Impact of temperature dependent growth rate of phytoplankton, salinity induced grazing rate of zooplankton and gestation delay of phytoplankton and zooplankton differently are considered in the present study. Reproduction of predator after consuming prey is not instantaneous, but mediated by some time lag required for gestation. Here first gestation delay is considered in the numerical response term of phytoplankton and second one is at the numerical response term of zooplankton respectively. There is a very interesting interplay between these two delays. In accordance with previous studies, it is observed that delay destabilizes the system, in general and stability loss occurs via Hopf-bifurcation. In particular, we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. All the stability results of the system significantly depend upon the temperature and salinity of the environment. Numerical examples are also support the model assumption and analytical results.
The Hooghly-Matla estuarine complex is the unique estuarine system of the world. Nutrient from the litterfall enrich the adjacent estuary through tidal influence which in turn regulate the phytoplankton, zooplankton and fish population dynamics. Environmental factors regulate the biotic components of the system, among which salinity plays a leading role in the regulation of phytoplankton, zooplankton and fish dynamics of the estuary. In this article, a P ZF model is considered with Holling type-II response function. The present model considers salinity based equations on plankton dynamics of the estuary. The interior equilibrium is considered as the most important equilibrium state of this model. The model equations are solved both analytically and numerically using the real data base of Hooghly-Matla estuarine system. The essential mathematical features of the present model have been analyzed thorough local and global stability and the bifurcations arising in some selected situations. A combination of set of values of the salinity of the estuary are identified that helped to determine the sustenance of fish population in the system. The ranges of salinity under which the system undergoes Hopf bifurcation are determined. Numerical illustrations are performed in order to validate the applicability of the model under consideration.Mathematics Subject Classification: 92D25, 92D30, 92D40
Impact of photosynthesis rate of phytoplankton, salinity induced grazing rate of zooplankton and gestation delay of zooplankton and fish population are considered in the present study. Models with delay are much more realistic, as in reality time delays occur in almost every biological situation and assume to be one of the reasons of regular fluctuations in population density. Reproduction of predator after consuming prey is not instantaneous, but mediated by some time lag required for gestation. It is observed that there is stability switches and Hopf bifurcation occurs when the delay crosses some critical value. It is observed that the quantitative level of abundance of system population depends crucially on the delay parameter if the gestation period exceeds some critical value. Also, environmental stochasticity in the form of Gaussian whitenoise, plays a significant role to describe the system and its values. Numerical examples are also support the model assumption and analytical results.
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