Nonlinear diffusion effects on biological population spatial patterns

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density $u(x,t)$. The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form ${\partial }_{t}u(x,t)=D{\partial }_{xx}u{(x,t)}^{\nu }$ (with $D,\nu >0$), encompassing the cases where the state-dependent diffusion coefficient either increases ($\nu >1$) or decreases ($\nu <1$) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.

Figure 1

Long-time patterns obtained from numerical integration of Eq. (3), with $a=b=1$, $L=100$, $D=0.1$, $w=10$, and different values of $\nu$ indicated on the figure, in (a) linear and (b) logarithmic scales. In the inset, the long-term profiles in the full grid are shown. The profiles are plotted for $t=200$, but they remain unchanged after $t\simeq 100$.

Figure 2

Time evolution of the density profile obtained from numerical integration of Eq. (3) with $a=b=1$, $L=100$, $D=0.1$, $w=10$, and $\nu =4$, represented for different times $t$ indicated on the figure, in (a) linear and (b) logarithmic scales. The thick line corresponds to $t=200$.

Figure 3

Dispersion relation ${\Lambda }_k={\lambda }_k/a$ vs scaled $k$ (solid line), for $\beta =5\times {10}^{-4}$. The dotted line corresponds to the term $-\sin \left(wk\right)/\left(wk\right)$ and the dashed line to the zero line, drawn for comparison. In the inset, we show the position of the first maximum $k_0$ as a function of $\beta \equiv \nu D{u}_0^{\nu -1}/\left(a{w}^2\right)$. The vertical dotted line indicates the instability threshold.

Figure 4

Fourier spectra for the long-time patterns shown in Fig. 1. The vertical dotted line indicates the position of the dominant mode $k_0$.

Figure 5

(a) Long-time patterns obtained from numerical integration of Eq. (3) up to time $t=100$, with $a=b=1$, $\nu =1$, $w=5$, $L=100$, and different values of $D$ indicated on the figure. In the inset, the profiles in the full grid are shown. In (b) the time evolution for $D={10}^{-5}$ is shown in logarithmic scale. Times are indicated in the figure.

Figure 6

Number of maxima $m$ as a function of the nonlocality width $w$, for $a=b=1$, $L=100$, and $D=0.01$, and different values of $\nu$ indicated in the figure. The solid line corresponds to the approximate theoretical relation (10) and the symbols to the outcomes of numerical simulations.

Figure 7

Number of maxima $m$ as a function of $\nu$ for two different values of $w$ indicated in the figure, with $a=b=1$, $L=100$, and $D=0.01$. Dashed lines correspond to the theoretical prediction given by Eq. (10), with the additional condition (9), defining a critical value ${\nu }_c$ beyond which no patterns occur (we set $m=0$ in such case).

Figure 8

Phase diagram of pattern formation in the $\nu w$ plane, for $u_0=1$, following Eq. (9). Shadowed is the region where no patterns arise. The lines show the frontier of the phase diagram for other values of $u_0=a/b$. Patterns emerge for $w>w_c$ (above the critical line). Parameter values are $L=100$, $D=0.01$, and $a=1$.

Figure 9

Pattern amplitude $\Delta u\equiv u_{\text{max}}-u_{\text{min}}$ as a function of the interaction width $w$, for $a=b=1$, $L=100$, $D=0.01$, and different values of $\nu$ indicated in the figure. The dotted lines are a guide to the eyes; the vertical ones indicate the critical value predicted by Eq. (9).