Connecting The Dots: Using Cross-Correlations to Understand Population Connectivity

Populations in nature sort of function like a giant game of “Red Rover”. You have different groups, or populations, of individuals in different places. Occasionally, individuals run away from their population and, depending on a variety of factors, they are either absorbed by another population, or come back home to their original population. A population that accumulates more individuals is a stronger population and won’t disappear as quickly. A population that loses all of it’s individuals, loses the game.

This dynamic, where by multiple populations are interlinked by exchanging individuals, is what ecologists refer to as “population connectivity”. Just as in the Red Rover game, populations which have lots of individuals coming in, or immigrating, are more inclined to grow and do very well, and are much less likely to go extinct. Likewise, populations that have high birth rates - another way populations can add individuals - tend to do well. On the other hand, populations with lots of individuals leaving (i.e. emigrating) or dying are more inclined to disappear, since these two processes remove individuals from a population. Obviously, each population has a combination of addition and removal of individuals occurring, and it’s the net gain or loss that determines the success of the population.

Immigration + Birth > Emigration + Death = Successful Population

Immigration + Birth < Emigration + Death = Unsuccessful Population

So - immigration and emigration are important to understanding population dynamics because they help control addition and loss of individuals to and from a population. Understanding the specific connectivity patterns among populations, or knowing where the individuals are coming from and going to, is also important important though. Think of populations A and B of…. Oh, I don’t know… mussels (of course). Let’s start by assuming that the babies coming from Population A go back to Population A, and the babies coming from Population B go back to Population B - what we call “retention”.

If we were to start fishing Population B, we would create a feedback loop, where by each individual we removed from Population B meant less babies could be born to come back to Population B. The Population would quickly go extinct in this scenario.

BUT, if Population B is also getting babies from Population A, then the fishing you do in Population B doesn’t matter as much, since Population B is constantly being rescued by babies from Population A where you’re not fishing and reducing the population.

These sorts of principles are central to planning many Marine Protected Areas - creating lots of Population As to sustain and enhance our fishing production in Population Bs. Just as with species interactions, the consequences of population connectivity can be very complex and difficult to understand. Ecologists’ biggest challenge at the moment, though, is determining the actual links, or “connectivity”, among populations - especially in the ocean where we can’t easily track planktonic larvae of fish and shellfish.

As a postdoc in Downeast Maine, I worked with a group of ecologists and physical oceanographers to understand how separate mussel populations along the coast were connected by larval dispersal. Since mussel harvesting is still an active industry Downeast, our research could eventually be used for fishery and management purposes. But, it takes a long time to build a biophysical model of larval dispersal for mussels (just a complex term for a mathematical model that predicts where and when mussel babies are born, float to, settle down, and grow up). So, while we were working on the biophysical model, we wanted to develop some other methods that also estimate population connectivity - 1) so we could check the model predictions against these other estimates, and 2) because we were just curious, and had a bunch of data lying around.

One of the larger datasets I had collected over the years were time-series (repeated measurements though time) of how much mussel spawning and settlement was happening in multiple populations spread along 75 miles of coastline (as the bird flies). It occurred to my advisor and I that, if we knew when and where a bunch of mussel larvae were born (our spawning time-series data), around how long they floated around in the open ocean for (what we call the “pelagic larval duration” or PLD - 4 to 6 weeks for mussels), and when and where a bunch of larvae settled down (our settlement time-series data), we could see how the timing of spawning from each population matched up with the timing of settlement at other populations to estimate population connectivity. Depicting this visually, if we have Populations A, B, and C and we know populations A and B spawned at a different times. We also know when larvae settled in population C. We can project out when we would expect larvae from Populations A and B to settle down according to the PLD time.

If we look at how those expected settlement times for populations A and B matchup with actual settlement times in population C, we can draw conclusions as to where those settling larvae came from!

To do this analytically, we first had to account for the fact that populations within the same geographic region had very similar spawning times (see my blog post in the mussel population dynamics section), complicating any spatial resolution in our signal. To solve this, we grouped data by geographic region (which was consistent with univariate clustering results) and summarized regional spawning and settlement time-series with Local Regression (LOESS with smoothing parameters chosen via generalized cross-validation).

Then, to quantify how spawning and settlement times overlapped, we used cross-correlation analysis, setting a “lag” based on the PLD time for mussels. Cross-correlations are a method of seeing how two series of data line-up/overlap (0=they don’t line up at all, 1=they line up perfectly).

The results of the cross-correlation analysis were pretty interesting. Below you can see a heat map reflecting cross correlations (more red = more correlated, more white = less correlated) between regional spawning data (columns) and settlement data (rows). If you look across a region’s settlement row, the columns tell you the region larvae probably came from. Regions are ordered from most North (1) to most South (3). Based on our 2015 data, it seems that larvae are traveling from northern populations to southern populations - especially more immediately adjacent populations. This is probably because there is a strong ocean current (the Eastern Maine Coastal Current) that flows from northeast to southwest along the Maine coastline. There are a lot of region-specific connectivity patterns as well though. Region 1 seems to only get larvae from itself (high retention, but no immigration). Populations in Region 2 get larvae from themselves too, but also get larvae from populations in Region 1 (a combination of retention and immigration). Populations in Region 3 seem to get larvae from Regions 1 and 2, but not from themselves (no retention, but lots of immigration).

Based on our current subjective understanding of regional connectivity and ocean currents, the cross-correlation method seems to be pretty effective. As I mentioned before, the working group and I are quantifying population connectivity in a variety of ways - this cross-correlation analysis, that biophysical model of larval dispersal, and even with shell chemistry data. The hope is that all three methods give us similar answers - which would suggest we have a very good understanding of how mussel population connectivity is working in Downeast Maine. Consistent results would also mean we could use the biophysical model for years when we don’t have other data, and feel confident that the predictions are accurate.

All in all though, the method I outlined above could be very useful to ecologists who just want a coarse understanding of population connectivity in their geographic region. Spawning and settlement data are relatively easy to gather, and are probably already available for many years for many species because of other research. The cross-correlation analysis is also a relatively simple and straightforward method, meaning connectivity estimates could be just a few lines of code away.

And for a nice takeaway, I overlaid the connectivity patterns from the cross-correlation analysis onto a map of Downeast Maine. Enjoy!