At the end of 50's, P.H.~Leslie constructed and numerically
analyzed a kind of time-discrete two dimensional dynamical
system derived from Lotka-Volterra type of competing 2-species
ordinary differential equations.
Leslie's idea to derive the time-discrete model is specific,
different from the usual discretization scheme for ordinal
differential equation (for instance, by Euler method), and
is intuitive since it significantly depends on an idea of
mathematical modelling concerning to the original ODE system
in part. His time-discrete system succeeds in qualitatively
conserving well the characteristics of solution for the original
differential dynamical system.
In this paper, we consider some extensions of Leslie's idea to
the more general single-species population dynamics and derive
the time-discrete system which can robustly maintain the
qualitative natures of original ODE system, especially focusing
on the existence and the local stability of equilibria. Further,
we discuss the behaviour of solution near equlibrium, too.