A diffusive Monod-Haldane predator-prey system with Smith growth and a protection zone

In predation, too large intrinsic growth rate of the predator will usually drive prey to extinction. In this paper, we consider a diffusive Monod-Haldane predator-prey system with Smith growth to study the effect of an internal protection zone. We mainly focus on the corresponding steady system to locate a critical patch size of the protection zone. We find that if the size of the protection zone is below the critical patch size, then the system has no positive steady state solutions for too large intrinsic growth rate of the predator, while if it is above the critical patch size, then there exist positive steady state solutions no matter how large the intrinsic growth of the predator is. We further get the asymptotic behavior of the system when the protection zone is large enough, which indicates that the predator and the prey are spatially segregated as the intrinsic growth rate of the predator tends to infinity. (c) 2023 Elsevier Ltd. All rights reserved.