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In this paper, an Economic Order Quantity (EOQ) model of a fractional order with time-varying holding cost is discussed in detail with the help of numerical computations. As is well known, financial variables possess memories, which makes it appropriate for developing fractional modeling to describe the effect of memory in the financial system. This paper is concerned with the issues of deterioration of the items of the inventory system where the importance of time-varying holding cost on the memory-dependent inventory model has been studied. For the inclusion of the memory effect, fractional order derivative and fractional order integration have been used. Illustrative numerical examples and graphical presentations are exploited to verify the efficiency of the effect of fractional rate of change of inventory model and memory-affected costs. The numerical result shows that the output of memory-affected systems is different from the output of memory-less and low-memory-affected systems. Sensitivity analysis has been carried out in long memory effect as well as low memory effect to highlight the critical parameters of the two different systems. Sensitivity analysis clears that in the long memory effect, a small change in deterioration rate can disturb the system but in the low memory effect, such type of disturbances does not arises. At last, some conclusions are recommended. [ABSTRACT FROM AUTHOR]

In this research, we delve into the dynamics of an infected predator–prey system in the presence of fear and refuge, presenting a novel inclusion of treatment for infected individuals in this type of model. Through our analytical efforts, we establish a significant reproduction number that holds a pivotal role in determining disease extinction or persistence within the system. A noteworthy threshold value for this reproduction number delineates a boundary below which the infected population cannot endure in the system. It's important to note that a range of reproduction numbers leads to both disease-free and endemic scenarios, yet the stability of these situations is contingent upon the initial population sizes. Furthermore, our investigation extends to the exploration of various types of bifurcation-namely, Backward, Saddle-node, and Hopf bifurcations. These findings unravel the intricate and diverse dynamics of the system. Of particular significance is the derivation of an optimal control policy for treatment, augmenting the practical utility of our work. The robustness of our analytical findings is fortified through meticulous verification via numerical simulations. These simulations not only bolster the credibility of our analytical results but also enhance their accessibility. Our study unveils that fear, refuge, and treatment possess individual capabilities to eradicate the disease from the system. Notably, increasing levels of fear and refuge exert a passive influence on the elimination of the infected population, whereas treatment wields an active influence-a crucial insight that bolsters the foundation of our model. Furthermore, our investigation uncovers a spectrum of system dynamics including bistability, one-period, two-period, and multi-period/chaotic behavior. These discoveries contribute to a profound enrichment of the system's dynamic landscape. [ABSTRACT FROM AUTHOR]

In recent research, fear, refuge, and harvesting have been highlighted, but their combined impact must also be explored. We investigate the combined effects of these three ecologically important factors in the context of a prey–predator system using Holling type II as an interaction term. By using the parametric conditions, we can determine the existence of biologically feasible equilibria with their local and global stability. We investigate transcritical, Saddle‐node, Hopf, backward, Bogdanov–Takens bifurcations about different equilibria theoretically and numerically. We observe that the system is stable for lower and higher values of catchability effect and harvesting parameters. And the system shows stable behavior for a high value of fear and refuge parameters. Our model is extended by considering fear, refuge, and harvesting as time‐dependent parameters. Nonautonomous systems exhibit periodic solutions when the corresponding autonomous system is stable. Further, the nonautonomous system shows complex dynamics such as higher periods, chaos and bursting patterns whenever the associated autonomous system goes through limit cycle oscillations. [ABSTRACT FROM AUTHOR]

In this research, we delve into the dynamics of an infected predator–prey system in the presence of fear and refuge, presenting a novel inclusion of treatment for infected individuals in this type of model. Through our analytical efforts, we establish a significant reproduction number that holds a pivotal role in determining disease extinction or persistence within the system. A noteworthy threshold value for this reproduction number delineates a boundary below which the infected population cannot endure in the system. It's important to note that a range of reproduction numbers leads to both disease-free and endemic scenarios, yet the stability of these situations is contingent upon the initial population sizes. Furthermore, our investigation extends to the exploration of various types of bifurcation-namely, Backward, Saddle-node, and Hopf bifurcations. These findings unravel the intricate and diverse dynamics of the system. Of particular significance is the derivation of an optimal control policy for treatment, augmenting the practical utility of our work. The robustness of our analytical findings is fortified through meticulous verification via numerical simulations. These simulations not only bolster the credibility of our analytical results but also enhance their accessibility. Our study unveils that fear, refuge, and treatment possess individual capabilities to eradicate the disease from the system. Notably, increasing levels of fear and refuge exert a passive influence on the elimination of the infected population, whereas treatment wields an active influence-a crucial insight that bolsters the foundation of our model. Furthermore, our investigation uncovers a spectrum of system dynamics including bistability, one-period, two-period, and multi-period/chaotic behavior. These discoveries contribute to a profound enrichment of the system's dynamic landscape. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of a predator–prey model with cooperation and Allee effect considering both deterministic and stochastic approaches. The main aim of this paper is to investigate the effect of environmental fluctuation in a deterministic predator–prey model. During the analysis of the deterministic model, it is shown that the system has saddle-node point of co-dimension 1, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. To study the effect of environmental fluctuation, we use perturbation to the birth rate of prey and death rate of predator density by Gaussian white noise. The persistence of the model and the stationary distribution is shown by forming a suitable Lyapunov function. Finally, numerical simulations are performed to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (S I R) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov–Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended S I R model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript. [ABSTRACT FROM AUTHOR]

In the current study, we have formulated an S I epidemic model incorporating the influence of Allee effect on the dynamics of the model system in space and time. During the spread of the disease, inhibition among the susceptible and infected individuals is seen, which is measured using the Crowley–Martin type incidence function. Extensive analysis explores the system's stability and different bifurcation scenarios, including Hopf, saddle-node, transcritical, and Bogdanov–Takens. We also examine how these bifurcations react to parameter variations within the proposed model. The temporal model has further been extended to a spatial one to investigate the impact of the Allee effect on the formation of patterns for different values of the cross-diffusion coefficient. The well-posedness of the stationary solution and the global stability of the spatial system are derived. Also, the Turing instability, Hopf, and Turing bifurcation are calculated considering the transmission rate as critical parameter. In a heterogeneous environment, the spatial distribution of the susceptible population shows a complex structure with flat tabular surface and a few almost circular holes in surface plots for both strong and weak Allee effects. To further explore the role of Allee effect on pattern development for various cross-diffusion coefficient values, the temporal model has been spatially extended. Additionally, the transmission rate is taken into account while calculating the Turing instability and Hopf and Turing bifurcations. We ran numerical simulations to validate our analytical results for both spatial and nonspatial models. Our current model system can explain the recent occurrence of respiratory distress syndrome in endangered penguin species. [ABSTRACT FROM AUTHOR]