The variational iteration method (VIM) was used to find approximate numerical solutions of classical and fractional dynamical system equations. To the best of our knowledge, no work on the numerical treatment of q-nonlinear dynamic systems NLDSs is done in the literature. This motivated us to study the numerical solutions of this problem. In this paper, the VIM is extended to find the numerical solutions of q-NLDSs. The proof of the convergence theorem and the error bound analysis are presented. Exact and numerical solutions, by using the extended VIM, of the q-logistic and Lotka–Volterra equations are found. And the comparison shows an excellent matching between the exact and numerical solutions. Approximate numerical solutions of the NLDS of predator–prey with and without self (or cross) difference between two patches are found. In the case of Lotka–Volterra equation, it cannot be solved exactly. Numerical solutions are obtained and a good accuracy is found via evaluating the residual error function. The results show an excellent error tolerance after few iteration steps.

中文翻译：

---

q-动态方程的数值解

变分迭代法 (VIM) 用于寻找经典和分数动力系统方程的近似数值解。据我们所知，文献中没有对q -非线性动态系统 NLDS 进行数值处理的工作。这促使我们研究这个问题的数值解。在本文中，VIM 被扩展来寻找q -NLDSs 的数值解。给出了收敛定理的证明和误差界分析。通过使用扩展的 VIM，q的精确和数值解-logistic 和 Lotka-Volterra 方程被发现。并且比较显示了精确解和数值解之间的极好匹配。找到了两个补丁之间有和没有自我（或交叉）差异的捕食者 - 猎物的 NLDS 的近似数值解。在 Lotka-Volterra 方程的情况下，它无法精确求解。通过评估残差函数，获得了数值解并发现了良好的精度。结果显示经过几个迭代步骤后具有出色的容错性。