Dynamics of Prey-Predator and Amensal Model
A. V. Paparao *
Department of Mathematics, JNTU-GV, Collège of Engineering, Vizianagaram(A), A.P., India.
T. S. N. Murthy
Department of ECE, JNTU-GV, Collège of Engineering, Vizianagaram(A), Vizianagaram, A.P., India.
*Author to whom correspondence should be addressed.
Abstract
Predator-prey interactions are among the most common and crucial ecological phenomena in nature. Over the course of long-term evolution, prey populations have developed various anti-predation strategies to cope with the threat of predators, with population dispersal being one of the most common strategies. In traditional ecological models, the prey population is typically constrained by direct predation. The study aims to investigate how time delay (τ) affects the stability and dynamics of a predator–prey–amensal ecological system modelled by delay differential equations. The influence of time lag(τ) on the predator species in the prey-predator and amensal model was studied. In this model, the first species x represents the predator, the second species y represents the prey, and the third species z is amensal to the prey species. The mathematical model is formulated as a system of delay differential equations. A hyperbolic equilibrium point of the system is identified, and its stability analysis was carried out at τ = 0 and τ >0. The hyperbolic equilibrium point is asymptotically stable at τ = 0. However, for τ >0 the system loses its stability and undergoes Hopf bifurcation for some critical values of 'τ'. Further, the global stability analysis was carried out, and the sufficient condition for the existence of Hopf bifurcation is derived. The critical values of 'τ' are identified using simulation for the proposed system. These values are correlated with the analytical finding.
Keywords: Amensal, predator, prey, stability, Hopf bifurcation