An alternative delayed population growth difference equation model

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.