Browsing by Author "Dubey, Balram"

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Bifurcation analysis and spatiotemporal dynamics in a diffusive predator–prey system incorporating a holling type II functional response
(World Scientific, 2024) Dubey, Balram
This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems.
Bifurcation Analysis of a Leslie-Gower Prey-Predator Model with Fear and Cooperative Hunting
(Springer, 2022-10) Dubey, Balram
The current work examines the dynamical features of a Leslie-Gower prey-predator model. The effects of fear and group defense among prey with the mechanism of cooperative hunting by predators are incorporated. The existence and uniqueness of the interior equilibrium are explained. We obtained sufficient conditions for the local and global stability behavior. With regard to the fear parameter and cooperation strength parameter, the proposed system undergoes Hopf-bifurcation, transcritical bifurcation, and saddle-node bifurcation. Moreover, the system exhibits the property of bi-stability between two interior equilibrium points. The basin of attraction of these points is also plotted. All theoretical results are verified numerically by MATLAB R2021a.
Bifurcation and chaos in a delayed eco-epidemic model induced by prey configuration
(Elsevier, 2022-12) Dubey, Balram
The present study assumes that infectious disease among prey classifies them as susceptible (S) and infected (I) prey. When strong (susceptible) prey forms a herd to defend against the predator, it can reverse their role. This paper focuses on spotlighting the impact of disease, generalized herd shape, predator mortality due to prey group, the attack rate for healthy prey, and time delay. These factors crucially govern the system’s dynamics like Hopf-bifurcation, transcritical bifurcation, and chaos. The sketch of the maximum Lyapunov exponent confirms the chaotic nature. Extensive theoretical and numerical analysis reveals the existence and stability of steady-states in the presence or absence of delay. This study finds out that disease spread in prey can enhance the chances of predator survival. Furthermore, sensitivity analysis demonstrates the influence of some epidemic and ecological parameters on the reproduction numbers of the proposed eco-epidemic system
Bifurcation and Chaos in a Diffusive Prey–Predator Model Incorporating Fear Effect on Prey and Team Hunting by Predator with Anti-Predation Response Delay
(World Scientific, 2025) Dubey, Balram
In this paper, we scrutinize the dynamics of a temporal and spatiotemporal prey–predator model incorporating the fear effect on prey and team hunting by the predator. Additionally, we explore the influence of delayed anti-predation response. The analysis includes discussions on well-posedness, local stability, and various bifurcations such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcations. The impact of fear cost and delay parameters on model dynamics is investigated by considering them as bifurcation parameters. We investigate how bifurcation values change with varying parameters by exploring different bi-parameter planes. It is observed that the system transitions into chaotic behavior through Hopf bifurcation for significant anti-predation response delay. The positivity of the maximal Lyapunov exponent indicates the confirmed characteristics of chaotic behavior. Furthermore, within the spatiotemporal model framework, a thorough analysis of local and global stability is provided, including the establishment of criteria for identifying Turing instability in cases of self- and cross-diffusion. Various stationary and dynamic patterns are elucidated as diffusion coefficients vary, showcasing the diverse dynamics of the spatiotemporal model. In order to illustrate the dynamic characteristics of the system, a series of comprehensive numerical simulations are conducted. The discoveries outlined in this paper could prove advantageous for understanding the biological implications resulting from the examination of predator–prey relationships.
Bifurcation and chaotic dynamics in a spatiotemporal epidemic model with delayed optimal control, stochastic process, and sensitivity analysis
(AIP, 2025-03) Dubey, Uma S.; Dubey, Balram
This study introduces an epidemic model with a Beddington–DeAngelis-type incidence rate and Holling type II treatment rate. The Bedding- ton–DeAngelis incidence rate is used to evaluate the effectiveness of inhibitory measures implemented by susceptible and infected individuals. Moreover, the choice of Holling type II treatment rate in our model aims to assess the impact of limited treatment facilities in the context of disease outbreaks. First, the well-posed nature of the model is analyzed, and then, we further investigated the local and global stability analysis along with bifurcation of co-dimensions 1 (transcritical, Hopf, saddle-node) and 2 (Bogdanov–Takens, generalized Hopf) for the system. Moreover, we incorporate a time-delayed model to investigate the effect of incubation delay on disease transmission. We provide a rigorous demonstration of the existence of chaos and establish the conditions that lead to chaotic dynamics and chaos control. Additionally, sensitivity analysis is performed using partial rank correlation coefficient and extended Fourier amplitude sensitivity test methods. Furthermore, we delve into optimal control strategies using Pontryagin’s maximum principle and assess the influence of delays in state and control parame- ters on model dynamics. Again, a stochastic epidemic model is formulated and analyzed using a continuous-time Markov chain model for infectious propagation. Analytical estimation of the likelihood of disease extinction and the occurrence of an epidemic is conducted using the branching process approximation. The spatial system presents a comprehensive stability analysis and yielding criteria for Turing instability. Moreover, we have generated the noise-induced pattern to assess the effect of white noise in the populations. Additionally, a case study has been conducted to estimate the model parameters, utilizing COVID-19 data from Poland and HIV/AIDS data from India. Finally, all theo- retical results are validated through numerical simulations. This article extensively explores various modeling techniques, like deterministic, stochastic, statistical, pattern formation(noise-induced), model fitting, and other modeling perspectives, highlighting the significance of the inhibitory effects exerted by susceptible and infected populations.
Bifurcations and multi-stability in an eco-epidemic model with additional food
(Springer, 2022-01) Dubey, Balram
In this work, we propose a Leslie–Gower prey–predator model where prey is afflicted with an incurable illness, and the predator may choose to eat the provided extra food. Our study aims to control the existing disease in the system with the provision of alternative food. To achieve the goal, we investigate the suggested model and its disease-free subsystem theoretically and numerically. The scope of our analysis is broadened to encompass both local and global bifurcations. Hopf-bifurcation, transcritical bifurcation, saddle-node bifurcation, homoclinic bifurcation, heteroclinic bifurcation, all occur due to stability transitioning between steady states or cycles. Numerical results indicate that the additional food parameter αA contributes to the complex dynamics of the system. A slight modification in αA can significantly change the characteristics of the entire system. In a specific range of αA, all of these unanticipated changes render the system bi-stable and multi-stable. In such cases, we plot their basins of attraction. Consequently, a set of starting values for which the system is disease-free is obtained. We also illustrate the phenomenon of global stability toward the positive equilibrium. Furthermore, the infection rate is capable of altering the dynamics of the system. Through a subcritical Hopf-bifurcation, it can control the oscillations in species around their positive steady state. However, ample energy from the alternative food may lead to disease eradication even for higher infection rates.
Chaos control in a multiple delayed phytoplankton–zooplankton model with group defense and predator’s interference
(AIP, 2021-08) Dubey, Balram
Phytoplankton–zooplankton interaction is a topic of high interest among the interrelationships related to marine habitats. In the present manuscript, we attempt to study the dynamics of a three-dimensional system with three types of plankton: non-toxic phytoplankton, toxic producing phytoplankton, and zooplankton. We assume that both non-toxic and toxic phytoplankton are consumed by zooplankton via Beddington–DeAngelis and general Holling type-IV responses, respectively. We also incorporate gestation delay and toxic liberation delay in zooplankton’s interactions with non-toxic and toxic phytoplankton correspondingly. First, we have studied the well-posedness of the system. Then, we analyze all the possible equilibrium points and their local and global asymptotic behavior. Furthermore, we assessed the conditions for the occurrence of Hopf-bifurcation and transcritical bifurcation. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also studied. Various time-series, phase portraits, and bifurcation diagrams are plotted to confirm our theoretical findings. From the numerical simulation, we observe that a limited increase in inhibitory effect of toxic phytoplankton against zooplankton can support zooplankton’s growth, and rising predator’s interference can also boost zooplankton expansion in contrast to the nature of Holling type IV and Beddington–DeAngelis responses. Next, we notice that on variation of toxic liberation delay, the delayed system switches its stability multiple times and becomes chaotic. Furthermore, we draw the Poincaré section and evaluate the maximum Lyapunov exponent in order to verify the delayed system’s chaotic nature. Results presented in this article might be helpful to interpret biological insights into phytoplankton–zooplankton interactions.
Chaos in a seasonal food-chain model with migration and variable carrying capacity
(Springer, 2024-05) Dubey, Balram
The carrying capacity’s functional dependence illustrates the reality that any species’ activities can enhance or diminish its carrying capacity. Migration is the need of many species to achieve better opportunities for survival. In a tri-trophic system, the middle predator often immigrates to consume its prey and often emigrates to secure themselves from predators. This work deals with formulating and investigating a mathematical model reflecting the aforementioned ecological aspects. We perform a detailed analysis to prove the boundedness of the solutions. Further, we examine the existence and stability of equilibrium points, followed by the bifurcation analysis. We explore various global and local bifurcations like Hopf, saddle-node, transcritical, and homoclinic for the critical parameters (measuring the impact of prey activities on the carrying capacity) and (measuring the migration rate of a predator). Higher values of generate unpredictability, which helps explain the enrichment paradox. The presence of a chaotic attractor and bi-stability of node-node type is demonstrated via numerical simulation. The migratory behavior of middle predators can control chaos in the system. Furthermore, we study the proposed model in the presence of seasonal fluctuations. Persistence of the non-autonomous system, existence, and global stability of periodic solutions are analyzed theoretically. The seasonality in brings the bi-stability between chaotic and periodic attractors, and seasonality in growth rate of the prey causes bi-stability between 2-periodic and 4-periodic attractors. Moreover, the bi-stability in the autonomous system shifts to the global stability of an equilibrium in the seasonal model due to the seasonality in . When birth and death rates are seasonal along with , the extinction of one or more populations is possible. The non-autonomous system also exhibits bursting oscillations when seasonality is present in the death rate. Our findings reveal that the population’s intense constructive and destructive actions can allow the basal prey to thrive while eradicating both predators.
Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay
(Elsevier, 2022-07) Dubey, Balram
Plankton-fish interactions are the central topic of interest related to marine ecology. Apart from the direct predation (lethal effect) of zooplankton by fish, there are some non-lethal implications in the zooplankton-fish relationship. Due to induced fear of predation, there can be a reduced reproduction rate of zooplankton species, and the effect of this non-lethal interconnection can be carried over to subsequent seasons or generations. In the current study, we tend to analyse the role of fish-induced fear in zooplankton with its carry-over effects (COEs) and a corresponding discrete delay (COE delay) in a phytoplankton-zooplankton-fish population model. We use Holling type IV and II functional responses to model the phytoplankton-zooplankton and zooplankton-fish interplay, respectively. In the well-posedness of the present biological system, firstly, we evaluate an invariant set in which the solutions of the model remain bounded. Then we prove its persistence under some ecologically well-behaved conditions. Next, we establish the conditions under which the different feasible equilibrium points exist; the existence of various interior equilibria is also set up here. To study the system's dynamical behavior, local and global stability analyses for the equilibria mentioned above are also discussed. Further, the theoretical conditions for Hopf and transcritical bifurcations in non-delayed and delayed models are determined. Impacts of non-lethal parameters, fear, and its carry-over effects, on the population densities are studied analytically and supported numerically. For intermediate values of COEs parameter, we notice that the system behaves chaotically, and decreasing (or increasing) it to low (or high) values, solution converges to interior equilibrium point through period-halving. Calculation of the largest Lyapunov exponent and drawing of Poincaré map validate the chaotic nature of the system. The chaos for medium values of COEs parameter can also be controlled by decreasing the fear parameter. Next, we numerically validate the theoretical result for transcritical bifurcation. We also note that our system shows the phenomenon of enrichment of paradox, and the attribute of multistability. In the delayed model, we observe that increasing delay can eliminate chaotic oscillations through amplitude death phenomenon. We draw various types of graphs and diagrams to assist our results. Thus we can say that the present study has various interesting characters related to non-linear models and can help biologists to study the plankton-fish models in a more detailed and pragmatic manner.
Chaotic dynamics of a stage-structured prey-predator system with hunting cooperation and fear in the presence of two discrete delays
(Wiley, 2023) Dubey, Balram
Depending on behavioral differences, reproductive capability and dependency, the life span of a species is divided mainly into two classes, namely immature and mature. In this paper, we have studied the dynamics of a predator–prey system considering stage structure in prey and the effect of predator-induced fear with two discrete time delays: maturation delay and fear response delay. We consider that predators cooperate during hunting of mature prey and also include its impact in fear term. The conditions for existence of different equilibria, their stability analysis are carried out for non-delayed system and bifurcation results are presented extensively. It is observed that the fear parameter has stabilizing effect whereas the cooperative hunting factor having destabilizing effect on the system via occurrence of supercritical Hopf-bifurcation. Further, we observe that the system exhibits backward bifurcation between interior equilibrium and predator free equilibrium and hence the situation of bi-stability occurs in the system. Thereafter, we differentiate the region of stability and instability in bi-parametric space. We have also studied the system’s dynamics with respect to maturation and fear response delay and observed that they also play a vital role in the system stability and occurrence of Hopf-bifurcation is shown with respect to both time delays. The system shows stability switching phenomenon and even higher values of fear response delay leads the system to enter in chaotic regime. The role of fear factor in switching phenomenon is discussed. Comprehensive numerical simulation and graphical presentation are carried out using MATLAB and MATCONT
Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays
(Springer, 2023-11) Dubey, Balram
In this paper, we consider a two-species predator–prey model with fading memory, where the growth rate of prey species is subject to predation induced fear. Growth rate of predator species depends not only on the present density of prey but also on the past densities with diminishing impact. As the societal activities and behavioral practices influence carrying capacity of any species, we consider the density dependent carrying capacity of prey species instead of a constant. As fear on growth rate and societal activities on carrying capacity entail some time lags to show their effect, so we incorporate two delay parameters to corroborate this in the modeling phenomenon. Feasibility criteria of equilibria and their stability analysis are carried out. We observe that fear parameter and predation rate have destabilizing effect on the system’s dynamics, whereas parameter representing intensity of fading memory has stabilizing impact. We also distinguish stability and instability regions in different parametric planes. With increasing value of production factor from negative to positive, stability region decreases. The system also shows multiple stability switching phenomenon with respect to delay parameters. Solutions show chaotic behavior for a range of fear response delay both in the absence and presence of other delay parameter.
Complex dynamics of Leslie–Gower prey–predator model with fear, refuge and additional food under multiple delays
(World Scientific, 2022) Dubey, Balram
In this paper, we analyze a system of delay differential equations incorporating prey’s refuge, fear, fear-response delay, extra food for predators and their gestation lag. First, we examined the system without delay. The persistence, stability (local and global) and various bifurcations are discussed. We provide detailed analysis for transcritical and Hopf-bifurcation. The existence of positive equilibria and the stability of prey-free equilibrium are interrelated. It is shown that (i) fear can stabilize or destabilize the system, (ii) prey refuge in a specific limit can be advantageous for both species, (iii) at a lower energy level (gained from extra food), the system undergoes a supercritical Hopf-bifurcation and (iv) when the predator gains high energy from extra food, it can survive through a homoclinic bifurcation, and prey may become extinct. The possible occurrence of bi-stability with or without delay is discussed. We observed switching of stability thrice via subcritical Hopf-bifurcation for fear-response delay. On changing some parametric values, the system undergoes a supercritical Hopf-bifurcation for both delay parameters. The delayed system undergoes the Hopf-bifurcation, so we can say that both delay parameters play a vital role in regulating the system’s dynamics. The analytical results obtained are verified with the numerical simulation.
Consequences of fear effect and prey refuge on the Turing patterns in a delayed predator–prey system
(AIP, 2022) Dubey, Balram
This study presents a qualitative analysis of a modified Leslie–Gower prey–predator model with fear effect and prey refuge in the presence of diffusion and time delay. For the non-delayed temporal system, we examined the dissipativeness and persistence of the solutions. The existence of equilibria and stability analysis is performed to comprehend the complex behavior of the proposed model. Bifurcation of codimension-1, such as Hopf-bifurcation and saddle-node, is investigated. In addition, it is observed that increasing the strength of fear may induce periodic oscillations, and a higher value of fear may lead to the extinction of prey species. The system shows a bistability attribute involving two stable equilibria. The impact of providing spatial refuge to the prey population is also examined. We noticed that prey refuge benefits both species up to a specific threshold value beyond which it turns detrimental to predator species. For the non-spatial delayed system, the direction and stability of Hopf-bifurcation are investigated with the help of the center manifold theorem and normal form theory. We noticed that increasing the delay parameter may destabilize the system by producing periodic oscillations. For the spatiotemporal system, we derived the analytical conditions for Turing instability. We investigated the pattern dynamics driven by self-diffusion. The biological significance of various Turing patterns, such as cold spots, stripes, hot spots, and organic labyrinth, is examined. We analyzed the criterion for Hopf-bifurcation for the delayed spatiotemporal system. The impact of fear response delay on spatial patterns is investigated. Numerical simulations are illustrated to corroborate the analytical findings.
A delayed prey-predator model with Crowley-Martin type functional response including prey refuge
(Wiley, 2017-05) Dubey, Balram
In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin-type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations.
A delayed prey–predator model with Crowley–Martin-type functional response including prey refuge
(Wiley, 2017-05) Dubey, Balram
In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin-type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.
Diffusive three species plankton model in the presence of toxic prey: application to Sundarbans mangrove wetland
(World Scientific, 2017) Dubey, Balram
The bloom of toxin producing phytoplankton (TPP) is an environmental issue due to its negative impact on fresh water and marine ecology. In this paper, such a phenomenon is modeled using the reaction–diffusion equations. The spatiotemporal interaction among non-toxin producing phytoplankton (NTP), TPP, and zooplankton has been considered with Holling type II and III functional responses. The stability analysis for non-spatial and spatial model system is carried out and numerical simulations are performed for a fixed set of parameter values, which is realistic to planktonic dynamics. It has been observed that on increasing the reduction rate of zooplankton, the system shows cyclic to stable behavior. The result shows that the predators which avoid to toxic prey promote the bloom. Non-Turing patchy pattern has also been observed on time evolution. In this work, we have taken the case study of Sundarban mangrove wetland which is suffering from algal bloom due to the presence of toxic Dinoflagellates and Cyanophyceae. Through the numerical simulation, it has been shown that the higher value of reduction rate of zooplankton (ξ2) is responsible for bad health of the wetland system.
Dynamics of a stage-structured predator–prey system with fear-induced group defense in autonomous and nonautonomous settings
(AIP, 2024-06) Dubey, Balram
In this investigation, we construct a predator–prey model that distinguishes between immature and mature prey, highlighting group defense strategies within the mature prey. First, we embark on exploring the positivity and boundedness of the solution, unraveling sustainable equilibrium points, and deducing their stability conditions. Upon further investigation, we observe that the system exhibits diverse bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov–Takens bifurcations. The results reveal that heightened fear decreases mature prey density, potentially causing prey extinction beyond a certain threshold. Increased maturation rates lead to the coexistence of immature and mature prey populations and higher predator density. Stronger group defense boosts mature prey density, while weaker defense results in weak persistence. Lower values of the maturation rate of prey and the decline rate of predators sustain only the predator population, reliant on resources other than focal prey. Furthermore, our model demonstrates intriguing and diverse dynamical phenomena, including various forms of bistability across distinct bi-parameter planes. We also explore the dynamics of a related nonautonomous system, where certain parameters are considered to vary with time. In the seasonally forced model, we set out to define criteria regarding the existence and stability of positive periodic solutions. Numerical investigations into the seasonally forced model uncover a spectrum of dynamics, ranging from simple periodic solutions to higher periodicities, bursting patterns, and chaotic behavior
Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate
(AAM, 2015-12) Dubey, Uma S.; Dubey, Balram
In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. The existence of Hopf bifurcation of model is investigated by using Andronov-Hopf bifurcation theorem. Further, numerical simulations are done to exemplify the analytical studies.
Dynamics of Phytoplankton, Zooplankton and Fishery Resource Model
(AAM, 2014-06) Dubey, Balram
In this paper, a new mathematical model has been proposed and analyzed to study the interaction of phytoplankton- zooplankton-fish population in an aquatic environment with Holloing’s types II, III and IV functional responses. It is assumed that the growth rate of phytoplankton depends upon the constant level of nutrient and the fish population is harvested according to CPUE (catch per unit effort) hypothesis. Biological and bionomical equilibrium of the system has been investigated. Using Pontryagin’s Maximum Principal, the optimal harvesting policy is discussed. Chaotic nature and bifurcation analysis of the model system for a control parameter have been observed through a numerical simulation
Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays
(Springer, 2019-04) Dubey, Balram
This study proposes a three-dimensional prey–predator model with stage structure in prey (immature and mature) including maturation delay in prey population and gestation delay in predator population. It is assumed that the immature prey population is consumed by predators with Holling type I functional response and the interaction between mature prey and predator species is followed by Crowley–Martin-type functional response. We analyzed the equilibrium points, local and global asymptotic behavior of interior equilibrium point for the non-delayed system. Hopf-bifurcation with respect to different parameters has also been studied for the system. Further, the existence of periodic solutions through Hopf-bifurcation is shown with respect to both the delays. Our model analysis shows that time delay plays a vital role in governing the dynamics of the system. It changes the stability behavior of the system into instability, even with the switching of stability. The direction and stability of Hopf-bifurcation are also studied by using normal form method and center manifold theorem. Finally, computer simulation and graphical illustrations have been carried out to support our theoretical investigations.