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Oscillations of two kinds of delay Nicholson models with density constraint are mainly considered. The nonlinear models are transformed into the corresponding linear delay differential equation by the theory of oscillation. The difference equations are obtained by using the linear θ − method. Oscillations conditions of the analytical solution and numerical solution of models are obtained. The properties of non-oscillatory solutions are investigated. In order to better explain the results, some numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the unconditional stability and the generally unconditional stability of the Grünwald Letnikov method for fractional-order delay differential equations (FDDEs), which is the generalization of P -stability and G P -stability for classical integer-order delay differential equations. Using the Z -transform, an equivalent form of the discrete Laplace transform, we first show the unconditional stability of the Grünwald Letnikov method for any delay and any constraint mesh. Secondly, we also derive the generally unconditional stability of the Grünwald Letnikov method with a linear interpolation for approximating the delay term under a general uniform mesh. It is shown that the Grünwald Letnikov method for FDDEs preserves the stability for the analytical solution and hence naturally inherits the α -dependence. Finally, two numerical examples for FDDEs and time fractional-order diffusion equations with delay are presented to demonstrate the validity and effectiveness of theoretical results. • We establish a general framework of the numerical stability analysis for FDDEs. • The unconditional and general unconditional numerical stability of Grunwald Letnikov method are established. • Our results generalize the P-stability and GP-stability for classical DDEs. • We prove that the Grunwald Letnikov method for FDDEs preserves the stability and hence naturally inherit the α -dependence. [ABSTRACT FROM AUTHOR]