Complex dynamics of delay-induced plankton–fish interaction exhibiting defense

Abstract

In this paper, we propose a delayed plankton–fish model with Monod–Haldane-type functional response. Many prey organisms have developed defense mechanisms against predation. Predators also exhibit ways of coping with variable environmental conditions that enter in adaptation in the event of a shortage of prey. We assume prey and middle predators exhibit defense ability towards their predators. Time delay is incorporated in middle predator as an adaptation to cope with variable environmental conditions. Analytically, we study the feasibility and boundedness of the model system by comparison principle followed by local and global stability conditions. We also investigate the stability and direction of bifurcating periodic solution by normal form theory and center manifold arguments. Numerically, we study the effect of defense and time delay on the stability behavior of the system. Our numerical results reveal that the increase in time delay system becomes unstable as well as stable under different parametric restrictions. We observe that due to defense ability in prey and middle predator system shows extinction in top predator.