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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 Modeling the effect of vaccinations, hospital beds, and treatments on the dynamics of infectious disease with delayed optimal control and sensitivity analysis(Springer, 2024-08) Dubey, Uma S.; Dubey, BalramImmunization plays a vital role in eradicating infectious diseases, typically requiring multiple doses at specific time intervals. This study focuses on developing and analyzing an infectious disease model governed by a six-dimensional system of ordinary differential equations, considering the impact of first and second vaccination doses along with hospital beds and treatment. The model’s qualitative behavior is analyzed, including conditions for positive solutions, the invariant region of the solution, equilibrium points, and their stability. When the basic reproduction number () is less than one (), the disease will be eradicated; conversely, an epidemic occurs when . Moreover, the transcritical bifurcation of the system is examined using the center manifold theory. Interestingly, backward bifurcation is discovered, and it indicates that the disease is not entirely eradicated even when . We have investigated different bifurcations like saddle-node, transcritical, and Hopf bifurcations of codimension 1, as well as Generalized-Hopf (GH), Cusp (CP), and Bogdanov–Takens (BT) bifurcations of codimension 2. Additionally, a delayed epidemiological model is explored, assuming a lag in vaccination among the susceptible population. A Hopf-bifurcation is observed near the endemic equilibrium point, linked to critical parameter values during the latent period. Moreover, the model is calibrated using the least-squares technique, incorporating coronavirus-infected case data and vaccination information from India and Italy’s mass vaccination program between March 1, 2021, and May 30, 2021. Global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies crucial parameters affecting threshold quantities after fitting the model. The study highlights the significance of critical parameters such as the effective transmission rate, rates of first and second-dose vaccinations, and recovery rate due to double-dose vaccination. Further, delayed optimal control measures are determined using Pontryagin’s maximal principle to mitigate infection, prevention, and treatment burdens. Numerical simulations are conducted to understand the effect of these delayed control measures on disease progression and demonstrate the insights obtained through analytical investigations. The study indicates that implementing all control strategies effectively reduces the disease burden among the population. Accurate estimation of vaccine efficacy is crucial for disease prevention, underlining the importance of well-planned vaccination strategies. Moreover, the numerical simulations validate all the theoretical findings, emphasizing the validity of this model in a real-world situation. Relying solely on vaccination might not swiftly or completely control the disease. Complementary pharmaceutical and non-pharmaceutical measures are necessary to combat the infection effectively. Further limitations on medical resources could lead to a backward bifurcation. Simulation results suggest that delaying the implementation of control measures could exacerbate epidemic situations.Item Impact of chemo-immunotherapy on tumour-immune interactions: a non-autonomous model of tumor necrosis factor and T cell dynamics(2025) Dubey, Uma S.; Dubey, BalramThis study explores the interaction between cancer cells, helper T cells, cytotoxic T cells, and tumour necrosis factors in chemotherapy and immunotherapy treatment in the microenvironment [1]. The goal is to analyze the connection of helper and cytotoxic T-cell levels with the anti-tumour immune response and the impact of various dosing regimens when combined with immunotherapy and chemotherapy. These protocols aim to shorten the interval between treatment cycles from three to two weeks or less to prevent tumour regrowth and maximize its cell elimination by treatment. Motivated by clinical trials, we thoroughly compare procedures involving two medications supplied sequentially or simultaneously in a non-autonomous system. We discussed the positivity and boundedness of the model. Further, we analyze the biologically valid equilibria and investigate their local stability properties, examining transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations numerically and analytically [2]. Furthermore, direction and stability conditions for periodic solutions are determined.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 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 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.Item Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion(Elsevier, 2024-05) Dubey, BalramIn this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.Item Spatiotemporal dynamics of prey–predator model incorporating holling-type ii functional response with fear and its carryover effects(AIP, 2024-05) Dubey, BalramThe recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey–predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models.