We simulated two types of traits: one with a latitudinal trend in the mean and the other with a longitudinal trend. For each type of trait, we simulated a range of different degrees to which the geographical axis (latitude or longitude) contributed to the overall variance in the trait. Specifically, we let x′ and y′ be normalized latitudinal and longitudinal variables, respectively (that is, x′ = (x − x̄)/σx and y′ = (y − ȳ) σy, where x is a vector of each individual’s longitude, y is likewise for latitude, t̄ is the mean value of the elements of t, and σt is their standard deviation). We then simulated two phenotypes with the mean determined by x′ or y′: φx = x′ + εx and φy = y′ + εy, where ε is a vector of random normal deviates with mean 0 and variance s2. We let s2 take values of (1, 4, 19, 99), so that the resulting variance in the traits are approximately (2, 5, 20, 100), and the proportion of variance explained is approximately (50, 20, 5, 1) per cent.
