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    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, Balram
    This 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.
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    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.
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    Stability switching and chaos in a multiple delayed prey-predator model with fear effect and anti-predator behaviour, Mathematics and Computers in Simulation
    (Elsevier, 2021-10) Dubey, Balram
    Recent studies demonstrate that the density of prey population is not only affected by direct killing by the predator, but the fear in prey caused by predator also reduces it by cutting down the reproduction rate of prey community, and prey shows anti-predator behavior in response to this fear. In this study, we propose a prey–predator model with fear in prey due to predator and anti-predator behavior by prey against the predator with fear response delay and gestation delay. It is assumed that the predator consumes prey via simplified Holling Type-IV functional response. We evaluate the equilibrium points and study the local and global stability behavior of the system around them. It is observed that our system undergoes Hopf-bifurcation with respect to the fear parameter. Moreover, the system shows the attribute of bi-stability involving two stable equilibriums. Further, we study the dynamics of the delayed system by incorporating fear response delay and gestation delay. We observe that the delayed system suffers Hopf-bifurcation with respect to both delays. Using the normal form method and center manifold theory, the direction and stability of Hopf-bifurcation are studied. Chaotic behavior for delayed system is observed for large values of fear response delay. All these findings are supported by numerical simulation.
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    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.
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    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
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    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