We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system is ten times more efficient than the one originally proposed in [Breda et al, SIAM Journal on Applied Dynamical Systems, 2016], as it avoids the numerical inversion of an algebraic equation.

翻译：我们通过普通差异方程式提出非线性更新方程式的近似值。 我们认为整合状态是绝对连续的,符合延迟差异方程式。 通过对差异方程式的抽象配方法应用假光谱法,我们获得了普通差异方程式的近似平衡系统。 我们为平衡和相关的特征根基提供了趋同证据,我们使用生态学和流行病学的一些模型来说明对平衡和周期解决方案进行数字对齐分析的方法的好处。 数字模拟显示,新近似系统的实施效率比[Breda 等人,SIM杂志《应用动态系统》,2016年]中最初提出的效率高出十倍,因为它避免了代数方程式的翻版。