Title: Numerical analysis of a nonlocal Lotka-Volterra system for competing species
Abstract: A nonstandard finite difference scheme is proposed for numerical approximation of the Lotka-Volterra competition model
$$ \left\{\begin{array}{l} u_t=J*u-\left(\int_{\Omega}J\right)u+u(K_1-u-av),\\ v_t=J*v-\left(\int_{\Omega}J\right)v+v(K_2-v-bu),\\ u(0, \mathbf x)=u_0(\mathbf x), \ v(0,\mathbf x)=v_0( \mathbf x), \ {\bf x\in\Omega\subset \mathbb{R}^2}, \end{array} \right.$$ with nonlocal interaction between species. We show that the scheme is uniquely solvable, stable, and that the numerical solution approaches the true solution $(u(t,{\bf x}),v(t,\bf{x}))$ with rate $\mathcal{O}(\Delta t+|\Delta \bf x|^2)$ as $\Delta t,|\Delta {\bf x}|\rightarrow 0$, uniformly on a finite interval $ [0,T] $. This is joint work with Jianlong Han and Sarah Duffin.