Department of Mathematics
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Item Stability Switching in a Cooperative Prey-Predator Model with Transcritical and Hopf-bifurcations(Springer, 2022-10) Dubey, BalramIn nature organisms attempt to adopt new techniques to diminish the possibilities of being falling prey. Interspecies cooperation is one of these approaches which two different types of prey can use against a common predator. Inspired by this, we purpose a prey-predator model having two prey who cooperate with each other while interacting with a predator. For making the model more general and realistic, the interactions between prey and predator are handled through general Holling type-IV and Crowley-Martin functional responses. For well-posedness of the proposed model, firstly, its boundedness is investigated which is followed by the vigorous proofs for the existence of equilibrium points, their stability analysis, evaluation of conditions for occurrence of transcritical and Hopf-bifurcations. Numerically, we observe that as the inverse measure of predator’s immunity from first prey and coefficient of cooperation from first prey to second prey crosses some respective critical values, there is occurrence of Hopf-bifurcation.Transcritical bifurcation is also depicted numerically for the intrinsic growth rate of first prey and the death rate of predator species. Several phase portraits, bifurcation diagrams are drawn to support our analytical findings. We also endorse the attribute of bistability, and basins of attraction for both stable equilibrium points are also drawn.Item Bifurcation Analysis of a Leslie-Gower Prey-Predator Model with Fear and Cooperative Hunting(Springer, 2022-10) Dubey, BalramThe 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.Item A Prey-Predator Model with a Reserved Area(VUP, 2007) Dubey, BalramIn this paper, a mathematical model is proposed and analysed to study the dynamics of a prey-predator model. It is assumed that the habitat is divided into two zones, namely free zone and reserved zone. Predators are not allowed to enter into the reserved zone. Criteria for the coexistence of predator-prey are obtained. The role of reserved zone is investigated and it is shown that the reserve zone has a stabilizing effect on predator-prey interactions.Item Stability and Bifurcation of a Prey-Predator System with Additional Food and Two Discrete Delays(Tech Science Press, 2021) Dubey, BalramIn this paper, the impact of additional food and two discrete delays on the dynamics of a prey-predator model is investigated. The interaction between prey and predator is considered as Holling Type-II functional response. The additional food is provided to the predator to reduce its dependency on the prey. One delay is the gestation delay in predator while the other delay is the delay in supplying the additional food to predators. The positivity, boundedness and persistence of the solutions of the system are studied to show the system as biologically well-behaved. The existence of steady states, their local and global asymptotic behavior for the non-delayed system are investigated. It is shown that (i) predator’s dependency factor on additional food induces a periodic solution in the system, and (ii) the two delays considered in the system are capable to change the status of the stability behavior of the system. The existence of periodic solutions via Hopf-bifurcation is shown with respect to both the delays. Our analysis shows that both delay parameters play an important role in governing the dynamics of the system. The direction and stability of Hopf-bifurcation are also investigated through the normal form theory and the center manifold theorem. Numerical experiments are also conducted to validate the theoretical results.Item Complex dynamics of Leslie–Gower prey–predator model with fear, refuge and additional food under multiple delays(World Scientific, 2022) Dubey, BalramIn 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.Item Trade-off and chaotic dynamics of prey-predator system with two discrete delays(AIP, 2023-05) Dubey, BalramIn our ecological system, prey species can defend themselves by casting strong and effective defenses against predators, which can slow down the growth rate of prey. Predator has more at stake when pursuing a deadly prey than just the chance of missing the meal. Prey have to "trade off" between reproduction rate and safety and whereas, predator have to "trade off" between food and safety. In this article, we investigate the trade-off dynamics of both predator and prey when the predator attacks a dangerous prey. We propose a two-dimensional prey and predator model considering the logistic growth rate of prey and Holling type-2 functional response to reflect predator's successful attacks. We examine the cost of fear to reflect the trade-off dynamics of prey, and we modify the predator's mortality rate by introducing a new function that reflects the potential loss of predator as a result of an encounter with dangerous prey. We demonstrated that our model displays bi-stability and undergoes transcritical bifurcation, saddle node bifurcation, Hopf bifurcation, and Bogdanov-Taken bifurcations. To explore the intriguing trade-off dynamics of both prey and predator population, we investigate the effects of our critical parameters on both population and observed that either each population vanishes simultaneously or the predator vanishes depending on the value of the handling time of the predator. We determined the handling time threshold upon which dynamics shift, demonstrating the illustration of how predators risk their own health from hazardous prey for food. We have conducted a sensitivity analysis with regard to each parameter. We further enhanced our model by including fear response delay and gestation delay. Our delay differential equation system is chaotic in terms of fear response delay, which is evidenced by the positivity of maximum Lyapunov exponent. We have used numerical analysis to verify our theoretical conclusions, which include the influence of vital parameters on our model through bifurcation analysis. In addition, we used numerical simulations to showcase the bistability between co-existence equilibrium and prey only equilibrium with their basins of attraction. The results that are reported in this article might be useful in interpreting the biological insights gained from studying the interactions between prey and predator.