determine parameter ranges for rescaling

[petrelharp]

Our parameters are:

• θ: proportional to inverse difference in per-capita birth and death rates
• σ = 1/sqrt(θ): dispersal distance
• ε: interaction distance
• N: population density per unit area
• N_loc = ε^d N: neighborhood size (up to a constant)
• T = 1/θ: one unit of rescaled time, the time scale on which population dynamics happen

As we take θ to infinity, how should we change these? Let's fix the spatial scale to

• W = 40: width of the range

Let's aim to scale

• θ between 1 and 100 (where θ=1 probably has a difference between birth and death rates of around 0.1), so
• σ goes from 1 to 0.1

We'll have two conditions:

• (local): ε=σ, so from 1 to 0.1, or
• (nonlocal): ε=10σ, so from 10 to 1.

We also want to have θ/N going to either zero (deterministic) or a nonzero (random) limit, so:

• (deterministic): N = sqrt(θ)^3, so from 1 to 1000
• (random): N = θ, so from 1 to 100

In these four cases we have maximum neighborhood sizes of:

• (local, determinstic): N_loc = 100 (d=1) or = 10 (d=2)
• (local, random): N_loc = 10 (d=1) or = 1 (d=2)
• (nonlocal, deterministic): N_loc = 1000 (d=1 or d=2)
• (nonlocal, random): N_loc = 100 (d=1 or d=2)

[petrelharp]

Terence writes:

1) θ is proportional to inverse difference in per-capita birth and death rates: To be more precise, for the pre-limit of PDE/ SPDEs with logistic term, the difference between per-capita birth and death rate is proportional to 1/theta but also dependent on the epsilon-neighbourhood size. For the PME case, birth rate will be N_loc (which corresponds to local neighbourhood density I suppose?) + 1/\theta, and death rate will be N_loc + N_loc/\theta. For FKPP case, it would be easier and we should see birth rate = 1+ 1/ \theta, death rate = 1+ N_loc / \theta 2) For the local / non-local condition on epsilon, we may want to consider the case where epsilon is fixed throughout for the non-local case. It may be also helpful to consider scaling \epsilon^d at the same rate at \theta to take into account dimensionality of our model. 3) How does SLiM really calculate interaction strength? Does it just count the number of points within a circle of epsilon radius around a point? That might be slightly different from our actual mathematical model, so I would like to know more to ensure we will have good simulation results.

[petrelharp]

1. Right - well, the way I do this is to have the arguments to gamma, mu, and r in units of population density, so birth rate at x is some function of x and \varrho_\epsilon * \eta (x) / N. (We probably should have defined the convolution to include this dividing by N, actually?)
2. Scaling \epsilon^d - so, that might correspond to keeping N_loc fixed? Or taking N_loc to infinity? This seems like a good idea, but I haven't figured out what hte motivation is. \epsilon and \sigma can clearly be compared, and we know that qualitatively different things happen if \epsilon > \sigma.
3. SLiM actually integrates the point measure against a kernel (we're using Gaussian, but we could use the indicator of a circle...)