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This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

Corona virus disease which has caused frustration in the human community remains the concern of the globe as every government struggles to defeat the pandemic. In this manuscript, we have extensively studied a new deterministic SEQIHR model of deadly Corona virus pandemic governed by nonlinear ordinary differential equations to provide deep insight into the dynamics of the disease. The purpose is to perform a complete mathematical analysis and the design of an optimal control strategy for the developed deterministic model by utilizing the preventive measures of quarantine and hospitalization. Some comprehensive mathematical techniques are employed to demonstrate the positivity and boundedness of solutions. Two main equilibrium points of the pandemic model are stated. To handle the future dynamical behavior of the pandemic, the threshold parameter value is computed using the next-generation method. The local and global asymptotic stability conditions of both the equilibria are obtained successfully. A numerical analysis to observe the effectiveness of quarantine and hospitalization strategies is performed and illustrated by graphs. We implemented the well known Non-Standard Finite Difference (NSFD) method to find the numerical solution of model. Numerical simulations are conducted to support our analytic results. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. The bifurcation nature of the developed model is examined. An optimal control problem is introduced for the proposed model to determine the best controls for the implemented hospitalization and quarantine strategies. With the attention of reducing the number of exposed and infected persons, an optimum control problem and its derived associated optimality conditions of Pontryagin type are explored. An important feature of this study is to employ NSFD method backward in time for the first time to solve optimal control problem instead of other standard methods. The extremals are obtained numerically. Moreover, we have shown the effectiveness of optimal controls on the human community to view more features of the state variables in the proposed model. • We develop a Covid-19 SEQIHR pandemic model and derive analytically several essential properties. • Necessary and sufficient conditions for the stability of equilibrium points are established. • The impact of quarantine and hospitalization on human community is demonstrated numerically. • An optimal control problem is introduced to determine the best controls for both the strategies. • Numerical simulations using an efficient NSFD scheme confirm the derived analytical results. [ABSTRACT FROM AUTHOR]

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin's maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level. [ABSTRACT FROM AUTHOR]

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique. [ABSTRACT FROM AUTHOR]