Dynamical study of fractional order differential equations of predator‐pest models

Abstract

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.

Time delays and economic profit in fractional differential‐algebraic predator–prey model

“…On the other hand, since fractional-order differential operators can fit the memory and heritability of population growth well, making the models established by fractional-order derivatives more accurate than integer-order models, scholars began to use fractional-order differential equations to explore the dynamics of population ecosystems and broaden the applicability of population models. Mandal et al [40] introduced the fractional-order Rosenzweig-MacArthur model to explore pest control strategies and found that the proposed fractional-order model is more stable and more suitable for pest management than the corresponding integer-order model. Yousef et al [41] established a fractional-order predator-prey system with fear effect, studied the effects of fear effect on the reproduction rate and mortality rate of the prey population, and found that the fractional derivative can stabilize the proposed system due to the memory effect.…”