Dynamical behaviour of a two prey and one predator system with indirect effect and time delay

The effect of time delay on a population model consisting of two preys and one predator is investigated. The interactions within the system are primarily governed by a Holling type II functional response, which describes the relationship between the predation rate and the density of the first prey. Additionally, the model accounts for the indirect effects of predation, capturing the complex dynamics arising from these interactions. By incorporating time delay, the analysis explores how the temporal lag in responses affects the stability and behavior of the populations, such as reproduction, predation, or resource regeneration. This approach provides insights into the conditions under which populations coexist, oscillate, or collapse, offering a deeper understanding of predator–prey dynamics in ecosystems influenced by delayed feedback mechanisms. Conditions under which an equilibrium point loses stability and bifurcates into cyclic oscillations are established. Our study sheds light on the intricate relationship between time delay and the emergent dynamics in this ecological system. Finally, the validity of the theoretical results is verified through several numerical examples.