An investigation into the impact of odour: A dynamical study of two predators and one prey model, taking into account both integer order and fractional order derivatives

This article presents a prey–predator system that takes into account two important factors: the negative impact of predator odour on its competitors and the positive impact of predator odour on the prey. The system is analysed using two models: one with ODEs and another with FDEs. We have extensively validated the model system biologically, ensuring that the solutions are both nonnegative and bounded. An in-depth investigation has been carried out to thoroughly investigate the stability of all potential equilibrium points of the model systems in a systematic manner. Our observations reveal that our model systems exhibit various types of bifurcations, including transcritical and Hopf bifurcations, around the interior equilibrium point for three distinct parameters. These parameters include the rate of conversion of the first predator $r_4$, the degree of resistance or avoidance exhibited by the prey due to predator odour $m$, and the level of disruption to competitors in predation due to the presence of predator odour $a$. A significant finding in this paper is that the resistance shown by prey towards the first predator in predation in reaction to the odour is vital for sustaining the population of the second predator. The survival of the second predator within the biosystem is heavily dependent on the growth rate of the first predator. The possibility of the second predator facing extinction becomes much less likely when the first predator is absent from the system, which is another significant result. Within the context of fractional order derivatives, the system dynamics demonstrate a higher level of stability in comparison to the traditional integer order derivative. It has been noticed that where the parameter values are identical, the fluctuations exhibited by the integer order system are stabilised in the fractional order system. Thus, the significance of predator odour and the effect of memory in the system have been thoroughly established. Ultimately, the study backs up the theoretical findings with convincing numerical simulations.