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In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In Ref. Upadhyay and Agrawal, 83(2016) 821-837, it was shown that the model possesses globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, even under the parametric restrictions derived in Ref. Upadhyay and Agrawal, 83(2016) 821-837. We prove blow-up in the delayed model, as well as the non-delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show that actually the initial data does not have to be very large, to induce blowup. The spatially explicit system is seen to possess non-Turing instability. We have also studied Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the complex dynamics of a one-dimensional spatial nonlinear coupled reaction-diffusion system with a Holling type IV functional response, akin to standard Michaelis-Menten inhibitory kinetics. Prey-taxis is included in a general reaction-diffusion equation to incorporate the active movement of predator species towards regions with high prey concentrations or if the predator is following some sort of cue (such as odor) to find the prey. We have carried out stability analysis of both the non-spatial model without diffusive spreading and of the spatial model. We performed extensive computer simulations to identify various parameter ranges for stable homogeneous solution. Our findings specifically elucidate the role of predator diffusion and prey-taxis in controlling emergent structures, and transitions towards spatio-temporal chaos. We observe that the increasing predator random movement and moderate value of prey-taxis stabilize the system. [ABSTRACT FROM AUTHOR]

A nonlinear mathematical model is proposed and analyzed to study the removal of primary and secondary air pollutants by precipitation in the atmosphere. The atmosphere, under consideration, consists of four interacting phases i.e. the rain droplets phase, the primary pollutants phase, the secondary pollutants phase and the combined phase of these pollutants absorbed in the rain droplets. The dynamics of these phases is governed by the ordinary differential equations with source, nonlinear interaction, conversion and removal terms. The proposed model is analyzed qualitatively using stability theory of differential equations. It is shown that under appropriate conditions, the pollutants can be removed from the atmosphere significantly and the removed amount would depend upon the rate of introduction of primary pollutants, rate of formation of secondary pollutants, rate of precipitation, rate of absorption and rate of falling rain droplets on the ground. Finally, computer simulations are performed to investigate the dynamics of model system. The results obtained in the paper are found to be in line with the experimental observations published in the literature. [ABSTRACT FROM AUTHOR]