On the structural sensitivity of some diffusion–reaction models of population dynamics

Abstract

In mathematical ecology, it is often assumed that properties of a mathematical model are robust to specific parameterization of functional responses, in particular preserving the bifurcation structure of the system, as long as different functions are qualitatively similar. This intuitive assumption has been challenged recently (Fussmann & Blasius, 2005). Having considered the prey–predator system as a paradigm of nonlinear population dynamics, it has been shown that in fact both the bifurcation structure and the structure of the phase space can be rather different even when the component functions are apparently close to each other. However, these observations have so far been largely limited to nonspatial systems described by ODEs. In this paper, our main interest is to investigate whether such structural sensitivity occurs in spatially explicit models of population dynamics, in particular those that are described by PDEs. We consider a prey–predator model described by a system of two nonlinear reaction–diffusion–advection equations where the predation term is parameterized by three different yet numerically close functions. Using some analytical tools along with numerical simulations, we show that the properties of spatiotemporal dynamics are rather different between the three cases, so that patterns observed for one parameterization may not occur for the other two ones.