  The paradox of enrichment was observed by M. Rosenzweig in a class of
predator-prey models. Two of the parameters in the models are crucial for the
paradox. These two parameters are the prey's carrying capacity and prey's
half-saturation for predation. Intuitively, increasing the carrying capacity
due to enrichment of the prey's environment should lead to a more stable
predator-prey system. Analytically, it turns out that increasing the carrying
capacity always leads to an unstable predator-prey system that is susceptible
to extinction from environmental random perturbations. This is the so-called
paradox of enrichment. Our resolution here rests upon a closer investigation on
a dimensionless number $H$ formed from the carrying capacity and the prey's
half-saturation. By recasting the models into dimensionless forms, the models
are in fact governed by a few dimensionless numbers including $H$. The effects
of the two parameters: carrying capacity and half-saturation are incorporated
into the number $H$. In fact, increasing the carrying capacity is equivalent
(i.e. has the same effect on $H$) to decreasing the half-saturation which
implies more aggressive predation. Since there is no paradox between more
aggressive predation and instability of the predator-prey system, the paradox
of enrichment is resolved.
  We also further explore spatially dependent models for which the phase space
is infinite dimensional. The spatially independent limit cycle which is
generated by a Hopf bifurcation from an unstable steady state, is linearly
stable in the infinite dimensional phase space. Numerical simulations indicate
that the basin of attraction of the limit cycle is riddled. This shows that
spatial perturbations can sometimes (neither always nor never) remove the
paradox of enrichment near the limit cycle!
