Are Diffusion Models too Simple? A Comparison with Telegraph Models of Invasion

Abstract

Diffusion models of animal movement are often criticized because they assume animals have infinite velocity and completely random motion. To investigate the impact of these assumptions, I compared a diffusion model with a telegraph model of dispersal The telegraph model assumes organisms have finite velocity and tend to maintain their direction. I compared the models in two settings: (i) as models for dispersal of nonreproducing organisms and (ii) as models for range expansion of organisms that simultaneously disperse and reproduce (so-called reaction-diffusion or reaction-telegraph models). Both models predict very similar dispersal patterns for nonreproducing organisms. In the case of reproducing organisms, however, they predict grossly different rates of range expansion for all but a small range of parameter values. The disparity is greatest for organisms with high population growth and low movement rates. To assess the magnitude of disparity for different organisms, I used published data to estimate model parameters for the cabbage butterfly (Pieris rapae), gypsy moth (Lymantria dispar), European starling (Sturnus vulgaris), collared turtledove (Streptopelia decaocto), Black Death (Yersinia pestis), and for rabies. All six cases fell within the narrow parameter range where the diffusion and telegraph models yield indistinguishable predictions regarding the rate of range expansion.

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

“…Obviously, regardless the choice of the initial population distribution f (x), the solution given by Equation (23) describes a population decay eventually leading, in the large-time limit, to population extinction. This is in agreement with intuitive expectations: since λ n ∼ L −2 , the condition given by Equation (20) means that the domain is 'too small' to support sustainable population dynamics for the given growth rate.…”

“…Questions arise here as to what may be the reason for the non-positivity and whether the model can possibly be amended to avoid this unrealistic behaviour. We mention here that the corresponding microscopic model that considers movement of individual 'particles' (e.g., animals) is positively defined [20]. The non-positivity of the solution is therefore an artefact of its mean-field counterpart rather than a genuine property of the movement-reproduction dynamics as such.…”

“…However, this is at odds with many observations as well as with common sense [16]: since the body of animals of all species has the front end and the rear end, they are more likely to choose the new movement direction (following re-orientation) close to the movement direction at the preceding moment. The corresponding movement pattern is known as the correlated random walk (CRW) [17] and the corresponding microscopic stochastic process as the telegraph process [18], and their mean-field counterpart is known as a telegraph equation [19][20][21][22][23]:…”

“…The precise meaning of parameter τ can be slightly different depending on the details of the microscopic model; for instance, in the telegraph movement process, it is the time over which the animal moves without changing its movement direction [18][19][20]. Brownian motion thus corresponds to the limit τ → 0 when Equation (1) turns into the diffusion equation.…”

“…Interestingly, in order to include them into the model, it is not enough to add the 'reaction' term (i.e., the population growth rate)-say, F(u)-into the right-hand side of Equation (1) (as is done in the case of diffusive animal movement [14]). An accurate account for birth and death events also modifies the factor in front of the first order derivative on the left-hand side [20,23]. The corresponding model is known as the reaction-telegraph equation:…”