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Island Biogeography functions

Survival function

We use the logarithm function as the basic survival function for an island $i$ with area $A_i$: $log(A_i)$

We standardize this by using proportions of total land area: $log(\frac{A_i}{A_{total}})$

The curve can be forced through (0,0) and (1,1): $log(\frac{A_i \times 9}{A_{total}} + 1)$

Increasingly steeper a (0,0) are $log(\frac{A_i \times 99}{A_{total}} + 1) / 2$ and $log(\frac{A_i \times 999}{A_{total}} + 1) / 3$

Generalizing, the basic survival curve is: $log(\frac{A_i \times (10^b - 1)}{A_{total}} + 1) / b$

Finally, a further parameter, $p_{surv}$ sets the survival for $A_{total}$:

Survival probability $= p_{surv} \times log(\frac{A_i \times (10^b - 1)}{A_{total}} + 1) / b$

In the input data, $p_{surv}$ is set with probSurvA and shape parameter $b$ is set with probSurvB. Values of ‘1.0’ for probSurvA and ‘2.0’ for probSurvB are appropriate starting values. You can view the shape of the curve with the ‘-l’ flag.

Dispersal function

For dispersal, we use a negative power function dependent on the distance between two places ( $d_{i,j}$), standardized by maximum distance between any two places ( $d_{max}$): $10^{(-1 \times \frac{d_{i,j}}{d_{max}})}$

The concavity of the curve (steepness at (0,1)) can best be adjusted with a parameter $b$ (as opposed to varying the ‘base,’ i.e. 10): $10^{(-1 \times b \times \frac{d_{i,j}}{d_{max}})}$

Finally, a further parameter, $p_{disp}$ sets the dispersal probability for distance zero, i.e., when landmasses are touching.

Dispersal probability $= p_{disp} \times 10^{(-1 \times b \times \frac{d_{i,j}}{d_{max}})}$

In the input data, $p_{disp}$ is set with probDispA and shape parameter $b$ is set with probDispB. Values of ‘1.0’ for probDispA and ‘2.0’ for probDispB are appropriate starting values. You can view the shape of the curve with the ‘-l’ flag.