Difference equations with the Allee effect and the periodic Sigmoid Beverton-Holt equation revisited

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton-Holt equation x(n+1) = a(n)x(n)(delta n)/1+x(n)(delta n), x(0) > 0, n = 0,1,0, ... , where the a(n) and delta(n) are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable delta. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass.