Global Dynamics of Delayed Sigmoid Beverton-Holt Equation

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation x(n+1) = f(x(n),x(n-1)), n=0,1, horizontal ellipsis , where f is decreasing in the variable xn and increasing in the variable xn-1. As a case study, we use the difference equation x(n+1) = (x(n-1)(2)/cx(n-1)(2)+dx(n)+f)), n=0,1, horizontal ellipsis , where the initial conditions x(-1), x(0)>= 0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.