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.