High Resolution High Order Schemes For A Hierarchical Size-Structured Model

Hierarchical size-structured population model is an important structured population model in mathematical biology. This model mainly describes the evolution of hierarchically size-structured population at a given time. Hierarchical size-structured population model has been used in modeling many biology problems such as modeling the competition for sunlight in a forest and modeling the competition for food and the advantage of reproduction among some kind of animals. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model with nonlinear growth, mortality and reproduction rates.In this paper we develop and discuss three explicit finite difference schemes, namely a first order upwind scheme, a second order high resolution scheme and a fifth order weighted essentially non-oscillatory (WENO) scheme for solving the hierarchical sizestructured population model with nonlinear growth, mortality and reproduction rates.For the first order upwind scheme and the second order high resolution scheme, we prove their TVB (Total Variation bounded) property. Then we prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions.Secondly we develop a high order explicit finite difference WENO scheme for solving the model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high order WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.