Udai Kumar

@vitbhopal.ac.in

Assistant Professor Mathematics
VIT Bhopal University



           

https://researchid.co/udaikumar

EDUCATION

B, Hindu University Varanasi, 2010.
M. and Computing-Indian Institute of Technology, Guwahati, 2012.
M.Tech-Mathematics and Computing-Indian Institute of Technology, Patna, 2015.
Mathematics-National Institute of Technology, Patna, 2021.

RESEARCH INTERESTS

Applied Mathematics; Mathematical Biology, Disease, Modeling, Dynamical systems, Stochastic modeling

10

Scopus Publications

Scopus Publications



  • Combined impact of cannibalism and allee effect on the dynamics of a prey-predator model
    UDAI KUMAR and PARTHA SARATHI MANDAL
    World Scientific Pub Co Pte Ltd
    In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system.


  • SIRS epidemiological model with ratio-dependent incidence: Influence of preventive vaccination and treatment control strategies on disease dynamics
    Udai Kumar, Partha Sarathi Mandal, Jai Prakash Tripathi, Vijay Pal Bajiya, and Sarita Bugalia
    Wiley
    In this paper, we study an SIRS epidemic model with ratio‐dependent incidence rate function describing the mechanisms of infectious disease transmission. Impacts of vaccination and treatment on the transmission dynamics of the disease have been explored. The treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. Analysis shows that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, and (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the infective population. It is also obtained that model undergoes transcritical and saddle‐node bifurcations with respect to disease contact rate. Moreover, in the presence of treatment strategy, the model has multiple endemic equilibria and undergoes a backward bifurcation. The maximal capacity of treatment plays important roles on the disease dynamics of the model. The number of infected individuals decreases with respect to the maximal capacity of treatment, and the disease completely dies out from the system for the large capacity of the treatment when constant treatment strategy is applied. Further, it is also found that the spread of disease can be suppressed by increasing treatment rate. From sensitivity analysis, we have observed that the transmission and treatment rates are most sensitive parameters. The effects of different parameters on the disease dynamics have also been investigated via numerical simulation.


  • Global Dynamics in a Beddington–DeAngelis Prey–Predator Model with Density Dependent Death Rate of Predator
    Koushik Garain, Udai Kumar, and Partha Sarathi Mandal
    Springer Science and Business Media LLC

  • Impact of Additive Allee Effect on the Dynamics of an Intraguild Predation Model with Specialist Predator
    Udai Kumar and Partha Sarathi Mandal
    World Scientific Pub Co Pte Lt
    Many important factors in ecological communities are related to the interplay between predation and competition. Intraguild predation or IGP is a mixture of predation and competition which is a very basic three-dimensional system in food webs where two species are related to predator–prey relationship and are also competing for a shared prey. On the other hand, Allee effect is also a very important ecological factor which causes significant changes to the system dynamics. In this work, we consider a intraguild predation model in which predator is specialist, the growth of shared prey population is subjected to additive Allee effect and there is Holling-Type III functional response between IG prey and IG predator. We analyze the impact of Allee effect on the global dynamics of the system with the prior knowledge of the dynamics of the model without Allee effect. Our theoretical and numerical analyses suggest that: (1) Trivial equilibrium point is always locally asymptotically stable and it may be globally stable also. Hence, all the populations may go to extinction depending upon initial conditions; (2) Bistability is observed between unique interior equilibrium point and trivial equilibrium point or between boundary equilibrium point and trivial equilibrium point; (3) Multiple interior equilibrium points exist under certain parameters range. We also provide here a comprehensive study of bifurcation analysis by considering Allee effect as one of the bifurcation parameters. We observed that Allee effect can generate all possible bifurcations such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Taken bifurcation and Bautin bifurcation. Finally, we compared our model with the IGP model without Allee effect for better understanding the impact of Allee effect on the system dynamics.

  • Allee effect can simplify the dynamics of a prey-predator model
    Partha Sarathi Mandal, Udai Kumar, Koushik Garain, and Rakhi Sharma
    Springer Science and Business Media LLC

  • Impact of Allee effect on an eco-epidemiological system
    Udai Kumar, Partha Sarathi Mandal, and E. Venturino
    Elsevier BV