Oscillation and stability of nonlinear discrete models exhibiting the Allee effect

Abstract

Abstract In this paper, we consider the discrete nonlinear delay population model exhibiting the Allee effect (*) $$x_{n + 1} = x_n \exp \left( {a + bx_{n - \tau }^p - cx_{n - \tau }^q } \right),$$ where a, b and c are constants and p, q and τ are positive integers. First, we study the local stability of the equilibrium points. Next, we establish some oscillation results of nonlinear delay difference equations with positive and negative coefficients and apply them to investigate the oscillatory character of all positive solutions of equation (*) about the positive steady state x * and prove that every nonoscillatory solution tends to x *.