12 results on '"UPADHYAY, RANJIT KUMAR"'
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2. Salton Sea: An ecosystem in crisis.
- Author
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Upadhyay, Ranjit Kumar, Kumari, Sarita, Kumari, Sangeeta, and Rai, Vikas
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ECOSYSTEMS , *HYDRAULICS , *MATHEMATICAL models , *BIFURCATION theory - Abstract
Salton Sea (a destination resort) is to be saved from being converted into a skeleton-filled wasteland. The critical amount of water flowing into the sea to maintain its level and salinity has been diverted since January 2018. This will lead to shrinking volumes and increasing salinities. Ecological consequences and public health impacts of altered conditions will be phenomenal. We design and analyze a minimal eco-epidemiological model to figure out future journey of this sea; a way station for fish-eating migratory birds. The mathematical model has been assembled in terms of prey-predator interaction. The salient feature of the proposed model is its seasonally varying contact rate which represents rate of conversion of susceptible fishes into infectives. We have analytically investigated the global stability, disease persistence and periodic solutions of the proposed model system. Susceptible prey-induced periodic solution is globally asymptotically stable when R ̄ 0 1 < 1 , otherwise unstable and hence disease persists for R ̄ 0 1 > 1. Global stability and Hopf bifurcation (HB) analysis help us extract parameter values to explore the dynamical behavior of the model system. Two-dimensional parameter scans and bifurcation diagrams reveal that the model displays propensity towards chaotic dynamics, which is associated with extinction-sized population densities. In the presence of stochastic external forces, this implies extinction of most of the fish species. This, in turn, suggests that resident birds will have to migrate to other destinations. The fish-eating migratory birds will be forced to switch over to invertebrates. Ecological consequences and public health impacts of this transition would be severe. Conservation groups are solicited to draw attention of the government to avert this impounding danger. It is important to plan for an ecosystem-wide transition such that impacts on birds and on human inhabitants living adjacent to the shrinking and salinizing sea are minimized. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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3. On the explosive instability in a three-species food chain model with modified Holling type IV functional response.
- Author
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Parshad, Rana D., Upadhyay, Ranjit Kumar, Mishra, Swati, Tiwari, Satish Kumar, and Sharma, Swarnali
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MATHEMATICAL models , *DIFFERENTIAL equations , *HOPF bifurcations , *MATHEMATICAL analysis , *PARAMETERS (Statistics) - Abstract
In earlier literature, a version of a classical three-species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
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- 2017
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4. Ecological chaos and the choice of optimal harvesting policy.
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Upadhyay, Ranjit Kumar and Tiwari, S.K.
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CHAOS theory , *ENVIRONMENTAL health , *FISHERY resources , *MATHEMATICAL analysis , *MATHEMATICAL models - Abstract
Harvested populations fluctuate because of two primary reasons: the inherent nonlinearity contained in the interactions among the constituent species and the forces of harvesting acting on the oscillatory dynamics resulting from species interactions. During the course of these fluctuations, population densities make excursions to low densities. When the ecological system executes chaotic motion, extinction-sized densities are common. Thus, it is imperative to design harvesting strategies which aim at maximizing economic gains giving due consideration to the ecological health of the concerned ecological system. The present study was designed and performed to figure out how to set harvesting strategies which optimize the economic gain. The choice of optimal harvesting policy can be made only if dynamical features of the concerned ecological system are well understood. In this paper, we have consider the temporal and spatiotemporal interactions among phytoplankton, zooplankton and fish population with Holling type II and Holling type III functional responses. We have calculated stability analysis of the model system and performed the numerical simulations for both non-spatial and spatial models to figure out the parameters that are responsible for chaotic dynamics of the model system. The temporal model system shows rich dynamics including limit cycles and chaos whereas spatial model shows different types of patterns for population distribution. In this work, we have taken the case study of Sundarban wetland ecosystem. We have carried out the analysis of maximum sustainable yield and identified the parameters that are responsible for good health of wetland ecosystem through numerical simulation results. [ABSTRACT FROM AUTHOR]
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- 2017
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5. A PREDATOR-PREY INTERACTION MODEL WITH SELF- AND CROSS-DIFFUSION IN AQUATIC SYSTEMS.
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UPADHYAY, RANJIT KUMAR, PATRA, ATASI, DUBEY, B., and THAKUR, N. K.
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AQUATIC ecology , *PREDATION , *COMPUTER simulation , *TURING test , *PREDATORY animals , *PREDATORS of fishes , *MATHEMATICAL models - Abstract
In this paper, the complex dynamics of a spatial aquatic system in the presence of self- and cross-diffusion are investigated. Criteria for local stability, instability and global stability are obtained. The effect of critical wavelength which can drive a system to instability is investigated. We noticed that cross-diffusion coefficient can be quite significant, even for small values of off-diagonal terms in the diffusion matrix. With the help of numerical simulation, we observed the Turing patterns (spots, strips, spot-strips mixture), regular spiral patterns and irregular patchy structures. The beauty and complexity of the Turing patterns are attributed to a large variety of symmetry properties realized by different values of predator's immunity, rate of fish predation and half saturation constant of predator population. [ABSTRACT FROM AUTHOR]
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- 2014
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6. Deterministic Chaos Versus Stochastic Oscillation in a Prey-Predator-Top Predator Model.
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Upadhyay, Ranjit Kumar, Banerjee, Malay, Parshad, Rana, and Raw, Sharada Nandan
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DETERMINISTIC chaos , *BIFURCATION theory , *PREDATION , *STOCHASTIC analysis , *OSCILLATIONS , *MATHEMATICAL models , *STABILITY (Mechanics) , *EQUILIBRIUM - Abstract
The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings. [ABSTRACT FROM AUTHOR]
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- 2011
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7. Modeling the spread of bird flu and predicting outbreak diversity
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Upadhyay, Ranjit Kumar, Kumari, Nitu, and Rao, V. Sree Hari
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STOCHASTIC processes , *MATHEMATICAL models , *EPIDEMIOLOGY , *AVIAN influenza - Abstract
Abstract: Avian influenza, commonly known as bird flu, is an epidemic caused by H5N1 virus that primarily affects birds like chickens, wild water birds, etc. On rare occasions, these can infect other species including pigs and humans. In the span of less than a year, the lethal strain of bird flu is spreading very fast across the globe mainly in South East Asia, parts of Central Asia, Africa and Europe. In order to study the patterns of spread of epidemic, we made an investigation of outbreaks of the epidemic in one week, that is from February 13–18, 2006, when the deadly virus surfaced in India. We have designed a statistical transmission model of bird flu taking into account the factors that affect the epidemic transmission such as source of infection, social and natural factors and various control measures are suggested. For modeling the general intensity coefficient , we have implemented the recent ideas given in the article Fitting the Bill, Nature [R. Howlett, Fitting the bill, Nature 439 (2006) 402], which describes the geographical spread of epidemics due to transportation of poultry products. Our aim is to study the spread of avian influenza, both in time and space, to gain a better understanding of transmission mechanism. Our model yields satisfactory results as evidenced by the simulations and may be used for the prediction of future situations of epidemic for longer periods. We utilize real data at these various scales and our model allows one to generalize our predictions and make better suggestions for the control of this epidemic. [Copyright &y& Elsevier]
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- 2008
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8. Dynamics and patterns of species abundance in ocean: A mathematical modeling study.
- Author
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Kumari, Sarita, Upadhyay, Ranjit Kumar, Kumar, Pramod, and Rai, Vikas
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MATHEMATICAL models , *FICK'S laws of diffusion , *SPATIAL systems , *PLANT size , *OCEAN , *PLANT competition - Abstract
In the complex and competitive world of oceans, different size of plants and animals exist. All of them compete for the limited resources; e.g., nutrients, sunlight, minerals etc. Size-specific and intraspecific predation is common among zooplankton. We design a model food chain and explore dynamics, and patterns of species abundance in ocean. The proposed mathematical model is based on a parameter; exponent of closure, m. A value of m less than 1 represents both size-specific and intraspecific predation among zooplankton. The mathematical model has been extended to include random movements of all the constituent populations by adding Fickian diffusion. Eigenvalues and amplitude equations are used to figure out relevant parameter spaces for numerical exploration. An analysis of the spatial system in the neighborhood of a critical parameter is performed using amplitude equation. Choosing appropriate control parameter from the Turing space, existence conditions for stable patterns are derived. Equal density contours were plotted for all the constituents of the model food chain. Epidemiological significance of these spatial patterns is provided. [ABSTRACT FROM AUTHOR]
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- 2021
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9. Restoration and recovery of damaged eco-epidemiological systems: Application to the Salton Sea, California, USA
- Author
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Upadhyay, Ranjit Kumar, Raw, S.N., Roy, P., and Rai, Vikas
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EPIDEMIOLOGY , *SYSTEMS biology , *CHAOS theory , *MATHEMATICAL models , *LYAPUNOV exponents , *NONLINEAR analysis - Abstract
Abstract: In this paper, we have proposed and analysed a mathematical model to figure out possible ways to rescue a damaged eco-epidemiological system. Our strategy of rescue is based on the realization of the fact that chaotic dynamics often associated with excursions of system dynamics to extinction–sized densities. Chaotic dynamics of the model is depicted by 2D scans, bifurcation analysis, largest Lyapunov exponent and basin boundary calculations. 2D scan results show that μ, the total death rate of infected prey should be brought down in order to avoid chaotic dynamics. We have carried out linear and nonlinear stability analysis and obtained Hopf-bifurcation and persistence criteria of the proposed model system. The other outcome of this study is a suggestion which involves removal of infected fishes at regular interval of time. The estimation of timing and periodicity of the removal exercises would be decided by the nature of infection more than anything else. If this suggestion is carefully worked out and implemented, it would be most effective in restoring the health of the ecosystem which has immense ecological, economic and aesthetic potential. We discuss the implications of this result to Salton Sea, California, USA. The restoration of the Salton Sea provides a perspective for conservation and management strategy. [Copyright &y& Elsevier]
- Published
- 2013
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10. The role of top predator interference on the dynamics of a food chain model
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Upadhyay, Ranjit Kumar, Naji, Raid Kamel, Raw, Sharada Nandn, and Dubey, Balram
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PREDATION , *PERFORMANCE evaluation , *FOOD chains , *MATHEMATICAL models , *BIFURCATION theory , *EXISTENCE theorems , *LYAPUNOV exponents - Abstract
Abstract: In this paper, the effects of top predator interference on the dynamics of a food chain model involving an intermediate and a top predator are considered. It is assumed that the interaction between the prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food depends on Beddington–DeAngelis type of functional response. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points are established. Number of the bifurcation and Lyapunov exponent bifurcation diagrams is established. It is observed that, the model has different types of attracting sets including chaos. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system. [Copyright &y& Elsevier]
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- 2013
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11. Modeling spatiotemporal dynamics of vole populations in Europe and America
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Upadhyay, Ranjit Kumar, Kumari, Nitu, and Rai, Vikas
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MATHEMATICAL models , *VOLES , *DIFFUSION , *ECOLOGY of predatory animals , *DATA analysis - Abstract
Abstract: The mathematical models proposed and studied in the present paper provide a unified framework to understand complex dynamical patterns in vole populations in Europe and North America. We have extended the well-known model provided by Hanski and Turchin by incorporating the diffusion term and spatial heterogeneity and performed several mathematical and numerical analyses to explore the dynamics in space and time of the model. These models successfully predicted the observed rodent dynamics in these regions. An attempt has been made to bridge the gap between the field and theoretical studies carried out by Turchin and Hanski (1997) and Turchin and Ellner (2000) . Simulation experiments, mainly two-dimensional parameter scans, show the importance of spatial heterogeneity in order to understand the poorly understood fluctuations in population densities of voles in Fennoscandia and Northern America. This study shed new light upon the dynamics of voles in these regions. The nonlinear analysis of vole data suggests that the dynamical shift is from stability to chaos. Diffusion driven model systems predict a new type of dynamics not yet observed in the field studies of vole populations carried out so far. This has been termed as chaotic in time and regular in space (CTRS). We observed CTRS dynamics in several simulation experiments. This directs us to expect that dynamics of this animal would be de-correlated in time and simultaneously mass extinctions might be possible at many spatial locations. [Copyright &y& Elsevier]
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- 2010
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12. Short-term recurrent chaos and role of Toxin Producing Phytoplankton (TPP) on chaotic dynamics in aquatic systems
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Upadhyay, Ranjit Kumar and Rao, V. Sree Hari
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PHYTOPLANKTON populations , *CHAOS theory , *MATHEMATICAL models , *PLANT toxins , *ZOOPLANKTON , *AQUATIC plants , *AQUATIC biology - Abstract
Abstract: We propose a new mathematical model for aquatic populations. This model incorporates mutual interference in all the three populations and an extra mortality term in zooplankton population and also taking into account the toxin liberation process of TPP population. The proposed model generalizes several other known models in the literature. The principal interest in this paper is in a numerical study of the model’s behaviour. It is observed that both types of food chains display same type of chaotic behaviour, short-term recurrent chaos, with different generating mechanisms. Toxin producing phytoplankton (TPP) reduces the grazing pressure of zooplankton. To observe the role of TPP, we consider Holling types I, II and III functional forms for this process. Our study suggests that toxic substances released by TPP population may act as bio-control by changing the state of chaos to order and extinction. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
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