In this paper, we discuss high order finite difference weighted essentially non-oscillatory (WENO) schemes, coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintanance of third order spatial / temporal accuracy when the limiters are applied to a third order finite difference scheme and third order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth order accuracy.

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In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \mathcal{A} to an appropriate q-polynomial algebra. In the case that \mathcal{A} is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \mathcal{A} is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.