Mehbuba Rehim

Boundedness and stability for nonautonomous cellular neural networks with delay

In this paper, a class of nonautonomous cellular neural networks is studied. By constructing a suitable Liapunov functional, applying the boundedness theorem for general functional-differential equations and the Banach fixed point theorem, a series of new criteria are obtained on the boundedness, global exponential stability, and existence of periodic solutions.

PERMANENCE, AVERAGE PERSISTENCE AND EXTINCTION IN NONAUTONOMOUS SINGLE-SPECIES GROWTH CHEMOSTAT MODELS

Non-autonomous single-species growth chemostat models with general response functions are considered. In the models, the dilution rate and removal rate are allowed to be different from each other. A series of new criteria on the boundedness, permanence, persistence, average persistence and extinction of the population is established. In particular, when models degenerate into the almost periodic case, the equivalences of the permanence, persistence and average persistence of species are obtained. These results improve and extend some well-known corresponding results obtained in Refs. 1, 14 and 17.

On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models.

Mathematical analysis of a nutrient–plankton system with delay

A mathematical model describing the interaction of nutrient–plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.