Relationship between exploitation, oscillation, MSY and extinction

• MSY may not exist in single species model.
• Prey harvesting to MSY may cause extinction of both prey and predator.
• Predator harvesting to MSY is stable in a prey–predator system.
• Harvesting the top predator to MSY may cause extinction in a tritrophic food chain.
• Increase in predator harvesting can remove oscillations.

We give answers to two important problems arising in current fisheries: (i) how maximum sustainable yield (MSY) policy is influenced by the initial population level, and (ii) how harvesting, oscillation and MSY are related to each other in prey–predator systems. To examine the impact of initial population on exploitation, we analyze a single species model with strong Allee effect. It is found that even when the MSY exists, the dynamic solution may not converge to the equilibrium stock if the initial population level is higher but near the critical threshold level. In a prey–predator system with Allee effect in the prey species, the initial population does not have such important impact neither on MSY nor on maximum sustainable total yield (MSTY). However, harvesting the top predator may cause extinction of all species if odd number of trophic levels exist in the ecosystem. With regard to the second problem, we study two prey–predator models and establish that increasing harvesting effort either on prey, predator or both prey and predator destroys previously existing oscillation. Moreover, equilibrium stock both at MSY and MSTY level is stable. We also discuss the validity of found results to other prey–predator systems.