<abstract><p>Many studies have shown that faced with an epidemic, the effect of fear on human behavior can reduce the number of new cases. In this work, we consider an SIS-B compartmental model with fear and treatment effects considering that the disease is transmitted from an infected person to a susceptible person. After model formulation and proving some basic results as positiveness and boundedness, we compute the basic reproduction number $ \mathcal R_0 $ and compute the equilibrium points of the model. We prove the local stability of the disease-free equilibrium when $ \mathcal R_0 < 1 $. We study then the condition of occurrence of the backward bifurcation phenomenon when $ \mathcal R_0\leq1 $. After that, we prove that, if the saturation parameter which measures the effect of the delay in treatment for the infected individuals is equal to zero, then the backward bifurcation disappears and the disease-free equilibrium is globally asymptotically stable. We then prove, using the geometric approach, that the unique endemic equilibrium is globally asymptotically stable whenever the $ \mathcal R_0 > 1 $. We finally perform several numerical simulations to validate our analytical results.</p></abstract>
<abstract><p>This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, to the predator that is in cooperation with its species to hunt the preys. At first, we show that the system has non-negative solutions. The existence and uniqueness of the established fractional-order differential equation system were proven using the Lipschitz Criteria. In applying the theory of Routh-Hurwitz Criteria, we determine the stability of the equilibria based on specific conditions. The discretization of the fractional-order system provides us information to show that the system undergoes Neimark-Sacker Bifurcation. In the end, a series of numerical simulations are conducted to verify the theoretical part of the study and authenticate the effect of fear and fractional order on our model's behavior.</p></abstract>
Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, the modeling study is necessary to investigate the effects of global warming in predator–prey dynamics. This research paper analyzed the memory effect evaluated by Caputo fractional derivative on predator–prey interaction using an exponential fear function with a Holling-type II function in the presence of global warming effect on prey and predator species. It is assumed that the densities of prey and predator species decrease due to the increase of global warming. It is considered that both prey and predator species are contributing to the increase of global warming. Also, it is considered that global warming is increasing constantly and decreasing due to the natural decay rate. All possible equilibria of the system are determined, and the stability of the system around all equilibria points is investigated. Around the interior equilibrium point, the Hopf bifurcation is also theoretically and numerically studied. A number of numerical simulation results are presented to demonstrate the impacts of fear, fractional order and global warming on the behavior of the model. It is observed that the global warming effect on predator species may destabilize the system but ultimately the system may become stable. Again, it is obtained that the natural decay rate of global warming can stabilize the system initially but a higher decay rate may destabilized the system. It is also found that the fractional-order model is determined to be more stable than the integer-order model.
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