  We propose an alternative delayed population growth difference equation model
based on a modification of the Beverton-Holt recurrence, assuming a delay only
in the growth contribution that takes into account that those individuals that
die during the delay, do not contribute to growth. The model introduced differs
from existing delay difference equations in population dynamics, such as the
delayed logistic difference equation, which was formulated as a discretization
of the Hutchinson model. The analysis of our delayed difference equation model
identifies an important critical delay threshold. If the time delay exceeds
this threshold, the model predicts that the population will go extinct for all
non-negative initial conditions and if it is below this threshold, the
population survives and its size converges to a positive globally
asymptotically stable equilibrium that is decreasing in size as the delay
increases. Firstly, we obtain the local stability results by exploiting the
special structure of powers of the Jacobian matrix. Secondly, we show that
local stability implies global stability using two different techniques. For
one set of parameter values, a contraction mapping result is applied, while for
the remaining set of parameter values, we show that the result follows by first
proving that the recurrence structure is eventually monotonic in each of its
arguments.
