopy groups and by invoking results of Adamaszek–Adams and Serre, we completely characterize them in the case of the circle. Finally, we establish that persistent homotopy groups enjoy stability in the Gromov–Hausdorff sense. We then discuss several implications of this result including that the critical spectrum of Plaut et al. is also stable under this notion of distance.

Point location is a fundamental primitive in Computational Geometry. In the plane it is stated as follows: Given a subdivision ℛ of the plane and a query point q, determine the region of ℛ containing q. We survey the work that has led to practical algorithms for the static version of the problem, and discuss current research on the corresponding dynamic algorithms.

Multiobjective oligopoly models are constructed. The objective of the first two models are to maximize profits and to maximize sales. In the third model, the objectives are to maximize profits and to minimize risk. Giving more weight to risk minimization decreased the profits. In all the three models, we found that the weight of the profit maximization has to be higher than a given threshold. Sufficient conditions for persistence of some multiobjective oligopolies are derived. Again, they require that the weight of profit maximization to be higher than certain value.

In the present paper, we will study the general telegraph coupled map lattice and finite amplitude instability will be discussed. The persistence in telegraph reaction diffusion equation (TRD) will be studied in two dimensional systems and we show the critical size effect where phenomena persist only if the domain is large enough. Some applications are introduced and we made a simulation on the small world network, which is more realistic than the ordinary lattices in many cases.

The Sato–Crutchfield equations are analytically and numerically studied. The Sato–Crutchfield formulation corresponds to losing memory. Then the Sato–Crutchfield formulation is applied for some different types of games including hawk–dove, prisoner's dilemma and the battle of the sexes games. The Sato–Crutchfield formulation is found not to affect the evolutionarily stable strategy of the ordinary games. But choosing a strategy becomes purely random, independent of the previous experiences, initial conditions, and the rules of the game itself. The Sato–Crutchfield formulation for the prisoner's dilemma game can be considered as a theoretical explanation for the existence of cooperation in a population of defectors.

Soap froths as typical disordered cellular structures, exhibiting spatial and temporal evolution, have been studied through their distributions and topological properties. Recently, persistence measures, which permit representation of the froth as a two-phase system, have been introduced to study froth dynamics at different length scales. Several aspects of the dynamics may be considered and cluster persistence has been observed through froth experiment. Using a direct simulation method, we have investigated persistent properties in 2D froth both by monitoring the persistence of survivor cells, a topologically independent measure, and in terms of cluster persistence. It appears that the area fraction behavior for both survivor and cluster persistence is similar for Voronoi froth and uniform froth (with defects). Survivor and cluster persistent fractions are also similar for a uniform froth, particularly when geometries are constrained, but differences observed for the Voronoi case appear to be attributable to the strong topological dependency inherent in cluster persistence. Survivor persistence, on the other hand, depends on the number rather than size and position of remaining bubbles and does not exhibit the characteristic decay to zero.

We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to five. The model includes a "social" local field which contains the magnetization at time t. The nearest neighbour quenched interactions are drawn from a binary distribution which is a function of the bond concentration, p. The decay of the persistence probability in the model depends on both the spatial dimension and p. We find no evidence of "blocking" in this model. We also discuss the implications of our results for possible applications in the social and economic fields. It is suggested that the absence, or otherwise, of blocking could be used as a criterion to decide on the validity of a given model in different scenarios.

In contact processes, the population can have heterogeneous recovery rates for various reasons. We introduce a model of the contact process with two coexisting agents with different recovery times. Type A sites are infected with probability $p$, only if any neighbor is infected independent of their own state. The type $B$ sites, once infected recover after $\tau$ time-steps and become susceptible at $(\tau +1)\mathrm{\text{th}}$ time-step. If susceptible, type $B$ sites are infected with probability $p$, if any neighbor is infected. The model shows a continuous phase transition from the fluctuating phase to the absorbing phase at $p=p_c$. The model belongs to the directed percolation universality class for small $\tau$. For larger values of $\tau =8,16$, the model belongs to the activated scaling universality class. In this case, the fraction of infected sites of either type shows a power-law decay over a range of infection probability $p<p_c$ in the absorbing phase. This region of generic power laws is known as the Griffiths phase. For $p>p_c$, the fraction of infected sites saturates. The local persistence $P_l(t)$ also shows a power-law decay with continuously changing exponent for either type of agent. Thus, the quenched disorder in timescales can lead to the temporal Griffiths phase in models that show a transition to an absorbing state.

Tasks of different nature and difficulty levels are a part of people’s lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable and justifiable tasks.

In this paper, we develop a model of business cycle fluctuations between two interacting open economies within the disequilibrium or non-market clearing paradigm. We analyze the main feedback mechanisms (Keynes, Mundell, Rose and Dornbusch) driving the dynamics and the conflict between their stabilizing and destabilizing tendencies and how these depend on certain key speeds of adjustment in the real and foreign exchange sectors. We explore numerically a variety of situations of interacting price cycles in the two countries, where the steady state is locally repelling, but where the overall dynamics are bounded in an economically meaningful domain by assuming downward money wage rigidity.

This paper investigates the persistency in the ex-post real interest rates in the presence of endogenous structural breaks for Australia, Austria, Belgium, Canada, Denmark, France, Germany, Ireland, Italy, the Netherlands, New Zealand, Norway, Switzerland, the UK and the USA using seasonally adjusted quarterly data. The procedure used in this study extends the previous research in the respect of investigating degree of persistency of the ex-post real interest rates series by allowing for possible process shifts at endogenously determined more than two structural breaks dates following the principles suggested by Lumsdaine and Papell (1997). The results from the study show that real interest rates are very persistent when such breaks are not taken into account. However, the findings also indicate low persistency in real interest rates for all countries when such breaks are allowed in the data-generating process. We find that endogenously determined structural breaks substantially reduce the degree of persistency of the real interest rate series, which has important theoretical implications as well.

By employing the factor augmented vector autoregression (FAVAR) model, this paper compares the role of macroeconomic and sector-specific factors in price movements for China and India, taking into account the features unique to developing economies. We find that fluctuations in the aggregated prices in China are more persistent than the underlying disaggregated prices. Compared to China, prices in India respond more promptly to macroeconomic and monetary policy shocks. We also show that the urban CPI in China responds more sharply than rural CPI when facing sector-specific shocks, while the opposite is valid for India.

In this paper, several sufficient conditions are established for the persistence and extinction in a Lotka–Volterra system with time delay. Based on the use of Lyapunov functionals techniques, necessary and sufficient conditions are also given for global asymptotic stability of the positive equilibrium for autonomous systems.

In this paper, a model of predator-prey with disease in food chain is investigated — where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis.