In this article a multi-team delayed predator-prey model has been considered. There are two preys and one predator species in this model and the time delay appears for gestation of the predator. The essential mathematical features of the proposed model around the interior equilibrium point are studied in terms of local asymptotic stability by constructing a suitable Lyapunov functional and the condition for existence of Hopf-bifurcation is derived. By the assumption that the prey teams may help each other the effect of the rate of cooperation on the stability of the predator-prey model has been observed. Numerically a critical value for the delay parameter is obtained as a condition for Hopf-bifurcation.

In this work the (2+1) dimensional integrable Maccari system is studied. An effective algorithmic method—the multiplier approach for finding the conservation laws of system of partial differential equations is discussed and used to find the conservation laws for this system. Infinite number of conserved vectors are found which strongly support the integrability aspects of the Maccari system. Also new exact solution for this system is derived by using the extended homogeneous balance method. The obtained solutions are plotted and they show bright and dark soliton nature.

Dust magnetoacoustic waves have been examined in an inhomogeneous, bounded, cylindrical dusty plasma containing oppositely polarized dust particles. Considering polarization force, dust dynamics in r − θ plane is studied in the presence of inhomogeneous external magnetic field along z axis. At equilibrium, the dusty plasma components are supposed to follow Gaussian density distribution. Using reductive perturbation method (RPM), a variable coefficient cylindrical Kadomtsev–Petviashvili (VCCKP) equation has been derived. For weak azimuthal perturbation, an analytical solution, obtained by Zhang (“Exact solutions of a kdv equation with variable coefficients via exp-function method,” Nonlinear Dynam., vol. 52, nos. 1–2, pp. 11–17, 2008) using Exp-function method, is chosen. Phase velocity of dust magnetoacoustic wave is found to be modified by the density inhomogeneities, polarization force, dust charge state ratio and ion-to-electron temperature ratio. Spatio-temporal evolution of the dust number densities has been noticed. Existence of the compressive electromagnetic solitary waves is observed numerically for the chosen dusty plasma parameter range. The impacts of the inhomogeneity, polarization force, dust charge state ratio and ion-to-electron temperature ratio on the relative amplitude of the dust magnetoacoustic wave are also discussed.