  We employ partial integro-differential equations to model trophic interaction
in a spatially extended heterogeneous environment. Compared to classical
reaction-diffusion models, this framework allows us to more realistically
describe the situation where movement of individuals occurs on a faster time
scale than the demographic (population) time scale, and we cannot determine
population growth based on local density. However, most of the results reported
so far for such systems have only been verified numerically and for a
particular choice of model functions, which obviously casts doubts about these
findings. In this paper, we analyse a class of integro-differential
predator-prey models with a highly mobile predator in a heterogeneous
environment, and we reveal the main factors stabilizing such systems. In
particular, we explore an ecologically relevant case of interactions in a
highly eutrophic environment, where the prey carrying capacity can be formally
set to 'infinity'. We investigate two main scenarios: (i) the spatial gradient
of the growth rate is due to abiotic factors only, and (ii) the local growth
rate depends on the global density distribution across the environment (e.g.
due to non-local self-shading). For an arbitrary spatial gradient of the prey
growth rate, we analytically investigate the possibility of the predator-prey
equilibrium in such systems and we explore the conditions of stability of this
equilibrium. In particular, we demonstrate that for a Holling type I (linear)
functional response, the predator can stabilize the system at low prey density
even for an 'unlimited' carrying capacity. We conclude that the interplay
between spatial heterogeneity in the prey growth and fast displacement of the
predator across the habitat works as an efficient stabilizing mechanism.
