3 May 2017

Literature on theory on stochastic spread

Melbourne & Hastings (2009)

Melbourne & Hastings (2009) has this brief review of theory (p. 1537):

Stochastic models are needed to assess uncer- tainty in the rate of spread because traditional deterministic models provide no information about variance. Analyses have shown that linear sto- chastic models have the same rate of spread as the equivalent deterministic model, whereas sto- chasticity slows spread in nonlinear (density- dependent) models (9, 10). Investigations of the variance of spread rate have largely relied on nu- merical simulations and suggest that variance be- tween realizations can be high, especially when birth rates are low, variation in birth rates between individuals is high, or dispersal includes infrequent long-distance events (11–15). This work sug- gests that great uncertainty in an invasion forecast can be expected because of inherent biological variability.

The rest of the models in the paper are simulation-based. Here are the references cited above:

  1. M. A. Lewis, J. Math. Biol. 41, 430 (2000).
  2. D. Mollison, Math. Biosci. 107, 255 (1991).
  3. J. S. Clark, M. Lewis, J. S. McLachlan, J. HilleRisLambers, Ecology 84, 1979 (2003).
  4. G. J. Gibson, C. A. Gilligan, A. Kleczkowski, Proc. R. Soc. London B Biol. Sci. 266, 1743 (1999).
  5. G. J. Gibson et al., Stat. Comput. 16, 391 (2006).
  6. M. A. Lewis, S. Pacala, J. Math. Biol. 41, 387 (2000).
  7. Analytic solutions for some special cases are provided in (14).

Skellam (1951)

Going back to Skellam (1951) (also cited by BM) it starts out looking promising as it is based on “random dispersal”—but in fact he is using microscale randomness to get to diffusion, and the description of spread is deterministic.

Lewis (2000)

Lewis (2000) studies an integro-difference framework: continuous space. His measure of spread is the “expectation velocity:”

This can be described as the speed of movement \(c\) of a point \(x_t\) beyond which the expected number of individuals \(n_e\) is fixed. Thus \(x_t\) is defined so that the integral of the expected density over the interval \((x_t,\infty)\) is \(n_e\). (p. 432)

He does note other studies have looked at furthest extent:

[Other definitions of spread rates are given by] The rate of change of the average location of the farthest forward individual with respect to time [6]. McKean [24] showed that, in certain linear stochastic processes, the distribution of the furthest forward individual can be modeled by a nonlinear deterministic reaction-diffusion equation (KPP or Fisher equation), which in turn exhibits traveling wave solutions.

The paper defies a quick read, but appears to be focused on the expected abundance at a location and the covariance in abundances between two locations (although the covariance might be in the location of pairs of ineividuals). I don’t see a straightforward way to connect this to our problem.

  1. Biggins, J.D.: How fast does a general branching random walk spread? In: Ath- reya, K.B. and Jagers, P. (eds) Classical and Modern Branching Processes, pp 19–39. Springer, 1998

24 McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov- Petrovskii-Piscounov, Commun. Pure Appl. Math, 28, 323–331 (1975)

Mollison (1991)

This paper is primarily about showing how deterministic linear models can approximate the mean spread in stochastic and nonlinear models. The stochasticity is again at the micro scale: birth-death with Brownian movement of individuals.

Clark et al (2003)

This paper looks at long distance dispersal (LDD). It is mostly about estimating parameters from data, and describing the LDD model (although I think it mostly comes from prior papers by Clark). They say repeatedly that spread in this case is “erratic” and “unpredictable” but don’t quantify it. In three numerical examples (Fig. 2c, 4) they plot medians, 50% and 90% intervals for cumulative distance spread through time. It’s not clear whether these are based on full population models or just draws from the distribution of jump distances. The intervals grow though time, but are smooth only in the case that lacks LDD (Fig. 4b), suggesting that they are not using enough draws to really characterize the distribution. It’s hard to tell how the width of the intervals scales with time.

Possible sources of the LDD model:

  • Clark, J. S., M. Lewis, and L. Horvath. 2001c. Invasion by extremes: population spread with variation in dispersal and reproduction. American Naturalist 157:537–554.
  • Clark, J. S., M. Silman, R. Kern, E. Macklin, and J. HilleRisLambers. 1999. Seed dispersal near and far: gen- eralized patterns across temperate and tropical forests. Ecology 80:1475–1494.

Clark et al (1999) introduces the “2Dt” dispersal kernel. Clark et al (2001) shows how LDD affects mean speed, and that variability in reproduction in the furthest extremens model slows spread. They do not address the variance in spread, although it is illustrated in a numerical example (fig. 5).

Lewis & Pacala (2000)

This paper is a more extensive treatment of hte models in Lewis (2000). Basically, the covariance is of interest because it describes the shape of the wavefront and characterises the extent of patchiness. Variability in spread is not addressed that I can see.

Stochastic models of population spread have been widely studied in the context of the spread of an infection. An introduction to the early work in this area can be found in a review by Mollison [26].

  1. Denis Mollison. Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B, 39(3), 283–326 (1977)

9.7 Summary

I haven’t read all the papers in BM yet, and I haven’t done any other literature search, but so far I haven’t found any characterizations of the variability in cumulative spread. Mark Lewis has a new (2016) book that will probably have anything that exists in the theory literature - I need to digest that next.