Department of Mathematics

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Author: search.filters.author.Dubey, BalramSubject: search.filters.subject.Mathematics

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Now showing 1 - 10 of 78
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    Exploring turing pattern formation in a seasonally forced predator-prey model incorporating fear effects and prey refuge
    (Springer, 2025-07) Dubey, Balram
    Seasonal 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.
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    Role reversal in a stage-structured prey–predator model with fear, delay, and carry-over effects
    (AIP, 2023-09) Dubey, Balram
    The 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 harmony
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    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.
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    Trade-off dynamics and chaotic behavior in nonautonomous prey-predator model with group defense
    (Springer, 2023-11) Dubey, Balram
    Ecological “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.
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    Eco-epidemiological model of predator-prey with two-strain infections: the impact of herd behavior
    (2024) Dubey, Balram
    This 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.
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    Spatiotemporal and trade-off dynamics in prey–predator model with domed functional response and fear effect
    (World Scientific, 2024) Dubey, Balram
    In 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.
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    Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion
    (Elsevier, 2024-05) Dubey, Balram
    In 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.
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    Spatiotemporal dynamics of prey–predator model incorporating holling-type ii functional response with fear and its carryover effects
    (AIP, 2024-05) Dubey, Balram
    The 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.
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    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.
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    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