  Reaction-diffusion analytical modeling of predator-prey systems has shown
that specialist natural enemies can slow, stop and even reverse pest invasions,
assuming that the prey population displays a strong Allee effect in its growth.
Few additional analytical results have been obtained for other spatially
distributed predator-prey systems, as traveling waves of non-monotonous systems
are notoriously difficult to obtain. Traveling waves have indeed recently been
shown to exist in predator-prey systems, but the direction of the wave, an
essential item of information in the context of the control of biological
invasions, is generally unknown. Preliminary numerical explorations have hinted
that control by generalist predators might be possible for prey populations
displaying logistic growth. We aimed to formalize the conditions in which
spatial biological control can be achieved by generalists, through an
analytical approach based on reaction-diffusion equations. The population of
the focal prey - the invader - is assumed to grow according to a logistic
function. The predator has a type II functional response and is present
everywhere in the domain, at its carrying capacity, on alternative hosts.
Control, defined as the invader becoming extinct in the domain, may result from
spatially independent demographic dynamics or from a spatial extinction wave.
Using comparison principles, we obtain sufficient conditions for control and
for invasion, based on scalar bistable partial differential equations (PDEs).
The searching efficiency and functional response plateau of the predator are
identified as the main parameters defining the parameter space for prey
extinction and invasion. Numerical explorations are carried out in the region
of those control parameters space between the super-and subso-lutions, in which
no conclusion about controllability can be drawn on the basis of analytical
solutions. The ability of generalist predators to control prey populations with
logistic growth lies in the bis-table dynamics of the coupled system, rather
than in the bistability of prey-only dynamics as observed for specialist
predators attacking prey populations displaying Allee effects. The
consideration of space in predator-prey systems involving generalist predators
with a parabolic functional response is crucial. Analysis of the ordinary
differential equations (ODEs) system identifies parameter regions with
monostable (extinction) and bistable (extinction or invasion) dynamics. By
contrast, analysis of the associated PDE system distinguishes different and
additional regions of invasion and extinction. Depending on the relative
positions of these different zones, four patterns of spatial dynamics can be
identified : traveling waves of extinction and invasion, pulse waves of
extinction and heterogeneous stationary positive solutions of the Turing type.
As a consequence, prey control is predicted to be possible when space is
considered in additional situations other than those identified without
considering space. The reverse situation is also possible. None of these
considerations apply to spatial predator-prey systems with specialist natural
enemies.
