Explicit impacts of harvesting in delayed predator-prey models

Modeling population dynamics using delay differential equations and exploring the impacts of harvesting in predator-prey systems are among few of the thrust areas of research in theoretical and applied ecology. Many results are established to understand distinct dynamics under population harvesting. However, comparatively less attention is paid to explain the explicit harvesting effects when populations fluctuate due to time delay. In this contribution, two well known Lotka–Volterra (LV) type and Rosenzweig–MacArthur (RM) predator-prey models incorporating time delay into the logistic growth term are considered. The analysis and the obtained results are summarized as follows. (a) Firstly, the dynamics of both the models, by considering the time delay as the bifurcation parameter, is analyzed. Some of the parameter conditions for the delay induced stability switching are improved and corrected in comparison to the earlier works. The delay induced stability results are derived and found to be similar for both the models. (b) We investigate whether harvesting of either prey or predator can locally stabilize (respectively, destabilize) the system when the unharvested system dynamics is at non-equilibrium (respectively, stable steady state) mode due to time delay. It is observed that harvesting can induce stability and instability switching depending upon the dynamics mode of the unharvested system. (c) In the same framework, we examine if a stable steady state can be obtained when predator is harvested towards Maximum Sustainable Yield (MSY) level. Unlike the case of non-delayed LV type and RM predator-prey models, it is found that harvesting the predator towards MSY in the delayed models does not guarantee a stable stock. The new results compared to the existing literature might contribute in enriching fishery management policy and theoretical ecology as a whole.