Ecological Archives E084-029-A2

Robin E. Snyder. 2003. How demographic stochasticity can slow biological invasions. Ecology 84:1333–1339.

Appendix B. A description of the definitions and methods for invasion speeds.

The equations I used to find the asymptotic deterministic speeds yield "expected density speeds". The expected density speed is the speed of a point moving with the wave so that the expected population density at the point remains constant. For the stochastic models, I used the furthest forward speed, which is the distance between furthest forward individuals in successive time steps. It is possible to measure expected density speeds for stochastic models; however, to do so risks introducing statistical biases, since one must choose a scale over which to measure population density and then smooth the data. Fortunately, Mollison and Daniels have proven that the expected density speed equals the furthest forward speed for continuous time models without density-dependence (Mollison and Daniels 1993), and this result seems likely to hold for discrete time models also. (See Lewis 2000 for a good discussion of different speed definitions for stochastic models.)

I found speeds by running each stochastic model 2000 times and performing a linear regression of furthest forward location against time for each run. The coefficient of time (the slope of the best-fit line plotting distance vs. time) gave me the speed for each run, and I averaged the speeds. I ran models with density-dependent reproduction for 100 time steps. Memory constraints forced me to stop models with density-independent reproduction after 17 or 18 time steps, since it is necessary to keep track of every individual's location, and the number of individuals quickly becomes very large when reproduction is density-independent. In both cases, I discarded the initial 5 time steps to reduce the effects of transients. (Discarding more than this did not yield significantly different speeds.)

Literature cited

Lewis, M. A. 2000. Spread rate for a nonlinear stochastic invasion. Journal of Mathematical Biology 41:430–454.

Mollison, Denis, and Henry Daniels. 1993. The "deterministic simple epidemic" unmasked. Mathematical Biosciences 117(1–2):147–153.


[Back to E084-029]