Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level

Abstract

The characterization of the dispersal of populations of non-identical individuals is relevant to most ecological and epidemiological processes. In practice, the movement is quantified by observing relatively few individuals, and averaging to estimate the rate of dispersal of the population as a whole. Here, we show that this can lead to serious errors in the predicted movement of the population if the individuals disperse at different rates. We develop a stochastic model for the diffusion of heterogeneous populations, inspired by the movement of the parasitic nematode Phasmarhabditis hermaphrodita. Direct observations of this nematode in homogeneous and heterogeneous environments reveal a large variation in individual behaviour within the population as reflected initially in the speed of the movement. Further statistical analysis shows that the movement is characterized by temporal correlations and in a heterogeneously structured environment the correlations that occur are of shorter range compared with those in a homogeneous environment. Therefore, by using the first-order correlated random walk techniques, we derive an effective diffusion coefficient for each individual, and show that there is a significant variation in this parameter among the population that follows a gamma distribution. Based on these findings, we build a new dispersal model in which we maintain the classical assumption that individual movement can be described by normal diffusion, but due to the variability in individual dispersal rates, the diffusion coefficient is not constant at the population level and follows a continuous distribution. The conclusions and methodology presented are relevant to any heterogeneous population of individuals with widely different diffusion rates.

Figure 1

Different patterns in P. hermaphrodita movement in (a) homogeneous and (b) heterogeneous environments. Trails labelled (i)–(iv) vary in nematode speed: (i)=fast through to (iv)=slow.

Figure 2

The distribution of individual's mean speed (mm s⁻¹) in homogeneous (dotted bars) and heterogeneous (hatched bars) environments obtained from 25 replicates.

Figure 3

The distribution of the turning angle corresponding to nematode movement in homogeneous (dotted bars) and heterogeneous (hatched bars) environments. The error bars represent standard errors for n=18 and 17 in the homogeneous and heterogeneous environments, respectively.

Figure 4

(a) Gamma distribution (dashed curve) with parameters ν=0.57 and λ=0.21 fitted to the distribution of individuals' diffusion coefficients (solid line). (b) The corresponding cumulative distribution functions used to perform the Kolmogorov–Smirnov goodness-of-fit test. It is shown that the gamma distribution (dashed curve) fits the data more accurately compared to the best-fit normal distribution (dotted curve) with parameters μ=0.118 and σ=0.187; solid line, diffusion coefficient distribution.

Figure 5

The modelled profile of PDF of nematode dispersal with parameters ν=0.57 and λ=0.21, at three different time intervals corresponding to 5 (solid curve), 10 (dashed curve) and 15 (dotted curve) min.

Figure 6

The modelled profile of the PDF of nematode dispersal at t=15 min, with parameters ν=0.57 and λ=0.21 (solid curve) compared to the PDF profile of a normal diffusion with diffusion coefficient D=0.118 mm² s⁻¹ (dotted curve) with an inset of the log tail of the leptokurtic dispersal provided by the model (solid curve) compared to the log-tail of the normal diffusion (dotted curve).