Impacts of inclusion of time delay in efforts on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature

Abstract

To understand the impacts of efforts applied to increase the density of forestry biomass on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature a nonlinear mathematical model is proposed and analysed. The mathematical model involves four dynamical variables namely: the density of forestry biomass, efforts applied to increase the density of forestry biomass, concentration of greenhouse gases and elevated environmental temperature. Since, there is always a time lag between implementation of efforts and its outcome, therefore we extend our model by introducing time delay in efforts. It is found that as delay in efforts crosses a critical value, the delay model loses its stability and undergoes Hopf bifurcation. The stability and direction of Hopf bifurcation are analysed using normal form method and central manifold theory. Numerical simulations are performed to verify and validate our analytical results. It is observed that if efforts are implemented for appropriate time, the density of forestry biomass can be conserved but implementation of efforts with longer time delay has destabilizing effect on the system. Increasing rate of atmospheric temperature due to greenhouse gases, decreases the density of forestry biomass but this decrease can also be maintained by implementation of efforts. Therefore, implementation of efforts for sufficient period of time, plays a very vital role in increasing the density of forestry biomass and decreasing the concentration of greenhouse gases and elevated temperature.