An analysis of the stability and bifurcation of a discrete-time predator-prey model with the slow-fast effect on the predator.

In many environments, predators have significantly longer lives and meet several generations of prey, or the prey population reproduces rapidly. The slow-fast effect can best describe such predator-prey interactions. The slow-fast effect ε can be considered as the ratio between the predator's linear death rate and the prey's linear growth rate. This paper examines a slow-fast, discrete predator-prey interaction with prey refuge and herd behavior to reveal its complex dynamics. Our methodology employs the eigenvalues of the Jacobian matrix to examine the existence and local stability of fixed points in the model. Through the utilization of bifurcation theory and center manifold theory, it is demonstrated that the system undergoes period-doubling bifurcation and Neimark-Sacker bifurcation at the positive fixed point. The hybrid control method is utilized as a means of controlling the chaotic behavior that arises from these bifurcations. Moreover, numerical simulations are performed to demonstrate that they are consistent with analytical conclusions and to display the complexity of the model. At the interior fixed point, it is shown that the model undergoes a Neimark-Sacker bifurcation for larger values of the slow-fast effect parameter by using the slow-fast effect parameter ε as the bifurcation parameter. This is reasonable since a large ε implies an approximate equality in the predator's death rate and the prey's growth rate, automatically leading to the instability of the positive fixed point due to the slow-fast impact on the predator and the presence of prey refuge.

Steady Squeezing Flow of Magnetohydrodynamics Hybrid Nanofluid Flow Comprising Carbon Nanotube-Ferrous Oxide/Water with Suction/Injection Effect.

The main purpose of the current article is to scrutinize the flow of hybrid nanoliquid (ferrous oxide water and carbon nanotubes) (CNTs + Fe3O4/H2O) in two parallel plates under variable magnetic fields with wall suction/injection. The flow is assumed to be laminar and steady. Under a changeable magnetic field, the flow of a hybrid nanofluid containing nanoparticles Fe3O4 and carbon nanotubes are investigated for mass and heat transmission enhancements. The governing equations of the proposed hybrid nanoliquid model are formulated through highly nonlinear partial differential equations (PDEs) including momentum equation, energy equation, and the magnetic field equation. The proposed model was further reduced to nonlinear ordinary differential equations (ODEs) through similarity transformation. A rigorous numerical scheme in MATLAB known as the parametric continuation method (PCM) has been used for the solution of the reduced form of the proposed method. The numerical outcomes obtained from the solution of the model such as velocity profile, temperature profile, and variable magnetic field are displayed quantitatively by various graphs and tables. In addition, the impact of various emerging parameters of the hybrid nanofluid flow is analyzed regarding flow properties such as variable magnetic field, velocity profile, temperature profile, and nanomaterials volume fraction. The influence of skin friction and Nusselt number are also observed for the flow properties. These types of hybrid nanofluids (CNTs + Fe3O4/H2O) are frequently used in various medical applications. For the validity of the numerical scheme, the proposed model has been solved by another numerical scheme (BVP4C) in MATLAB.

Natural convection heat transfer in an oscillating vertical cylinder.

This paper studies the heat transfer analysis caused due to free convection in a vertically oscillating cylinder. Exact solutions are determined by applying the Laplace and finite Hankel transforms. Expressions for temperature distribution and velocity field corresponding to cosine and sine oscillations are obtained. The solutions that have been obtained for velocity are presented in the forms of transient and post-transient solutions. Moreover, these solutions satisfy both the governing differential equation and all imposed initial and boundary conditions. Numerical computations and graphical illustrations are used in order to study the effects of Prandtl and Grashof numbers on velocity and temperature for various times. The transient solutions for both cosine and sine oscillations are also computed in tables. It is found that, the transient solutions are of considerable interest up to the times t = 15 for cosine oscillations and t = 1.75 for sine oscillations. After these moments, the transient solutions can be neglected and, the fluid moves according with the post-transient solutions.