This paper investigates influence of fractional order on the dynamic behavior of a cross-reaction diffusion model. Stability analysis determines the existence conditions of equilibrium points, and bifurcation analyses are performed. A numerical method combining high-order time-precision generating functions and a nine-point difference spatial discretization scheme is proposed, incorporating the short memory principle to improve computational efficiency. Numerical simulations demonstrate the spatial pattern formation of the system at different parameters, validating the effectiveness of the theoretical results. Some novel patterns are shown.
Citation: Hao Lu Zhang, Xiao Yu Li, Zhi Yuan Li. Influence of fractional order on the dynamic behavior of a cross-reaction diffusion model[J]. Networks and Heterogeneous Media, 2026, 21(6): 2310-2328. doi: 10.3934/nhm.2026086
This paper investigates influence of fractional order on the dynamic behavior of a cross-reaction diffusion model. Stability analysis determines the existence conditions of equilibrium points, and bifurcation analyses are performed. A numerical method combining high-order time-precision generating functions and a nine-point difference spatial discretization scheme is proposed, incorporating the short memory principle to improve computational efficiency. Numerical simulations demonstrate the spatial pattern formation of the system at different parameters, validating the effectiveness of the theoretical results. Some novel patterns are shown.
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Figures(8) / Tables(3)
Hao Lu Zhang, Xiao Yu Li, Zhi Yuan Li. Influence of fractional order on the dynamic behavior of a cross-reaction diffusion model[J]. Networks and Heterogeneous Media, 2026, 21(6): 2310-2328. doi: 10.3934/nhm.2026086
Stability and eigenvalue distribution of System (1.1) at the equilibrium $E_*(3.4957, 8.3977)$ for $r = 1.04$, $K = 8.0$, $\beta = 0.8$, $e = 0.4$, $c = 1.6$, $\zeta = 0.8$, $s = 0.39$, $d = 0$, $d_1 = 0$, $d_2 = 0$, $\gamma = 2.1$. The dots denote the eigenvalues $\lambda_i$ of the characteristic equation, the dashed lines mark the critical angles $\pm\pi/(2m)$, the shaded sectors indicate the stable region $|\arg(\lambda)| > \pi/(2m)$ and the unstable region $|\arg(\lambda)| < \pi/(2m)$. The unit circle is included only as a visual reference and does not enter the stability criterion
Phase portraits of System (1.1) at different fractional orders, with $d_1 = 0$, $d_2 = 0$, $d = 0$, $r = 0.8$, $K = 10.0$, $\beta = 0.8$, $e = 0.4$, $c = 1.6$, $\zeta = 0.8$, $s = 0.4$, $\gamma = 2.1$. Panel (a) corresponds to $K = 10.0$ and panel (b) to $K = 15.0$
The critical wave-number at $r = 0.85, K = 10.0, \beta = 0.4, e = 0.5, c = 1.6$, $\zeta = 0.4, d s = 0.8, d = 0.9, d_1 = 0.2, d_2 = 2.5, \gamma = 2.1.$
Stability and eigenvalue distribution of System (1.1) at the equilibrium $E_*(3.4957, 8.3977)$ for $r = 0.85$, $K = 8.0$, $\beta = 0.8$, $e = 0.4$, $c = 1.6$, $\zeta = 0.4$, $s = 0.8$, $d = 0.9$, $d_1 = 0.2$, $d_2 = 2.5$, $\gamma = 2.1$. The three panels correspond to different wave-numbers $k$ and different fractional orders $\alpha_1 = \alpha_2$, as indicated in the sub captions. The dots denote the eigenvalues $\lambda_i$ of the characteristic equation, the dashed lines mark the critical angles $\pm\pi/(2m)$, the shaded sectors indicate the stable region $|\arg(\lambda)| > \pi/(2m)$ and the unstable region $|\arg(\lambda)| < \pi/(2m)$. The unit circle is included only as a visual reference and does not enter the stability criterion
Patterns at $ \alpha_1 = \alpha_2 = 1.2 $, $ t = 20,000 $. Parameters (from left to right and top to bottom) are shown in Table 1
Patterns at $ \alpha_1 = \alpha_2 = 1.3 $, $ t = 5000 $. Parameters (from left to right and top to bottom) are shown in Table 2
Patterns at different $ \alpha_1 = \alpha_2 $. Parameters (from left to right) are shown in Table 3
Effect of the sign of the cross-diffusion coefficient $ d $ on pattern formation in System (1.1) at $ \alpha_1 = \alpha_2 = 1.3 $, $ d_1 = 0.2 $, $ d_2 = 2.0 $, $ r = 0.8 $, $ K = 8.1 $, $ e = 0.9 $, $ c = 1.8 $, $ \zeta = 0.5 $, $ s = 0.8 $, $ \beta = 0.5 $, $ \gamma = 2.5 $: (a) $ d = +0.8 $ (positive cross-diffusion), producing large-scale stripe--patch patterns; (b) $ d = -0.8 $ (negative cross-diffusion), producing small-scale spot patterns. From left to right, $ t = 5000 $, $ 10,000 $, $ 15,000 $, $ 20,000 $
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