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Computational treatment of traveling-wave solutions of a population model with square-root dynamics

Keywords: Population equation , square-root dynamics , exact nonlinear scheme , non-negativity , boundedness , monotonicity

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Abstract:

Motivated by a recent report by R. E. Mickens, we design an efficient,non-standard, two-step, nonlinear, explicit, exact finite-differencemethod to approximate solutions of a population equation with squarerootreaction law. Mickens’ report establishes the existence of nonnegative,traveling-wave solutions of that model which are boundedfrom above by 1, and which are spatially and temporally monotone.As its analytic counterpart, the computational technique proposed inthe present manuscript is capable of preserving the non-negativity andthe boundedness of initial profiles under suitable and flexible conditionson the computational parameters. We provide theoretical results on theexistence and uniqueness of non-negative and bounded solutions of themethod, and we establish that our technique conditionally preserves thespatial and temporal monotonicity of the approximations. The numericalsimulations obtained through a computer implementation of ourfinite-difference scheme support the fact that the method preserves allof the mathematical characteristics of approximations mentioned above.

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