The interaction between prey and predator is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of predator and prey interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for a class of predator-prey systems with Holling type-III functional response. Positivity, boundedness, and persistence of solutions are investigated. Analysis of existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes a Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes a Neimark-Sacker bifurcation at its interior fixed point. A numerical simulation is provided to strengthen our theoretical discussion.

An Orthotropic Kelvin-like model is developed here to study wave dispersion relation along microtubules when they are embedded in viscoelastic material. Owing anisotropicity of elastic shell like microtubules, an orthotropic elastic shell model is derived while the surrounding environment of microtubules is modeled as Kelvin like material. Symmetrical and asymmetrical waves are studied in embedded microtubules. We compared the wave velocities for embedded and free microtubules as well as the comparison of wave velocities for isotropic and orthotropic microtubules are also given. Longitudinal, Torsional and Radial wave velocities are obtained, noticing that torsional and radial wave velocities are lower in embedded microtubules as compared to longitudinal wave velocities in embedded microtubules. The radial wave frequency is considerably low because in cylindrical microtubules, pressure is exerted from the surroundings. The decrease in wave velocities is due to strong mechanical coupling of microtubules with surrounding medium and this decrement is more obvious when the wave length is long.

Microtubules, the key components of cytoskeleton of all living cells, are important for maintaining the cell shape and transporting the cellular organelles. Understanding the mechanics of microtubules is very important for these functions. Mechanics of these components are greatly affected when they are embedded in cells. To understand the mechanical properties of microtubules in living cells, we developed an orthotropic-Kelvin like model and investigated the vibrational behavior when they are embedded in surrounding elastic medium. We considered them as orthotropic elastic shell and its surrounding elastic matrix as Kelvin model. We found that due to mechanical coupling of these components with the elastic medium, the flexural vibration is increased and radial frequencies in all modes are increased considerably while other vibrational modes are not affected that much.