Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

► We study spatial models of predator-prey systems. ► Nonlocal intraspecies prey competition is modelled with integral kernels. ► Behind predator invasions there can be different spatiotemporal patterns. ► Linear stability analysis correctly predicts patterns near instabilities. ► Kurtosis of the integral kernel plays an important role in selecting patterns.