The complex dynamical behaviour of fractal-fractional forestry biomass system

Abstract

In this research, we aim to propose a non-integer order mathematical model that illustrates the detrimental impact of illegal logging on forest biomass. The model is based on the idea that soil nutrients play a vital role in the growth of forestry biomass. However, the growth of forestry biomass is hindered by the harvesting of trees for industrial expansion. To develop the model, we use a fractional order approach, which allows us to explore the impact of conservation efforts on the extension of industries. If the amount of forestry biomass that is illegally harvested exceeds a certain threshold, the system becomes unstable. To investigate the proposed model's solution, we use fixed-point theory, which is highly effective in determining the existence and uniqueness of the solution. We also use Ulam-Hyers stability to analyse the stability of the proposed model. We have also employed new versions of fractional numerical techniques using power law and exponential decay. Our proposed model provides more details through numerical calculations with fractional orders different from integer ones. Moreover, we have addressed all significant solution behaviours in the fractal-fractional system and parameter values. The graphical representation offers a better understanding of the forestry biomass model, allowing us to identify the direction and stability of solutions. From various perspectives, we analysed the features of solutions to this fractal-fractional model for stopping illegal logging.

Keywords

• Forestry biomass model
• fractal-fractional Caputo derivative
• fractal-fractional Caputo-Fabrizio derivative
• existence and uniqueness
• Ulam-Hyers stability
• numerical method