  Mathematical modelling of the evolution of the size-spectrum dynamics in
aquatic ecosystems was discovered to be a powerful tool to have a deeper
insight into impacts of human- and environmental driven changes on the marine
ecosystem. In this article we propose to investigate such dynamics by
formulating and investigating a suitable model. The underlying process for
these dynamics is given by predation events, causing both growth and death of
individuals, while keeping the total biomass within the ecosystem constant. The
main governing equation investigated is deterministic and non-local of
quadratic type, coming from binary interactions. Predation is assumed to
strongly depend on the ratio between a predator and its prey, which is
distributed around a preferred feeding preference value. Existence of solutions
is shown in dependence of the choice of the feeding preference function as well
as the choice of the search exponent, a constant influencing the average volume
in water an individual has to search until it finds prey. The equation admits a
trivial steady state representing a died out ecosystem, as well as - depending
on the parameterregime - steady states with gaps in the size spectrum, giving
evidence to the well known cascade effect. The question of stability of these
equilibria is considered, showing convergence to the trivial steady state in a
certain range of parameters. These analytical observations are underlined by
numerical simulations, with additionally exhibiting convergence to the
non-trivial equilibrium for specific ranges of parameters.
