O. P. Misra

Role of Delay on Planktonic Ecosystem in the Presence of a Toxic Producing Phytoplankton

A mathematical model is proposed to study the role of distributed delay on plankton ecosystem in the presence of a toxic producing phytoplankton. The model includes three state variables, namely, nutrient concentration, phytoplankton biomass, and zooplankton biomass. The release of toxic substance by phytoplankton species reduces the growth of zooplankton and this plays an important role in plankton dynamics. In this paper, we introduce a delay (time-lag) in the digestion of nutrient by phytoplankton. The stability analysis of all the feasible equilibria are studied and the existence of Hopf-bifurcation for the interior equilibrium of the system is explored. From the above analysis, we observe that the supply rate of nutrient and delay parameter play important role in changing the dynamical behaviour of the underlying system. Further, we have derived the explicit algorithm which determines the direction and the stability of Hopf-bifurcation solution. Finally, numerical simulation is carried out to support the theoretical result.

Mathematical study of a Leslie–Gower-type tritrophic population model in a polluted environment

In this paper, the effect of toxicant on the dynamics of Leslie–Gower tritrophic food chain population model is studied. Two models have been studied to visualize the dynamical behaviour of prey and predator populations under toxicant stress. All the feasible equilibria of the systems are obtained and using stability theory the conditions are derived for the survival or extinction of species. From the analysis of the models it is observed that the effect of toxicant on predator population increases the equilibrium density of prey population. Finally, we support our analytical findings with numerical simulations.

Dynamics of one-prey two-predator system with square root functional response and time lag

Animals grouping together is one of the most interesting phenomena in population dynamics and different functional responses as a result of prey–predator forming groups have been considered by many authors in their models. In the present paper we have considered a model for one prey and two competing predator populations with time lag and square root functional response on account of herd formation by prey. It is shown that due to the inclusion of another competing predator, the underlying system without delay becomes more stable and limit cycles do not occur naturally. However, after considering the effect of time lag in the basic system, limit cycles appear in the case of all equilibrium points when delay time crosses some critical value. From the numerical simulation, it is observed that the length of delay is minimum when only prey population survives and it is maximum when all the populations coexist.

A mathematical model for the conservation of forestry biomass with an alternative resource for industrialization: a modified Leslie Gower interaction

Abstract In this paper we have considered an age structure forestry biomass characterized as pre-mature and mature population stages and industrialization as a state variable. Industries prefer to harvest trees as its favorite raw material to be used, but it may depend on the alternatives in the shortage or low growth of mature trees, based on this assumption a modified Leslie–Gower interaction is considered for industrial growth. We have obtained sufficient conditions for the persistence and global attractivity of the system by applying the differential inequality theory. Theory of differential equation is used to establish the stability of the system. Pontryagin’s principle is applied for the optimal control and found the solution in the interior equilibrium. For the verification of analytic results we have done numerical simulation, sensitivity of parameters involved in interior equilibrium is also examined.

Fate of dissolved oxygen and survival of fish population in aquatic ecosystem with nutrient loading: a model

In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of dissolved oxygen and survival or extinction of fish population in a nutrient enriched aquatic ecosystem. It is assumed in the model that there is an external constant input of nutrients (phosphorus and nitrogen) in the water body on account of anthropogenic activities. Stability analysis of the equilibria of the model is carried out and from the analysis it is shown that the fish population will survive at very low equilibrium level due to reduced concentration of dissolved oxygen and excessive presence of algal biomass on account of nutrient loading. Further, it is shown in this paper that the fish population tend to extinction due to decrease in the concentration of dissolved oxygen from its threshold level. Numerical simulations are also carried out in this paper to support the analytical results.

Modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey

In this paper, a mathematical model is proposed and analyzed to study the effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey population. The model is formulated by using the system of non linear ordinary differential equations. In the model, it is assumed that the growth rate of prey population is affected by organismal toxicant. In this paper, we have introduced distributed delay in the environmental toxicant in the model. The distributed delay differential equations, though simple in structure, possess a rich array of solutions. The models are being analyzed by using variational matrix and Liapunov functions. The conditions for local and global stability of the equilibrium points are obtained. A region of attraction is being found for global asymptotic stability of the equilibrium points. Also, a Hopf bifurcation analysis has been performed with respect to key parameters for non-trivial equilibrium points. Furthermore, we support our analytical findings with numerical simulations.

Dynamics of cholera epidemics with impulsive vaccination and disinfection

Waterborne diseases have a tremendous influence on human life. The contaminated drinking water causes water-borne disease like cholera. Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. A waterborne disease like cholera can also be controlled by using impulse technique. In this paper, we have proposed a delayed SEIRB epidemic model with impulsive vaccination and disinfection. We have studied the pulse vaccination strategy and sanitation to control the cholera disease. The existence and stability of the disease-free and endemic periodic solution are investigated both analytically and numerically. It is shown that there exists an infection-free periodic solution, using the impulsive dynamical system defined by the stroboscopic map. It is observed that the infection-free periodic solution is globally attractive when the impulse period is less than some critical value. From the analysis of the model, we have obtained a sufficient condition for the permanence of the epidemic with pulse vaccination. The main highlight of this paper is to introduce impulse technique along with latent period into the SEIRB epidemic model to investigate the role of pulse vaccination and disinfection on the dynamics of the cholera epidemics.