When ecologists build simulations of populations of plants or animals, we usually add in randomness, because populations are always changing in ways that are impossible to fully pre-determine. This randomness is usually white noise, which means that the random values at each point in time are uninformed by the values that came before. Each time, the slate is wiped clean and a new value is drawn from scratch. This is a nice and easy way to generate randomness, but the problem is that populations usually don’t fluctuate with white noise, but with red noise.
Red noise is a kind of randomness where each random value is positively correlated with the ones that came before. That is to say, it’s randomness with a memory: when it starts to go on a downward trend, it tends to continue that downward trend. You can see this in the images above, where the red noise is much less “wiggly” than the white noise. Blue noise is the opposite of red noise: it also has a memory, but it is negatively correlated to previous points and tends toward the opposite of what happened before.
Population ecologists have already studied colored noise in population growth. Populations usually exhibit red noise in their fluctuations. This is important to know because red noise tends to increase extinction risk: if a population size happens to wiggle downward, then it tends to continue in that direction, until it might eventually crash. But overall population size is just one part of the picture. Population size comes from the combination of births (fertility) and deaths (mortality) of individuals of different ages and life stages. In our study, we wanted to find out if fertility and mortality in different life stages exhibit colored noise in their year-to-year fluctuations, and if they do, how that might change the way we simulate populations.
Nature is full of colored noise, but it can be hard to measure, because you need at least fifteen continuous time points to get an accurate read on noise color, and most datasets on populations are short-term. Fortunately, the COMADRE and COMPADRE databases have detailed population data on a variety of organisms, some of them long-term. We gathered all the long-term datasets we could find, and measured the noise color in fertility and mortality at different life stages. Here are two examples.
The one on the left is the mountain gorilla, and the one on the right is a plant called velvety goldenrod. Each column in the matrix represents an age, youngest at the left and oldest at the right (velvety goldenrod usually only lives one or two years.) The first row of the matrices is fertility at that age, and the other rows are survival. In both of these species, you can see that fertility and survival have very different noise colors.
Each matrix is also outlined in a color that indicates the noise color for overall population growth. You can see that the noise color for the population growth isn’t always obvious from the noise color of individual fertility and survival squares within the matrix. In the gorilla, the population growth is strongly red noise, like the fertility rates, while in the goldenrod the growth rate has nearly white noise, somewhere in between the blue noise of the fertility and the red noise of survival.
We hope our work will open the door for other researchers to learn more about colored noise in fertility and survival. To that end, we created a package in R for modeling populations with colored noise. It’s called, appropriately, colorednoise.
The findings and applications of this work can be found in a recent publication here:
Pilowsky & Dahlgren. 2019. Incorporating the temporal autocorrelation of demographic rates into structured population models. Oikos DOI 10.111/oik.06438