Figure 3

Simulation for the delay prey–predator model of variable order (2.8) of the Atangana–Baleanu–Caputo fractional derivative numerically mentioned as Eq. (4.5). In (a)–(b) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for \(\epsilon = \vert \sin (\epsilon \tau ) \vert \), \(\varsigma =\tanh (3-\tau )\), and \(\kappa =0.01\) seconds. In (c)–(d) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for \(\epsilon =1-\frac{1}{10}\exp (-\frac{1}{2} \tau )\), \(\varsigma =\frac{\sqrt{t}}{2}\), and \(\kappa =0.03\) seconds. In (e)–(f) phase portrait and time series \(\mathbf{x}(\tau )\) and \(\mathbf{y}(\tau )\) for \(\epsilon = \vert \sin (\epsilon \tau ) \vert \), \(\varsigma =1-\frac{1}{10}\exp (-\frac{1}{2} \tau )\), and \(\kappa =0.5\) seconds