Dimensionally homogeneous fractional order Rosenzweig–MacArthur model: a new perspective of paradox of enrichment and harvesting

Abstract

The classical (ODE framework) Rosenzweig–MacArthur (RM) predator–prey model is a well-established ecological system. This model is modified further by incorporating time delay, diffusion, stochasticity, etc., in exploring several ecological aspects. However, investigations of this RM model in the fractional order differential equations are limited in the literature. More importantly, we have proposed a dimensionally consistent Caputo fractional order RM model subject to harvesting. The existence and uniqueness of solutions are established first in a feasible domain. Then, non-negativity, boundedness, and positivity of the solutions are demonstrated. Later, the local and global stability of all the equilibria is studied, and proper co-relations between stability zones are established in a two-parameter space. The theory of paradox of enrichment is investigated, and we present our insightful perspective on this ecological theory. The smaller carrying capacity shows a decreased mean population, which can be referred to as a paradox, but further increasing carrying capacity in our model has proved to be beneficial for the ecosystem. Hence, there is no paradox of enrichment in the system in our perspective. In the classical RM model, predator harvesting never changes the stability of a locally stable equilibrium, and the hydra effect is exhibited on predators at unstable coexisting equilibrium. For the first time, we have uncovered that the fractional order model could cause instability of a stable equilibrium through a supercritical Hopf-bifurcation and stabilize it again through a second supercritical Hopf-bifurcation, leading to stability switching and a bubble structure formation. Furthermore, predator harvesting exhibits the hydra effect at the stable state at two non-overlapping effort intervals. This study reveals several hidden results in the fractional order RM model compared to the classical RM model, which acts as the building blocks of ecological communities. © The Author(s), under exclusive licence to Springer Nature B.V. 2024.