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 $\alpha \in (0,1)$alpha is an element of(0,1). The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near $t = 0.$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.