Delay-induced multiple stability scenarios, species coexistence, and predator extinction in an ecological system

Time delays are integral to ecological processes. Population models incorporate time delays to account for the time required for maturation, gestation, dispersal, and many more. Time delay can induce various stability dynamics, including (i) stability invariance, (i) stability change, (iii) stability switching, (iv) instability invariance, and (v) instability switching. Even one of these dynamics can occur with multiple mechanisms based on the distribution of critical time delays. Generally, two or three types of dynamics are detected in many population models, but exhibiting all the above dynamics is not observed. In an ecological system, species form groups to improve their chances of survival. Taking inspiration from tuna’s forging behavior Cosner et al. (1999) developed the Cosner functional response. In this study, we propose a delayed predator–prey model with Cosner functional response. The non-delayed model can have up to four equilibria, two coexisting equilibria (anti-saddle and saddle), along with trivial and boundary equilibria. The stability of all equilibria is analyzed with time delay. Under certain parameter conditions, the boundary equilibrium remains globally stable for all delays. For increasing delay, the anti-saddle equilibrium may: (i) remain stable, (ii) undergo stability change (two possible scenarios), (iii) undergo stability switching, (iv) remain unstable (two possible scenarios), or (v) undergo instability switching. These seven stability scenarios are verified to exhibit, while an additional instability invariance scenario, where no critical delay exists, is analytically shown to be non-existent. Showing all these mentioned stability scenarios in a predator–prey model with a single delay is a novelty of this paper. If the anti-saddle equilibrium is stable in the absence of delay, then the degenerate case may occur, which implies the local stability between any two consecutive delay thresholds. Moreover, we have analytically proved that the degenerate case is not possible if the anti-saddle equilibrium is unstable in the absence of delay, which is a new observation in population dynamics. We have computed species survival basin for increasing delay. Our investigation reveals that increasing delay can change the shape and size of the basin, making delay beneficial or harmful for the species’ survival, depending on the initial populations of species. Finally, we have proposed an open question and outlined a couple of potential directions for future research.