## Preprint 2013-03

## Raimund Bürger, Ricardo Ruiz-Baier, Canrong Tian:

### Stability analysis and finite volume element discretization for delay-driven spatial patterns in a predator-prey model

### Abstract:

Time delay is an essential ingredient of spatio-temporal predator-prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by some constant time lag for the gestation of predators. Specifically, time delay is considered within a predator-prey reaction-diffusion system. A stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations obtained by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that the delay can induce spatial patterns when the carrying capacity of the prey is large. The numerical method consists of a finite volume element (FVE) method for the spatial discretization of the model combined with a Runge-Kutta scheme for its time discretization.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Ricardo RUIZ-BAIER, Canrong TIAN: ***Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model*. Mathematics and Computers in Simulation, vol. 132, pp. 28-52, (2017).