  Spatio-temporal complexity of ecological dynamics has been a major focus of
research for a few decades. Pattern formation, chaos, regime shifts and long
transients are frequently observed in field data but specific factors and
mechanisms responsible for the complex dynamics often remain obscure. An
elementary building block of ecological population dynamics is a prey-predator
system. In spite of its apparent simplicity, it has been demonstrated that a
considerable part of ecological dynamical complexity may originate in this
elementary system. A considerable progress in understanding of the
prey-predator system's potential complexity has been made over the last few
years; however, there are yet many questions remaining. In this paper, we focus
on the effect of intraspecific competition in the predator population. In
mathematical terms, such competition can be described by an additional
quadratic term in the equation for the predator population, hence resulting in
the variant of prey-predator system that is often referred to as Bazykin's
model. We pay a particular attention to the case (often observed in real
population communities) where the inherent prey and predator timescales are
significantly different: the property known as a `slow-fast' dynamics. Using an
array of analytical methods along with numerical simulations, we provide
comprehensive investigation into the spatio-temporal dynamics of this system.
In doing that, we apply a novel approach to quantify the system solution by
calculating its norm in two different metrics such as $C^0$ and $L^2$. We show
that the slow-fast Bazykin's system exhibits a rich spatio-temporal dynamics,
including a variety of long exotic transient regimes that can last for hundreds
and thousands of generations.
