Department of Mathematics
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Item Spatio-temporal dynamics of an ecological model with Cosner's functional response and prey taxis in networked vs. non-networked environments(Springer, 2025-05) Dubey, Balram; Dubey, Uma S.This study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.Item Impact of hospital bed availability on infectious disease management: a stochastic and optimal control approach(Springer, 2025-05) Dubey, BalramThis study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.Item Exploring turing pattern formation in a seasonally forced predator-prey model incorporating fear effects and prey refuge(Springer, 2025-07) Dubey, BalramSeasonal variations critically influence species movement and migration, with profound implications for ecological stability as evidenced by numerous natural phenomena. In this work, we modify the traditional Lotka-Volterra model by incorporating three key mechanisms: predator-induced fear effects on prey reproduction and mortality, prey refuge dynamics, and periodic environmental fluctuations. For the autonomous system, we conduct a comprehensive stability analysis and uncover rich dynamics, including key bifurcation such as saddle-node, Hopf, and codimension-two bifurcations specifically Bogdanov-Takens and cusp bifurcations as well as global homoclinic bifurcations. Building upon the temporal case, we explore the non-autonomous dynamics, by including seasonal changes in the fear and refuge parameters, where we establish criteria for permanence and the existence of globally attractive periodic solutions, highlighting how seasonal forcing can lead to ecological collapse by crossing extinction thresholds. We further analyze a reaction-diffusion system under both autonomous and non-autonomous frameworks to investigate the spatial distribution of species. For non-autonomous cases with time-varying cross-diffusion and periodic reaction rates, we derive Turing instability conditions using comparison principles, expressed through inequalities involving time-varying parameters and their derivatives. The autonomous case recovers classical Turing conditions, validating our generalized approach. Numerical simulations quantify how fear intensity and refuge availability modulate pattern formation, while seasonality induces complex dynamics such as periodic oscillations, chaotic regimes, and bursting behaviors. This study highlights the profound impact of seasonal variations on ecological stability and pattern formation, offering valuable tools for understanding non-autonomous systems in ecological modelling.Item Role reversal in a stage-structured prey–predator model with fear, delay, and carry-over effects(AIP, 2023-09) Dubey, BalramThe present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov–Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmonyItem Spatiotemporal dynamics of a multi-delayed prey–predator system with variable carrying capacity(AIP, 2023-11) Dubey, BalramThis paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species.Item Complex dynamics of a predator–prey system with fear and memory in the presence of two discrete delays(Springer, 2023-11) Dubey, BalramIn 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.Item Trade-off dynamics and chaotic behavior in nonautonomous prey-predator model with group defense(Springer, 2023-11) Dubey, BalramEcological “trade-off” is prioritising one trait over another. Predators put their lives at danger to pursue dangerous prey, and their injuries can reduce their chances of survival. Prey must “trade-off” between reproduction rate and safety, whereas predators must “trade-off” between food and safety. We present a two-dimensional prey and predator model that takes into account prey logistic growth rate and Monod-Haldane type functional response to reflect prey collective defense. We investigate the cost of fear in order to depict prey trade-off dynamics, and we change the predator’s mortality rate by incorporating a function that reflects predator loss as a result of encountering dangerous prey. Our model shows bistability and goes through transcritical bifurcation, saddle node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, Bautin bifurcation, Homoclinic bifurcation, and Limit point of cycle bifurcation. We investigated the effects of our critical parameters on both populations and discovered that predators become extinct if their loss of predator is too high due to encounters with dangerous prey, demonstrating how predators risk their own health for food. We find that fear can lead to global stability in a system by causing the stable and unstable limit cycles to collide. We also see that the degree of seasonality in the level of fear in the nonautonomous model might lead to chaos. Sensitivity analysis, the positivity of the maximal Lyapunov exponent, and the uneven distribution of points observed in the Poincaré map shown are the established signs of chaotic nature. We note that variations in intensity of seasonality in carry-over can cause a system to shift under many different periodic windows. The findings presented in this article may be beneficial in comprehending the biological insights derived from investigating prey-predator interactions.Item Eco-epidemiological model of predator-prey with two-strain infections: the impact of herd behavior(2024) Dubey, BalramThis study presents an eco-epidemiological model exploring a prey population infected by two distinct pathogen strains in the presence of an unaffected predator population. The model investigates how prey herding behavior provides protection against predation under multi-strain infections. A well-posedness and boundedness analysis of the populations ensures the robustness of the model. Linear stability analysis reveals that, under specific herd shapes and predator mortality rates, prey infected with either strain can vanish. Bifurcation analysis uncovers critical dynamics: a supercritical Hopf bifurcation occurs at a threshold prey herd shape (k), indicating the onset of stable oscillatory population cycles. As predator mortality (δ) varies, both subcritical and supercritical Hopf bifurcations emerge, marking shifts between stable and unstable dynamics, potentially leading to prey extinction or sharp population collapses. The analysis further identifies a Generalized Hopf bifurcation, distinguishing between predictable cycles and more complex. Numerical simulations confirm these findings, offering insights into predator-prey dynamics in ecosystems subject to multi-strain infections. The results have potential implications for understanding disease control, population stability, and ecological resilience.Item The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis(Elsevier, 2024-03) Dubey, Balram; Dubey, Uma S.Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.Item Spatiotemporal and trade-off dynamics in prey–predator model with domed functional response and fear effect(World Scientific, 2024) Dubey, BalramIn the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.