We report some organized structures of two linearly coupled logistic maps with different harvesting. The coupled system exhibits chaos via period-bubbling and quasiperiodic routes for identical and weak coupling strength, in contrast to conventional period-doubling route for a simple logistic map. Studies reveal the existence of infinite families of periodic Arnold tongues and self-similar shrimp-shaped structures with period-adding sequences for periodic windows embedded in quasiperiodic and chaotic regions, respectively. Different Fibonacci-like sequences are formed leading to the Golden Mean. The shrimp-shaped structures maintain period 3-times self-similarity scaling. The quasiperiodicity route is the necessary condition for the occurrence of periodic Arnold tongues in this coupled system resulting in the appearance of shrimps in the chaotic region near the tongues. It is also revealed that the existence of shrimp implies the period-bubbling cascade but the reverse is not true. The bifurcation-induced hysteresis is born in a certain parameter range resulting in the birth of coexisting multiple attractors of different kinds. Basin sets of the coexisting attractors have either self-similar or intertwining fractal basin boundaries.

In the present paper, we investigate the impact of time delay during cooperative hunting in a predator–prey model. We consider that cooperative predators do not aggregate in a group instantly, but individuals use different stages and strategies such as tactile, visual, vocal cues, or a suitable combination of these to communicate with each other. We observe that delay in hunting cooperation has stabilizing as well as destabilizing effects in the system. Also, for an increase in the strength of the delay, the system dynamics switch multiple times and eventually become chaotic. We see that depending on the threshold of time delay, the system may restore its original state or may go far away from its original state and unable to recollect its memory. Furthermore, we explore the dynamics of the system in different bi-parameter spaces and observe that for a particular range of other parameter values, the system dynamics switch multiple times with an increase of delay in all the planes. Different kinds of multistability behaviors, the coexistence of multiple attractors, and interesting changes in the basins of attraction of the system are also observed. We infer that depending on the initial population size and the strength of cooperation delay, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species.