Parastoo Reihani - Department of Mathematics, Payame Noor University (PNU), P. O. Box: ۱۹۳۹۵-۴۶۹۷, Tehran, Iran.
Hossein Esmailpour - Department of Mathematics, Payame Noor University (PNU), P. O. Box: ۱۹۳۹۵-۴۶۹۷, Tehran, Iran.
Fahimeh Soltanian - Department of Mathematics, Payame Noor University (PNU), P. O. Box: ۱۹۳۹۵-۴۶۹۷, Tehran, Iran.

In the present study, we consider an important mathematical model of the spread of two competing species in an ecological system with two species considering the interactions between these species. This model is derived from a system of nonlinear reaction-diffusion equations. We investigate this model as an inverse problem. Using appropriate initial and boundary conditions, the finite difference method in the time variable and the Quartic Bspline collocation method in the spatial variable are used to develop a numerical method. The proposed numerical approach results in an ill-posed linear system of equations and to overcome the ill-posedness, the Tikhonov regularization method is implemented. An effective approach based on the ABC algorithm is established to determine the regularization parameter. To show the robustness and ability of the present approach, for a test case, the results are compared with the results of the L-curve and GCV methods.

Inverse problem, Reaction-Diffusion Problem, Regularization, ABS Algorithm