Mathematical analysis of a fractional resource-consumer model with disease developed in consumer

The research presents a qualitative investigation of a fractional-order consumer-resource system with the hunting cooperation interaction functional and an infection developed in the resources population. The existence of the equilibria is discussed where there are many scenarios that have been distinguished as the extinction of both populations, the extinction of the infection, the persistence of the infection, and the two populations. The influence of the hunting cooperation interaction functional is also investigated where it can influence the existence of equilibria and their stability. A proper numerical scheme is used for building a proper graphical representation for the goal of confirming the theoretical results.

SCREENING IN SPACE: RICH AND POOR CONSUMERS IN A LINEAR CITY

Unlike standard models of monopolistic screening (second-degree price discrimination), we consider a situation where consumers are heterogeneous not only vertically, in their willingness to pay, but also horizontally, in their tastes or "addresses'' a la Hotelling's Linear City. For such a screening game, a novel model is composed. We formulate the game as an optimization program, prove the existence of equilibria, develop a method to calculate equilibria, and characterize their properties. Namely, the solution structure of the resulting menu of contracts can be either a "chain of envy'' like in usual screening or a number of disconnected chains. Unlike usual screening, "almost all'' consumers get positive informational rent. Importantly, the model can be extended to oligopoly screening.

Multi-Time Generalized Nash Equilibria with Dynamic Flow Applications

We propose a multi-time generalized Nash equilibrium problem and prove its equivalence with a multi-time quasi-variational inequality problem. Then, we establish the existence of equilibria. Furthermore, we demonstrate that our multi-time generalized Nash equilibrium problem can be applied to solving traffic network problems, the aim of which is to minimize the traffic cost of each route and to solving a river basin pollution problem. Moreover, we also study the proposed multi-time generalized Nash equilibrium problem as a projected dynamical system and numerically illustrate our theoretical results.

Sliding Dynamics and Bifurcations in the Extended Nonsmooth Filippov Ecosystem

We propose a nonsmooth Filippov refuge ecosystem with a piecewise saturating response function and analyze its dynamics. We first investigate some key elements to our model which include the sliding segment, the sliding mode dynamics and the existence of equilibria which are classified into regular/virtual equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point. In particular, we consider how the existence of the regular equilibrium and the pseudo-equilibrium are related. Then we study the stability of the standard periodic solution (limit cycle), the sliding periodic solutions (grazing or touching cycle) and the dynamics of the pseudo equilibrium, using quantitative analysis techniques related to nonsmooth Filippov systems. Furthermore, as the threshold value is varied, the model exhibits several complex bifurcations which are classified into equilibria, sliding mode, local sliding (boundary node and focus) and global bifurcations (grazing or touching). In conclusion, we discuss the importance of the refuge strategy in a biological setting.

Competitive Equilibrium with Indivisible Goods and Generic Budgets

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible even in the simple two-agent, single-item market. Yet it is easy to see that, once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper, we aim to extend this approach beyond the single-item case and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that, for agents with equal budgets, making the budgets generic—by adding vanishingly small random perturbations—ensures the existence of equilibrium. We further consider agents with arbitrary nonequal budgets, representing nonequal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among nonequal agents and that it exists in cases of interest (such as when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of nonequals.

An Eco-Epidemic Predator–Prey Model with Allee Effect in Prey

This article describes the dynamics of a predator–prey model with disease in predator population and prey population subject to Allee effect. The positivity and boundedness of the solutions of the system have been determined. The existence of equilibria of the system and the stability of those equilibria are analyzed when Allee effect is present. The main objective of this study is to investigate the impact of Allee effect and it is observed that the system experiences Hopf bifurcation and chaos due to Allee effect. The results obtained from the model may be useful for analyzing the real-world ecological and eco-epidemiological systems.

Dynamics Analysis of a Betel Nut Addiction Spreading Model on Scale-Free Networks

Medical research has shown that overeating betel nut can be addictive and can damage health. More serious cases may cause mouth cancer and other diseases. Even worse, people’s behavior habit of chewing betel nut may influence each other through social interaction with direct or indirect ways, such as face-to-face communication, Facebook, Twitter, microblog, and WeChat, which leads to the spreading phenomenon of betel nut addiction. In order to investigate the dynamic spreading characteristics of betel nut addiction, a PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model is presented on scale-free networks. The basic reproductive number R0 and equilibria are derived. Theoretical results indicate that the basic reproductive number R0 is significantly dependent on the topology of the underlying networks, and some influence parameters. The existence of equilibria is determined by the basic reproductive number R0. Furthermore, we prove that if R0<1 the addiction-elimination equilibrium is globally asymptotically stable. If R0>1, the betel nut addiction spreading is permanent, and the addiction-prevailing equilibrium is globally attractive. Finally, numerical simulations confirm the theoretical analysis results.

Dynamics of Duopoly Models with Undecided Clients under Decentralized Affine Feedback Advertising Policies

This paper extends the Deal-Vidal-Wolfe and Lanchester models of duopoly dynamics, which involve two populations, by explicitly introducing a third population of undecided users. An analysis of these extended models establishes conditions for the existence of equilibria, as well as their stability properties under different classes of advertising policies. This analysis also leads to the surprising result that the extended Vidale–Wolfe and Lanchester models, despite having different dynamics, under the general class of decentralized affine feedback advertising policies have equilibria in identical locations, with the same stability properties.