Dynamic Characteristics of Four Systems of Difference Equations with Higher Order

In this paper, we explore the global dynamical characteristics, boundedness, and rate of convergence of certain higher-order discrete systems of difference equations. More precisely, it is proved that for all involved respective parameters, discrete systems have a trivial fixed point. We have studied local and global dynamical characteristics at trivial fixed point and proved that trivial fixed point of the discrete systems is globally stable under respective definite parametric conditions. We have also studied boundedness and rate of convergence for under consideration discrete systems. Finally, theoretical results are confirmed numerically. Our findings in this paper are considerably extended and improve existing results in the literature.

Comparison Between Different Growth Functions of the Jatropha Curcas Plant with Random Attack Pattern of Whitefly

We have proposed here two deterministic models of Jatropha Curcas plant and Whitefly that simulate the dynamics of interaction between them where the distribution of Whitefly on plant follows Poisson distribution.In the first model growth rate of the plant is assumed to be in logistic form whereas in the second model it is taken as exponential form. The attack pattern and the growth of the whitefly are assumed as Holling type II function.The first model results a globally stable state and in the second one we find a globally attracting steady state for some parameter values,and a stable limit cycle for some other parameter values. It is also shown that there exist Hopf bifurcation with respect to some parameter values. The paper also discusses the question about persistence and permanence of the model. It is found that the specific growth rate of both the population and attack pattern of the whitefly governs the dynamics of both the models.

Global Analysis of a Time Fractional Order Spatio-Temporal SIR Model

We deal in this paper with a diffusive SIR epidemic model described by reaction-diffusion equations involving a fractional derivative. The existence and uniqueness of the solution are shown, next to the boundedness of the solution. Further, it has been shown that the global behavior of the solution is governed by the value of R0 , which is known in epidemiology by the basic reproduction number. Indeed, using the Lyapunov direct method it has been proved that the disease will extinct for R0 < 1 for any value of the diffusion constants. For R0 > 1, the disease will persist and the unique positive equilibrium is globally stable. Some numerical illustrations have been used to confirm our theoretical results.Subject classification: 26A33; 34A08; 92D30; 35K57.

Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession*

In this paper, we investigate the propagation dynamics of a reaction–diffusion competition model with seasonal succession in the whole space. Under the weak competition condition, the corresponding kinetic system admits a globally stable positive periodic solution ( u ^ ( t ) , v ^ ( t ) ) . By the method of upper and lower solutions and the Schauder fixed point theorem, we first obtain the existence and nonexistence of traveling wave solutions connecting (0, 0) to ( u ^ ( t ) , v ^ ( t ) ) . Then we use the comparison arguments to establish the spreading properties for a large class of solutions.

Stability and Hopf Bifurcation Analysis of a Delayed Innovation Diffusion Model with Intra-Specific Competition

This article is concerned with the diffusion of a sport in a region, and the innovation diffusion model comprising of population classes, viz. nonadopters class, information class and adopters class. A qualitative analysis is carried out to assess the global asymptotic stability of the interior equilibrium for null delay. It has also been proved that the parameter [Formula: see text] (age gaps among sportspersons) in the intra-specific competition between the new players and the senior players can even destabilize the otherwise globally stable interior equilibrium state and the coexistence of all the populations is possible through periodic solutions due to Hopf bifurcation. With the help of normal form theory and center manifold arguments, the stability of bifurcating periodic orbits is determined. Numerical simulations have been executed in support of the analytical findings.

On Stochastic-User-Equilibrium-Based Day-to-Day Dynamics

Researchers have proposed many different concepts and models to study day-to-day dynamics. Some models explicitly model travelers’ perceiving and learning on travel costs, and some other models do not explicitly consider the travel cost perception but instead formulate the dynamics of flows as the functions of flows and measured travel costs (which are determined by flows). This paper investigates the interconnection between these two types of day-to-day models, in particular, those models whose fixed points are a stochastic user equilibrium. Specifically, a widely used day-to-day model that combines exponential-smoothing learning and logit stochastic network loading (called the logit-ESL model in this paper) is proved to be equivalent to a model based purely on flows, which is the logit-based extension of the first-in-first-out dynamic of Jin [Jin W (2007) A dynamical system model of the traffic assignment problem. Transportation Res. Part B Methodological 41(1):32–48]. Via this equivalent form, the logit-ESL model is proved to be globally stable under nonseparable and monotone travel cost functions. Moreover, the model of Cantarella and Cascetta is shown to be equivalent to a second-order dynamic incorporating purely flows and is proved to be globally stable under separable link cost functions [Cantarella GE, Cascetta E (1995) Dynamic processes and equilibrium in transportation networks: Towards a unifying theory. Transportation Sci. 29(4):305–329]. Further, other discrete choice models, such as C-logit, path-size logit, and weibit, are introduced into the logit-ESL model, leading to several new day-to-day models, which are also proved to be globally stable under different conditions.

Modeling the Phenomenon of Xenophobia in Africa

In this study, we applied the principle of a competitive predator-prey system to propose a prey-predator-like model of xenophobia in Africa. The boundedness of the solution, the existence and stability of equilibrium states of the xenophobic model are discussed accordingly. As a special case, the coexistence state was found to be locally and globally stable based on the parametric conditions of effective group defense and anti-xenophobic policy implementation. The system was further analyzed by Sotomayor’s theory to show that each equilibrium point bifurcates transcritically. However, numerical proof showed period-doubling bifurcation, which makes the xenophobic situation more chaotic in Africa. Further numerical simulations support the analytical results with the view that tolerance, group defense and anti-xenophobic policies are critical parameters for the coexistence of foreigners and xenophobes.

Oscillations in Planar Deficiency-One Mass-Action Systems

Whereas the positive equilibrium of a planar mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, with centers or multiple limit cycles.

Consensus towards Partially Cooperative Strategies in Self-regulated Evolutionary Games on Networks

Cooperation is widely recognized to be challenging for the well-balanced development of human societies. The emergence of cooperation in populations has been largely studied in the context of the Prisoner's Dilemma game, where temptation to defect and fear to be betrayed by others often activate defective strategies. In this paper we analyze the decision making mechanisms fostering cooperation in the two-strategy Stag-Hunt and Chicken games, which include the mixed strategy Nash equilibrium, describing partially cooperative behavior. We find the conditions for which cooperation is asymptotically stable in both full and partial cases, and we show that the partially cooperative steady state is also globally stable in the simplex. Furthermore, we show that the last can be more rewarding than the first, thus making the mixed strategy effective, although people cooperate at a lower level with respect to the maximum allowed, as it is reasonably expected in real situations. Our findings highlight the importance of Stag-Hunt and Chicken games in understanding the emergence of cooperation in social networks.