Temporal and Spatio‐Temporal Dynamics of Signaling Birds, Mammals and Its Predator With Anti‐Predator Behavior

ABSTRACT In order to explore the interaction between a mammal and its predator, a model is formulated by incorporating mutualistic interactions between the signaling bird and the mammal. It is assumed that the signaling bird feeds on the parasites present on the mammal's body and that the mammals show anti‐predator behavior towards their predator. We analyze the boundedness of the proposed system and establish the conditions for the existence and stability of ecologically feasible equilibrium states. The dynamical properties of the system, including the conditions for the occurrence of saddle‐node, transcritical, and Hopf‐bifurcation, are also investigated. To see the effect of signals, we obtain the conditions for the Hopf‐bifurcation by assuming the signaling coefficients as bifurcation parameters. We also determine (the basic reproduction ratio) that represents a threshold parameter and determines whether the predators and their prey can coexist. We further show that if , then the prey population may exist with signaling birds and the predator population goes extinct. If , then all the three population may coexist. This analysis helps us to choose two critical parameters for which the bifurcations of codimension‐2 may appear. The existence of Bogdanov–Takens and homoclinic bifurcations are ensured for suitable choices of parameters. To discuss the more natural scenarios, we further introduce diffusion in all three‐species to capture their movement. The feasibility of solutions for the spatial model is derived. The conditions for Turing and non‐Turing patterns are established with respect to the growth rate of birds, the saturation effect due to mammals and the signal given by birds to mammals. The corresponding patterns are plotted by using the five‐point stencil method with random perturbation.