Feasible Equilibrium Research Articles

Short- and long-term optimality under sustainable threats in Contest Theory models of advertising and short-run competition

Abstract We model advertising with effects on different time scales for a duopoly in imperfect substitutes using elements from Contest Theory. Firms additionally compete in a short-run strategic variable, here price or quantity, allowing simultaneous or sequential decisions, or collusion in endogenously changing stage games. Strategic variables range from ‘slow’ (advertising), over ‘moderate’ (quantities) to ‘fast’ (prices). We find feasible rewards and equilibria for the limiting average reward criterion. Uniqueness of equilibrium is not guaranteed, and we introduce two criteria which act as natural refinements. We impose stage-game rationality, i.e., the firms play optimally in each stage game. Furthermore, in establishing threats, we require that punishment is sustainable, i.e., the punisher must have nonnegative long term average own profits to avoid bankruptcy.

Role of food conversion efficiency due to predation of prey populations in an eco-epidemiological model with Allee and refugia

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.

Effective Pest Control Through Natural Predators: A Dynamical Plant–Pest–Predator Model

The impact of pests on crop yield is a significant worry for farmers, and finding an effective strategy to control insect population growth has become a pressing matter. This study explores the dynamic analysis of a model that incorporates natural predators to control crop pests. In the research, an analysis is conducted on the model’s positivity and boundedness, followed by an examination of the existence and stability of the equilibrium points. The study focused on analyzing bifurcations at a biologically feasible equilibrium point. The parameters considered for bifurcation are the predation rate by pests and the consumption rate of natural predators. In addition, an optimal control technique has been employed to enhance the growth of the plant population through the utilization of Pontryagin’s maximum principle. Moreover, the impact of the nine parameters on the model is examined through sensitivity analysis using the partial rank correlation coefficient method. Numerical simulations are conducted to validate the analytical findings, revealing the occurrence of bifurcation and the positive impact of control on plant population growth.

Adolescent vaping behaviours: Exploring the dynamics of a social contagion model

Vaping, or the use of electronic cigarettes (e-cigarettes), is an ongoing issue for public health. The rapid increase in e-cigarette usage, particularly among adolescents, has often been referred to as an epidemic. Drawing upon this epidemiological analogy between vaping and infectious diseases as a theoretical framework, we present a deterministic compartmental model of adolescent e-cigarette smoking which accounts for social influences on initiation, relapse, and cessation behaviours. We use results from a sensitivity analysis of the model’s parameters on various response variables to identify key influences on system dynamics and simplify the model into one that can be analysed more thoroughly. We identify a single feasible endemic equilibrium for the proportion of smokers that decreases as social influence on cessation increases. Through steady state and stability analyses, as well as simulations of the model, we conclude that social influences from and on temporary quitters are not important in overall model dynamics, and that social influences from permanent quitters can have a significant impact on long-term system dynamics. In particular, we show that social influence on cessation can induce persistent recurrent smoking outbreaks.

Dynamic complexities in a predator–prey model with prey refuge influenced by double Allee effects

Within the context of a two-dimensional framework encompassing interacting species, an examination is conducted in this study on the double Allee effect and prey refuge, considering both species in the interaction. The stability of the feasible equilibrium of the system and diverse bifurcation patterns including codimension-one and codimension-two bifurcations are scrutinized through theoretical and numerical investigations, which reveals the complex dynamics induced by saturated functional response and double Allee effects. Additionally, one-parameter bifurcation diagrams and two-parameter bifurcation diagrams are constructed to intricately evaluate the system’s dynamics indicative of the presence of multiple attractors like bi-stability and tri-stability. Lastly, the sensitivity analysis is performed to delve into the effect of system parameters on species density, which indicates that the parameter η proportional to the conversion rate is the most sensitive parameter. A brief discussion further reveals that the model without double Allee effect reduces dynamic complexity.

A peer‐to‐peer joint energy and reserve market considering renewable generation uncertainty: A generalized Nash equilibrium approach

AbstractPeer‐to‐peer (P2P) energy trading enhances distribution network resilience by reducing energy demand from central power plants and enabling distributed energy resources to support critical loads after extreme events. However, adequate reserves from main grids are still required to ensure real‐time energy balance in distribution networks due to the uncertainty in renewable generation. This paper introduces a novel two‐stage joint energy and reserve market for prosumers, wherein local flexible resources are fully utilized to manage renewable generation uncertainty. In contrast to cooperative optimization methods, the interactions between prosumers are modelled as a generalized Nash game (GNG), considering that prosumers are self‐interested and should follow distribution network constraints. Then, linear decision rules are employed to ensure a feasible market equilibrium and develop a privacy‐preserving algorithm to guide prosumers toward the market equilibrium with a proven convergence. Finally, the numerical study on a modified IEEE 33‐power system demonstrates that the designed market effectively manages renewable generation uncertainty, and that the algorithm converges to the market equilibrium.

When can higher-order interactions produce stable coexistence?

Most ecological models are based on the assumption that species interact in pairs. Diverse communities, however, can have higher-order interactions, in which two or more species jointly impact the growth of a third species. A pitfall of the common pairwise approach is that it misses the higher-order interactions potentially responsible for maintaining natural diversity. Here, we explore the stability properties of systems where higher-order interactions guarantee that a specified set of abundances is a feasible equilibrium of the dynamics. Even these higher-order interactions which lead to equilibria do not necessarily produce stable coexistence. Instead, these systems are more likely to be stable when the pairwise interactions are weak or facilitative. Correlations between the pairwise and higher-order interactions, however, do permit robust coexistence even in diverse systems. Our work not only reveals the challenges in generating stable coexistence through higher-order interactions but also uncovers interaction patterns that can enable diversity.

A comprehensive study of spatial spread and multiple time delay in an eco-epidemiological model with infected prey

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations.

Dynamical behavior of a diseased predator-prey model with fear effect and prey harvesting

This article consists of a three-species food web model that has been developed by considering the interaction between susceptible prey, infected prey and predator species. It is assumed that susceptible prey species grow logistically in the absence of predators. It is assumed that predators consume susceptible and infected prey . We consider the effect of fear on susceptible prey due to predator species. Again, the harvesting of susceptible and infected prey has been considered. Furthermore, the predator consumes its prey in the form of Holling-type interactions. The positive invariance, positivity, and boundedness of the system are discussed. The conditions of all biologically feasible equilibrium points have been examined. The local stability of the systems around these equilibrium points is investigated. Furthermore, the occurrence of Hopf-bifurcation concerning the harvesting (h) of the system has been investigated. Finally, we demonstrate some numerical simulation results to illustrate our main analytical findings.

Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations

This research article investigates the interaction between prey and a generalist predator, considering the effect of hunting cooperation. The predator–prey interaction is modeled using a predator dependent functional response, specifically the Crowley–Martin type. System’s dynamics are explored using both analytical and numerical techniques. Feasible equilibria are analyzed, and their local stability is determined. Various bifurcations in the system are explored, and one-and two-parameter bifurcation structures are constructed to unveil complex dynamical behaviors. Our findings reveal that both prey and predator face extinction when the predator growth rate from alternative food sources exceeds certain threshold values. However, prey extinction is driven by the higher levels of hunting cooperation among predators, while the availability of external food sources enhances the system’s stability and persistence. Furthermore, we enhance the model by introducing seasonal perturbations, acknowledging the influence of seasonal variations on ecological parameters. The system exhibits diverse patterns in response to seasonality, contributing to a deeper understanding of the dynamics in predator–prey interactions.

Mathematical modeling and bifurcation analysis of a delayed eco-epidemiological model with disease in predator and linear harvesting

ABSTRACT This study proposes a three-dimensional prey-predator model with disease in predators, including the time delay due to the gestation of the predator population. The positive invariance of the solutions and the existence of equilibria in the proposed system have been explained. We also analyse the local stability of all feasible equilibrium points for the delayed and non-delayed systems. Additionally, the system’s Hopf bifurcation with regard to several parameters has been investigated. Further, the existence of periodic solutions through Hopf bifurcation is shown with respect to time delay, and it is also shown that time delay is crucial in order to control the dynamics of the system. Explicit formulas are presented to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution using the normal form theory and the centre manifold theorem. The outcome of the theoretical investigation is ultimately validated through numerical simulation.

Dynamical analysis of a network-based SIR model with saturated incidence rate and nonlinear recovery rate: an edge-compartmental approach.

A new network-based SIR epidemic model with saturated incidence rate and nonlinear recovery rate is proposed. We adopt an edge-compartmental approach to rewrite the system as a degree-edge-mixed model. The explicit formula of the basic reproduction number $ \mathit{\boldsymbol{R_{0}}} $ is obtained by renewal equation and Laplace transformation. We find that $ \mathit{\boldsymbol{R_{0}}} < 1 $ is not enough to ensure global asymptotic stability of the disease-free equilibrium, and when $ \mathit{\boldsymbol{R_{0}}} > 1 $, the system can exist multiple endemic equilibria. When the number of hospital beds is small enough, the system will undergo backward bifurcation at $ \mathit{\boldsymbol{R_{0}}} = 1 $. Moreover, it is proved that the stability of feasible endemic equilibrium is determined by signs of tangent slopes of the epidemic curve. Finally, the theoretical results are verified by numerical simulations. This study suggests that maintaining sufficient hospital beds is crucial for the control of infectious diseases.

A Stage-structure Leslie-Gower Model with Linear Harvesting and Disease in Predator

The growth dynamics of various species are affected by various aspects. Harvesting interventions and the spread of disease in species are two important aspects that affect population dynamics and it can be studied. In this work, we consider a stage-structure Leslie–Gower model with linear harvesting on the both prey and predator. Additionally, we also consider the infection aspect in the predator population. The population is divided into four subpopulations: immature prey, mature prey, susceptible predator, and infected predator. We analyze the existences and stabilities of feasible equilibrium points. Our results shown that the harvesting in prey and the disease in predator impacts the behavioral of system. The situation in the system is more complex due to disease in the predator population. Some numerical simulations are given to confirm our results.

Complex Dynamics and Fractional-Order Optimal Control of an Epidemic Model with Saturated Treatment and Incidence

In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.

An intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model

Intuitionistic fuzzy sets cannot consider the degree of indeterminacy (i.e., the degree of hesitation). This study presents an intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model. We examine the proposed model by assuming generalized trapezoidal intuitionistic fuzzy numbers for the initial condition. Feasible equilibrium points, along with their stability criteria, are evaluated. We describe the characteristics of intuitionistic fuzzy solutions and clarify the difference between strong and weak intuitionistic fuzzy solutions. Numerical simulations are performed in MATLAB to validate the model results.

Mathematical Study of One Prey and Two Competing Predators Considering Beddington-DeAngelis Functional Response with Distributed Delay

A nonlinear prey predator model is suggested and discussed to analyze the one prey and two competing predators considering Beddington-DeAngelis functional response with distributed delay. The analysis of stability of models is executed and the sufficient controls and suitable rules have been discussed for populations with the Beddington-DeAngelis functional response with distributed delay. All the feasible equilibria are observed and carried out for the stability rules. In the study, it has been observed that without delay at certain level of values the system would be stable. Additionally, we have supported our analytical conclusions with numerical visualisations.

Simulation and forecasting CTLs response effectualness against global dynamics of SARS-CoV-2/HIV coinfection

Quantifying the dynamical traits and harshness of SARS-CoV-2/HIV coinfection, a modified deterministic seven compartmental mathematical model is framed at cellular level encapsulating the interrelationships between epithelial cells (predominant target cells of COVID-19 infection), CD4+ T cells (target cells of HIV/AIDS infection), SARS-CoV-2 virions, HIV virions and HIV-specific CTL response. As a way to comprehend the kinetics of SARS-CoV-2/HIV coinfection and to provide prediction of model outcomes, the positivity, boundedness and existence conditions of the biologically feasible equilibrium points have been studied. The criteria of local stability and global stability of the coinfection system around the non-negative equilibrium points have been investigated and the analytical results are enhanced numerically. To measure the influence of the model parameters in SARS-CoV-2/HIV transmission dynamics, sensitivity analysis has been performed. Semi-relative sensitivity analysis demonstrates the maximum deviation of the concentration of infected epithelial cells due to changes in the model parameters and states that six parameters are most sensitive. The numerical simulation stipulates that the COVID-19 infection is likely to be died out from a HIV patient body by increasing the apoptosis of infected epithelial cells and that is possible through the proliferation of the CD4+ T cells count of an individual living with HIV via antiretroviral therapy (ART). Additionally, elevated concentration of CD4+ T cells in people living with HIV assists in heightening the CTLs response to combat COVID-19 and other opportunistic infections.

Rich dynamics of a bidirectionally linked immuno-epidemiological model for cholera.

Cholera is an environmentally driven disease where the human hosts both ingest the pathogen from polluted environment and shed the pathogen to the environment, generating a two-way feedback cycle. In this paper, we propose a bidirectionally linked immuno-epidemiological model to study the interaction of within- and between-host cholera dynamics. We conduct a rigorous analysis for this multiscale model, with a focus on the stability and bifurcation properties of each feasible equilibrium. We find that the parameter that represents the bidirectional connection is a key factor in shaping the rich dynamics of the system, including the occurrence of the backward bifurcation and Hopf bifurcation. Numerical results illustrate a practical application of our model and add new insight into the prevention and intervention of cholera epidemics.

Hysteresis and Hopf bifurcation: Deciphering the dynamics of an in-host model of SARS-CoV-2 with logistic target cell growth and sigmoidal immune response

Mathematical models of in-host dynamics of SARS-CoV-2 may reveal the intricate underlying dynamics of host-pathogen interaction and guide crucial steps to curtail progression of COVID-19 and other similar infectious diseases. While the existing models on SARS-CoV-2 have considered a wide range of host-pathogen interactions, at their core, these models are remarkably similar, particularly in the choice of mathematical functions used in modeling the interactions. Specifically, these models have employed mass action kinetics while modeling target-cell interaction with virus particles and immune response to virus, thus overlooking the role of saturated evolution of target epithelial cells as well as immune cells. Here we fill this gap by developing a more comprehensive model of in-host dynamics of SARS-CoV-2 that concurrently models target epithelial cell growth using a logistic equation and virus-immune interaction using Michaelis–Menten kinetics. We identify a feasible disease-free equilibrium whose local stability depends on ”reproduction number” and global stability on the replenishment rate as well as the removal rate of immune cells. We also obtain multiple endemic equilibria and perform their local and global stability analysis. Our detailed numerical investigation demonstrates that the infection rate and reproduction number govern several interesting dynamical behaviors such as forward bifurcation with and without hysteresis, Hopf bifurcation, and multiple endemic equilibria. To analyze the impact of parameters on the model system and disease dynamics, we investigate several sensitive parameters and compute their sensitivity indices. Finally, to understand the impact of anti-viral therapy on viral load, we identify a novel threshold of replication rate that must be crossed to enable virus proliferation. Our results thus provide novel insights into the in-host dynamics of SARS-CoV-2 and we expect this model will help in the development of better treatment strategies for COVID-19 and other such infectious diseases.

QUALITATIVE STUDY OF CONTROLLING PEST OF JATROPHA CURCAS FOR LINEAR FUNCTIONAL RESPONSE

In today's world, Jatropha curcas plants are a renewable energy feedstock for producing biodiesel to overcome the limitations of natural fuel. The alternative fuel biodiesel can be produced from the Jatropha plant as its seeds contain oil which is one of the important resources for biodiesel. Despite being toxic to many insects and animals, the Jatropha curcas plant is not pest and disease-free. In view of this, our research article presents the formulation and analysis of a mathematical model that describes the Jatropha plantation along with a prediction to control its natural pests by biological pesticides. We assume linear functional response of predators for susceptible pests, and infected pests as infected pests are weaker than susceptible pests and easy to catch. The dynamics of the system around each of the ecologically feasible equilibria are studied. The reduction of disease eradication and coexistence of predator and pest are observed around the predator-free and disease-free equilibrium respectively.