Numerical simulation for two species time fractional weakly singular model arising in population dynamics

In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order α∈(0,1). The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near t=0. Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme.