Stock patterns in a class of delayed discrete-time population models

Abstract

This paper highlights the behavior presented by a new class of discrete-time predator-prey models which have a time delay in the predation process. These models are formulated by discretizing continuous-time delayed Rosenzweig-MacArthur predator-prey systems. The stability analysis is performed through Routh-Hurwitz (RH) criteria, and bifurcation diagrams help to grasp the change in the dynamics of interacting species. First, we examine the impact of changing carrying capacity on (i) the stability of the equilibrium and (ii) the mean population level in delayed systems. When the delay is very large, the stability thresholds of the carrying capacity of different delayed models tend to converge. We compute the mean population size as a function of carrying capacity. Mean prey (resp. predator) stock increases (resp. decreases) with increasing carrying capacity of prey resulting in the paradox of enrichment in our discrete-time systems. For higher delayed models, the value of the mean prey (resp. predator) stock is higher (lower). Second, harvesting has a potential to stabilize an unstable mode of the coexisting equilibrium. When prey is harvested, the mean population of prey (resp. predator) decreases (increases). Therefore, no hydra effect is experienced by prey species. On the other hand, predator harvesting potentially increases its own mean stock, leading to hydra effects in predators. Most importantly, we have computed the mean stock with respect to harvesting. For both prey and predator harvesting, the mean prey size increases with an increase in delay, whereas the mean predator size decreases with an increase in delay. For the system with larger delay, the hydra effects on predators become more prominent in the sense that the rate of increase of mean predator stock is higher.