Exponential Runge-Kutta integrators for modelling Predator-Prey interacttions

Spatially explicit models consisting of reaction-diffusion partial differential equations are considered in order to model prey-predator interactions, since it is known that the role of spatial processes reveals of great interest in the study of the effectsof habitat fragmentations on biodiversity. As almost all of the realistic models in biology, these models are nonlinear and their solution is not known in closed form. Our aim is approximating the solution itself by means of exponential Runge-Kutta integrators. Moreover, we apply the shift-and-invert Krylov approach in order to evaluate the entire functions needed for implementing the exponential method. This numerical procedure reveals to be very efficient in avoiding numerical instability during the simulation, since it allows us to adopt high order in the accuracy.

Spatially explicit models consisting of reaction-diffusion partial differential equations are considered in order to model prey-predator interactions, since it is known that the role of spatial processes reveals of great interest in the study of the effects of habitat fragmentation on biodiversity. As almost all of the realistic models in biology, these models are nonlinear and their solution is not known in closed form. Our aim is approximating the solution itself by means of exponential Runge-Kutta integrators. Moreover, we apply the shift-and-invert Krylov approach in order to evaluate the entire functions needed for implementing the exponential method. This numerical procedure reveals to be very eff cient in avoiding numerical instability during the simulation, since it allows us to adopt high order in the accuracy. This work has received funding from the European Union's Seventh Framework Programme FP7/2007-2013, SPA.2010.1.1-04: "Stimulating the development of GMES services in specif c are", under grant agreement 263435, project title: Biodiversity Multi-Source Monitoring System:from Space To Species (BIOSOS) coordinated by CNR-ISSIA, Bari-Italy (http://www.biosos.eu). © 2012 American Institute of Physics.