Dipak Kesh

A Mathematical Approach to Control Cutaneous Leishmaniasis Through Insecticide Spraying

Leishmaniasis is one sort of the vector born diseases, which is transmitted to human or animals by sand-fly bites. Single cell parasite grounds skin infection called Cutaneous Leishmaniasis. In our current research article, we consider a simple mathematical model to analyze the disease dynamics of Cutaneous Leishmaniasis consisting susceptible and infected populations of human and vector. In this article, our focus is to reduce the vector population so that the disease may be controlled. Here we try to explore the effect of insecticide for controlling the vector population through impulsive mode. We have established the efficiency of the insecticide spraying, which contributes a greater impact on the dynamics to move the system towards disease free situation.

Succession processes in a food web of a two autotroph--one herbivore system.

This paper deals with the succession process of a food web model consisting of one herbivore, two autotrophs and available nutrient in the environment in a closed nutrient flux. The model provides a way of describing successional changes in the form of species replacement with increasing nutrient levels. It is shown that distinct threshold (with upper and lower) values of nutrient are required for progression of succession process.

Ratio dependent predation :A bifurcation analysis

In this study, we have considered a prey-predator model reflecting the predator interference with discrete time delay. This delay is regarded as the lag due to gestation. In absence of delay, the criteria for existence of interior equilibrium and its global stability are derived. By choosing the delay as a bifurcation parameter, we have shown that a Hopf bifurcation may occur when the delay passes its critical value. Finally, we have derived the criteria for stability switches and verified the results through computer simulation.

Succession in a three-species food-chain model

Abstract In this study we consider a succession process of three-species food-chain model in a closed biocenosis in the sense of available resources. Under this assumption it is shown that the dynamics results in the stable composition of abiotic and biotic components of the food chain with increase in resources. Here the uptake rate of limiting biogenic elements by the prey to be of Lotka-Volterra type and uptake function of predators to be in the general form. The qualitative character of the gradual changes in climacteric composition is determined from relationships among ecological parameters.

Deterministic and stochastic analysis of an eco-epidemiological model

Chronic wasting disease (CWD) is a contagious prion disease among the deer family that has the potential to disrupt the ecosystems where deer occur in abundance. To understand the dynamics of this emerging infectious disease, we consider a simple eco-epidemic model where the host population is infected by CWD. Boundedness of the system is established. The structure of equilibria and their linearized stability are investigated. The persistence condition is discussed. By constructing a suitable Lyapunov function, we discuss the global stability of the endemic equilibrium. Local bifurcation (transcritical) around the boundary equilibria is developed. Sufficient conditions for the existence of Hopf-bifurcation are derived. Further, we have also introduced white type of noise into the system to investigate stochastic stability. This suggests that the deterministic model is robust with respect to stochastic perturbation. Some numerical simulations are performed to validate our results.

Influence of competition in modelling dynamics of plant defense with induced volatile

Being sessile organism, plants have to deal with environmental stresses like herbivore attack, competition with neighboring plants in different ways. Plant produced volatile plays a major role in plant defense. After herbivore attack, plant induced volatile attracts the natural enemies of herbivore. So, natural enemies enter the plant herbivore system. There are lots of works on growth and defense for individual plants whereas it is essential to analysis the trade-off between growth and defense in context of plant as a member of plant community. In this paper we address the effect of plant–plant competition and herbivore induced volatile in four species plant–herbivore–carnivore system. We outline the various role of volatile in plant competition and defense. Besides local stability, we also study the global behavior at the densities of all population at equilibrium point. We discuss the conditions for long term survival of all population. We analyze different types of bifurcations at the various steady states. Finally, numerical simulations are carried out to illustrate our obtained results.

Persistence and global stability of a Leslie-Gower predator-prey refuge system with a competitor for the prey

This paper analyzes the effect of refuge on the dynamics of a Leslie-Gower predator-prey model in which one predator feeds on one of two competing species. Existence conditions for equilibrium points are discussed. By using differential inequality argument, we developed persistence criterion. Sufficient condition for global stability of the unique positive equilibrium point is derived. Different type of local bifurcation near the equilibrium points has been investigated. The role of refuges have been shown on equilibrium densities of prey, competitor for prey and predator respectively. The results establish the fact that the effects of refuges used by prey increase the equilibrium density of prey population under certain restrictions, whereas opposite hold for competitor of prey population. However equilibrium density of predator may decrease or increase by increasing the amount of prey refuge. Some numerical simulations are performed to validate the results obtained. https://doi.org/10.28919/cmbn/3382

Qualitative analysis of two resource-based food-web models of three interacting species

Abstract Two models are developed and analysed for resource-based food webs. In model 1, the food web originates from abiotic resources but the environment is closed with respect to it; while, in model 2, the available resource is biotic in nature but the environment is open with respect to resource supply. Community structure as well as the physical properties are taken into consideration in developing these models. It is shown that model I is more fragile in nature, the model may even be impermanent and the dynamics of the system depend critically on the available conserved resources. In model 2, it is shown that the model is globally asymptotically stable depending on the parameters of the community structure.