Consider the difference equation x k+1 = x k e α−x n−d where α is a positive parameter and d is a nonnegative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d ≥ 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.

We consider the global asymptotic stability of the trivial fixed point of the difference equation x n+1 = mxn − αϕ(x n−1), where (α, m) ∈ R 2 and ϕ is a real function satisfying the discrete Yorke condition: min{0, x} ≤ ϕ(x) ≤ max{0, x} for all x ∈ R. If ϕ is bounded then (α, m) ∈ [|m| − 1, 1] × [−1, 1], (α, m) = (0, −1), (0, 1) is necessary for the global stability of 0. We prove that if ϕ(x) ≡ tanh(x), then this condition is sufficient as well.

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We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.