Clustering, advection, and patterns in a model of population dynamics with neighborhood-dependent rates

We introduce a simple model of population dynamics which considers reproducing individuals or particles with birth and death rates depending on the number of other individuals in their neighborhood. The model shows an inhomogeneous quasistationary pattern with many different clusters of particles arranged periodically in space. We derive the equation for the macroscopic density of particles, perform a linear stability analysis on it, and show that there is a finite-wavelength instability leading to pattern formation. This is responsible for the approximate periodicity with which the clusters of particles arrange in the microscopic model. In addition, we consider the population when immersed in a fluid medium and analyze the influence of advection on global properties of the model, such as the average number of individuals.

Figure 1

Spatial configurations for the BB model at two different times. Left: configuration after $100$ steps, with a large number of surviving clusters. Right corresponds to a single cluster remaining after $3000$ steps. The value of the parameters are ${\lambda }_0={\beta }_0=0.5$, $D={10}^{-5}$, and the initial population is of $N_0=1500$ bugs randomly distributed.

Figure 2

(a) Total number of particles, $N\left(t\right)$, vs time for the BB model and three different values of the control parameter $\mu$: From top to bottom: $\mu =5\times {10}^{-4}$, $\mu =0$, and $\mu =-5\times {10}^{-4}$; here $D={10}^{-5}$. (b) Idem for the ND model and four different values of $\mu$: From top to bottom, $\mu =0.7$, $\mu =0.5$, $\mu =0.4$, and $\mu =0.3$. Two are above critical $\left({\mu }_c\approx 0.4\right)$ and two below it. The other parameter values are $R=0.1$, $N_s=50$, and $D={10}^{-5}$.

Figure 3

Long-time average number of particles $N\left(t\right)$ vs $\mu$. Left panel corresponds to $D={10}^{-4}$ and right to $D={10}^{-5}$, and the other parameters as indicated. The average is taken from the instantaneous particle numbers at times between $1000$ and $10000$ steps; the error bar indicates the standard deviation of the instantaneous fluctuations around this mean value.

Figure 4

Long-time spatial structures for the ND model. Left column corresponds to two patterns with the same value of $D={10}^{-4}$, and two different values of $\mu =0.5$ (up) and $\mu =0.9$ (bottom). Right column corresponds to fixed $\mu =0.7$, and $D={10}^{-4}$ (upper), and $D={10}^{-5}$ (bottom). In all the plots, $N_s=50$ and $R=0.1$.

Figure 5

Structure factors (left panels) and corresponding height of the main peak (right) for different patterns in the ND model. Upper panels are for fixed value of $\mu =0.7$ and the values of $D$ shown in the legend box. Bottom panels are for $D={10}^{-5}$ and different $\mu$’s as shown in the legend box. In all the plots, $N_s=50$ and $R=0.1$. Note the different scales in all the plots.

Figure 6

Linear growth rate $\lambda$ vs wave number $K$ from (8) for different values of $\mu$ close to ${\mu }_P$. We take $R=0.1$ and $D={10}^{-5}$ so that ${\mu }_P=0.185$ and $K_m=47.79$.

Figure 7

Spherically averaged continuum structure function against $K$ for different values of the control parameter $\mu$. The other parameter values are $D={10}^{-5}$ and $R=0.1$.

Figure 8

Steady spatial pattern from the deterministic equation (7). $\mu =0.70$, $R=0.1$, $D={10}^{-5}$, and $N_s=50$. Note the strong similarity with the pattern in the bottom-right plot in Fig. 4.

Figure 9

$N\left(t\right)$, normalized with $\mu N_s∕\pi R^2$, vs time for different values of the external flow strength, $A$, and the two chaotic maps in the text. Left is for the Harper map and right for the Standard map. From top to bottom: $A=0$, $A=0.01$, $A=0.05$, and, fluctuating around the value $1$ (white line), $A=1$ and $A=3$. The $A=0.01$ and $A=0.05$ plots cannot be properly distinguished for the figure of the Standard map. The other parameters: $\mu =0.8$, $N_s=50$, $D={10}^{-5}$, and $R=0.1$.

Figure 10

Long-time spatial structure for the ND model with an external flow (Harper map). From top to bottom and left to right, $A=0$, $A=0.1$, $A=0.5$, and $A=1$. The other parameters: $\mu =0.9$, $D=5\times {10}^{-6}$, $N_s=50$, and $R=0.1$.

Figure 11

Structure factor for the patterns shown in Fig. 10.