Written by Jonathan Reynolds, Head of Predation Control Studies as part of a two-part series. Read part one here.

Part two - How was it done? 

We had data from 74 gamekeepers, all of whom favoured lamping as their principal means of fox control. The number of foxes seen on each lamping foray can reasonably be assumed to reflect the number of foxes present. You would expect the removal of foxes to result in fewer foxes being seen the next time. If not, there must have been replacement. The longer the time between successive lamping forays, the more likely that culled foxes will have been replaced. So, every removal is potentially ‘informative’ about both the number of foxes present and how quickly those killed are replaced.

However, it isn’t a matter of simple subtraction and addition. Even if the population was unchanged, the number of foxes seen would vary from night to night for many reasons, not least being weather and chance. This variability makes the data ambiguous: there are many conceivable scenarios that could have given rise to the same observations. But we can work out which is the most likely scenario. This is what Tom Porteus has done, acquiring his PhD in the process.

The puzzle is like that facing fishery scientists, where again the best information about the population comes from the catch. Combining catch data with a knowledge of how the fishing was done, it is possible to deduce what is happening to the fish stock. Our collaborator in all this, and Tom Porteus’ supervisor, is Murdoch McAllister, a recognised expert on modelling fish stocks around the world. It is from Murdoch’s team at the University of British Columbia, and fisheries research in general, that Tom has acquired his analytical methods.

First, Tom wrote a computer model representing the population processes – births and deaths, immigration and emigration – by which fox numbers changes through time. A second model describes the lamping process in quite mechanical terms like the time spent lamping, vehicle speed, visibility of foxes. The two models are linked because the number of foxes seen while lamping reflects both the number present at that time and searching efficiency.

We have no direct information on most of the processes represented in the population model. Some could perhaps never be measured directly, and it would be difficult to measure any of them while culling was ongoing. How frequently do dispersing foxes arrive on the estate looking for somewhere to live? How long would the foxes have lived if they hadn’t been culled? How many foxes could the estate hold before their intolerance for each other drove some away (we call this the ‘carrying capacity’)? How efficiently is the gamekeeper searching the estate, given the limitations of access, terrain and vegetation cover?

When computer models are used to make weather forecasts or financial forecast, you input relevant data (like weather station readings), and your model outputs predictions from those data. In this fox culling scenario, the model is used rather differently. We already have the outcome (the data recorded by each gamekeeper). Instead we want to know what input values the model would have needed to produce that outcome. So we try out millions of combinations of values – a modern computer can do that very swiftly – until the model’s outputs closely match the gamekeeper’s data. At that point the model is a fair representation of the actual situation, and its input values are the numbers we want to know.

This process, known as ‘fitting the model to the data’, must be steered a bit to prevent the computer running off into la-la land. There are of course biological limits to litter size, and to the natural longevity of foxes in the absence of culling. We can also put realistic limits on how many foxes are likely to be available to move onto the estate from the surrounding region. And we have a good understanding of the practical constraints on lamping, like vehicle speed and visibility of foxes. Where we have such prior knowledge, we can constrain the computer from trying impossible or unlikely values. Collating all the relevant knowledge from the literature was a major task in itself - listed here:

• Porteus, T.A., Reynolds, J.C., & McAllister, M.K. (2018). Establishing Bayesian priors for natural mortality rate in carnivore populations. Journal of Wildlife Management, 82: 1645-1657.
• Porteus, T.A., Reynolds, J.C., & McAllister, M.K. (2018). Quantifying the rate of replacement by immigration during restricted-area control of red fox in different landscapes. Wildlife Biology, 2018: wlb.00416: 1-9.
• Porteus, T.A., Reynolds, J.C., & McAllister, M.K. (2019). Modelling the rate of successful search of red foxes during population control. Wildlife Research, 46: 285-295.

The end result of this ‘fitting’ process is a set of realistic estimates of those things we couldn’t measure directly, which include carrying capacity, the number of cubs born, and the risk of death from natural causes; and – arguably most interesting of all – immigration rate. Comparing these estimates among different estates, we begin to understand what determines the success or otherwise of fox control. The model estimates the number of foxes within the estate, week by week, so we can see how it responded to removals by culling. And we can compare the fox density in any week against the carrying capacity, to see how much worse things could have been in the absence of control.

Read a summary of the paper by Porteus et al here or see the full paper here.