Nonlocal Interaction Effects on Pattern Formation in Population Dynamics

We consider a model for population dynamics such as for the evolution of bacterial colonies which is of the Fisher type but where the competitive interaction among individuals is nonlocal, and show that spatial structures with interesting features emerge. These features depend on the nature of the competitive interaction as well as on its range, specifically on the presence or absence of tails in, and the central curvature of, the influence function of the interaction.

Figure 1

Dramatic difference in the amplitude of the steady state patterns with Gaussian influence function and periodic boundary conditions, for $\Sigma /L=$ (a) $0.205$, (b) $0.245$, and (c) $12$. Here, $u(x)$ is plotted in units of $a/b$, and $x$ in units of $L$.

Figure 2

Dependence of the amplitude of the patterns on the ratio of the width of the influence function to the domain size. Inset, which depicts the small $\Sigma /L$ behavior on an expanded scale, shows an apparent transition around the value $2/9$ of $\Sigma /L$. The arrows mark the three patterns depicted in Fig. 1.

Figure 3

Separation of large-amplitude patterns (shadowed region) from negligible-amplitude patterns for the cutoff Gaussian influence functions with width $\Sigma$ and cutoff length $x_c$. The domain size is $L=100$. Units for $\Sigma$ and $x_c$ are arbitrary but identical.

Figure 4

Counterpart of Fig. 3 for the influence function of Eq. (6). The inset shows $f_{\sigma ,r}$ in the periodic boundary condition case for several values of $r=100$ (dashed), $1$ (dotted), and $0.01$ (full); respectively, $\Sigma =70$, $7$, and $0.7$. The square and the Gaussian emerge as particular cases. Units as in Fig. 3.

Figure 5

Typical steady-state pattern in a 2D system with periodic boundary conditions and a cutoff influence function. Lighter areas correspond to larger values of $u$. Parameters are arbitrary.

Figure 6

Typical steady state patterns with cutoff influence function and zero-flux boundary conditions.