Figure 2

Mathematical reproduction for the prey–predator design by means of the Atangana–Baleanu–Caputo fractional derivative with variable order (2.7) through the numerical plan indicated by Eq. (3.26). In (a)–(b) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for \(\epsilon = \vert \sin (\epsilon \tau ) \vert \) and \(\varsigma =\tanh (3-\tau )\). In (c)–(d) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for for \(\epsilon =1-\frac{1}{10}\exp (-\frac{1}{2} \tau )\) and \(\varsigma =\frac{\sqrt{\tau }}{2}\). In (e)–(f) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for \(\epsilon = \vert \sin (\epsilon \tau ) \vert \) and \(\varsigma =1-\frac{1}{10}\exp (-\frac{1}{2} \tau )\)