Incorporating overhead costs and handling times into invasive species control

I have a new paper out, focused on optimising resource allocation with multiple control methods. Invasive species eradications involve complicated logistics and cost structures, and my PhD optimal papers didn’t really consider this (though see my Ashmore Reef & fire ant chapter for an application of overhead and start-up costs to optimal detection). I tried to break down some of these assumptions to understand their effect on optimal management, with the help of Paul Armsworth and Suzanne Lenhart from the University of Tennessee and the National Institute for Mathematical and Biological Synthesis. In my PhD I looked at how to allocate resources for a single control method, but there are usually multiple options available, and it is common to use several in a single eradication. Hence, we asked: how do you split your budget between two different control methods.

The key difference between the control methods was that one had an overhead cost and the other had a handling time. Overhead costs are costs that must be constantly paid to have a control method available. These differ from start-up costs. A start-up cost is a once-off payment that allows a method to be used in the future; this models initial training or purchase costs. Overhead costs model things like aircraft rental for aerial baiting. The cost of the aircraft does not change with how much you use it (only fuel, bait and personnel costs). Handling times impose a limit to how quickly species can be removed, and they typically apply to ground-based control methods, such as trapping. If traps are checked daily, then a single trap cannot remove more than one individual per day, even if many more individuals encounter the trap. There is a fixed amount of time associated with each removal.

One of the reasons that I started this work was that I had understood that the standard wisdom was to start eradications with expensive broad-scale methods and then switch towards the end. While this makes some intuitive sense, I wondered why and whether it is the best strategy or not. Essentially, should you start by using methods with high overhead costs and switch at the end? The results from this paper would say ‘yes’, but there is some nuance. We always found that the high overhead cost method should be used from the start, and, depending on the magnitude of the overhead cost, be phased out at some stage. Though, for small overhead costs, the effort is just reduced towards the end, rather than a complete stop. The other result is that we found that the ground-based method with handling time should commence before the overhead cost method is ceased. So, rather than a direct switch, a gradual change from one method to the other is optimal.

overheads_finalplot

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PhD papers

My final PhD paper is finally out! So, I think this is the perfect time to post an overview of what I did & lessons learnt. While all of the papers focus on invasive species control, they vary from being quite theoretical to applied.

Chapter 1: Spatial control of invasive species in conservation landscapes

In this paper, we introduced the population dynamic model that I used for most of my thesis. The basic idea of this paper was to determine the best way to allocate control effort spatially in order to minimise the abundance of an invasive species at a specific location. We showed that the optimal allocation is related to the ratio of spread rate to growth rate. If this ratio is small, then control effort should be focused in a small area, while if the ratio Is large, then control effort should be spread out more widely in the landscape.

Chapter 2: Placing invasive species management in a spatiotemporal context

In this paper, we considered the spatial problem, along with the associated problem of removing an invasive species from an island. In the spatial problem, we looked at how a buffer zone heuristic compared to the optimal solution, and we found that an appropriately chosen buffer zone performed extremely well. We also found an interesting trade-off for island eradications, between the species growth rate and the diminishing marginal returns of control effort. In particular, the faster the species growth rate, the faster eradication should be completed. Even though it’s intuitive to think that a high growth rate means that eradications will take longer, it actually means that it’s imperative to complete them quickly.

Chapter 3: Target the Source: Optimal Spatiotemporal Resource Allocation for Invasive Species Control

In this paper, I solved for optimal spatio-temporal resource allocation. These results really pointed toward a principle of targeting the source of invasions. That is, the part of the landscape which is responsible for the majority of spread. It’s important to note that the location of the ‘source’ can change through time. High-density regions close to low-density regions are prime examples, while a high-density region surrounded by other high-density regions would not be a source, as it would not be able to effectively cause spread into new regions.

Chapter 4: Modelling tropical fire ant (Solenopsis geminata) dynamics and detection to inform an eradication project

In this paper, we focused on fire ant control at Ashmore Reef in the Timor Sea. There were two main aspects to this: understanding how different ant baits affect the population and modelling the detectability of fire ants. There are two classes of ant bait available: toxin and growth regulator. We used pilot data to quantify the effect of each bait and then used this to suggest that a combined use of both over two years would be required to complete eradication. Following any eradication attempt, it is vital to estimate the probability of success. We used detection experiments to quantify the probability of detection and then optimise the number of lures that should be used. Further, we considered the possibility of detector dogs, and we found that, provided sufficient use, canine detection would be a cost-effective option.

 

Biol Invasions_S geminata distribution_Figure

Native fire ant range (left) and the islands of Ashmore Reef Commonwealth Marine Researve (right)

 

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Ecosystem Modelling at SCBO16

This week I’m at the Society of Conservation Biology Oceania section meeting in Brisbane. This year I’m presenting on some of the work I’ve done about predicting how species reintroductions and eradications affect other species in an ecosystem. To see all the details, come to room P6 at 12.30pm on Friday. We’ve also started working on making the code to run the analysis available. It’s pretty basic, but we’ve written it in MATLAB and R, and by running this you can recreate some of the plots in my presentation. We’re still working on improving it and hopefully a more comprehensive program is not too far away. In the meantime, you can easily adjust the network structure to see how certain assumptions about the species interactions affect the predicted outcomes.

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PhD completion and next steps

It’s been some time since I updated this blog, almost two years. The time has flown by and has included lots of travel, hard work and a bit of fun along the way. This time next week I will have just finished by completion seminar, which, for those who are interested and in Melbourne, is at 4pm next Friday in G26, BioSciences 1. I’ll be submitting my thesis in the week or two following that, and then heading to Pennsylvania State University to join the Shea lab as a postdoc.

Last year, two papers that I led were published. I didn’t get around to writing about them at the time, so I’m going to give a quick recap.

Baker C. M. and Bode M. (in press). Placing invasive species management in a spatio-temporal context. Ecological Applications. [pdf]

As the title suggests, we used a spatiotemporal model in the paper, however, we didn’t solve for optimal management using the full model. Rather we focused on different variations of the model, specifically assuming either steady-state or no spatial variation, and solved for optimal management. We argue that, even though we didn’t solve management problems with it, using a spatiotemporal model is useful because it allows us to use more types of data to estimate parameter values and because considering different types of problems at once enables us to see similarities between them.

Baker C. M., Hughes B. D. and Landman K. A. (2015) Length-based connectivity metrics and their ecological interpretation. Ecological Indicators, Volume 58, pg. 192-198. [pdf]

In this paper we considered different metrics that one can use to quantify habitat fragmentation. This was motivated by the use of the ‘radius of gyration’ in the ecological literature. The radius of gyration links geometric properties of habitat pages to the dispersal range of a species that is confined to that habitat. Unfortunately, we show that the equation used in the ecological literature differs from the definition in every other field, and it therefore loses the above property. Anyone can easily invent a new habitat metric, but any arbitrary metric is unlikely to have a useful physical interpretation. We argue that this is one of the few metrics that does have a physical interpretation, and therefore it is important to use it correctly.

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Confirmation and Jervis Bay

It’s two weeks since I passed the one year mark of my phd. I’ve come a long way since joining the School of Botany last year: when people say ‘parameterise’, I think of the process of figuring out parameter values rather than an equation for a shape; I now know that what ecologists call a community matrix is really just what I know as a Jacobian; and when people say Xanthorrhoea, I know what they are talking about. Along with random ecological knowledge, a year as a PhD student in the School of Botany brings confirmation.

Confirmation is the time of your PhD when you sit down with your supervisors, plus some other academics, and convince them that you’ve made sufficient progress and you will be able to finish. Even though I was quite confident in passing, writing my confirmation report took a surprisingly long time (I tried really hard to keep it short and not spend too much time…).  After passing my confirmation, I had to rush of to a workshop at in Jervis Bay Territory, which is on the coast, pretty much directly east of Canberra.

Jervis Bay’s national park, Booderee, has a pretty big problem with invasive species, with over one hundred species present there. One of the more notorious is the red fox. About 10 years ago the park management started intensive fox control. Although some of the native animals in the park did well once fox numbers were reduced (e.g. bandicoots, wallabies), the fox cull has some unexpected consequences with a sharp decline in greater glider and ringtail possum numbers. These declines have been attributed to either an increase in powerful owls, or an increase in competition for tree hollows due to more brushtail possums (or possibly both). Whatever the reason, this has highlighted that unintended (and surprising) consequences can result from altering ecosystems.

On a tour of Booderee with senior ecologist, Nick Dexter

On a tour of Booderee with senior ecologist, Nick Dexter

So, why were we there? Well, park management is planning on introducing potoroos, and they want to know what possible unintended effects there could be. Although the ecosystem in Booderee is incredibly complex, we have a lof data about how the abundance of many of the species changed with a big reduction in fox numbers. Along with a good knowledge of which species interact (e.g. wallabies eat plants, owls eat bandicoots), this was enough information to give qualitative modelling a go.

The idea behind qualitative modelling is pretty simple. We don’t actually now exactly how all of the species interact, so we generate a model with a network of random interactions (that is consistent with what we know about their interactions). We then cull foxes in the model and look at how all the species react. If the reactions match the abundance data we keep the random network, otherwise it’s discarded. We repeat this until we have thousands of possible interactions. Using these, we can simulate what might happen when we make a change to the system (such as adding potoroos), and hopefully foresee any unacceptable consequences.

Steamers Beach. An evening swim in the warm water inside the bay was a perfect way to relax after a long day of workshopping.  Also if you visit, keep an eye out for stingrays.

Steamers Beach. An evening swim in the warm water inside the bay was a perfect way to relax after a long day of workshopping. Also if you visit, keep an eye out for stingrays.

One problem with this kind of modelling is: what if we failed to generate the ‘true’ network of interactions? There is really no way to prove that we have considered every possible type of network. But, at the moment it is the best way that we have to convert our knowledge of species interactions into predictions of what could happen. I guess our predictions are really a set of outcomes that we can’t rule out as not happening. A way to get an answer to this would be to simulate a system and then use qualitative modelling to try and figure out what is happening in the simulated system. Then we could actually get an idea of how much data is required to get decent predictions of a system.

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Spatial control of invasive species

Invasive predators are a major threat to Australia’s native species. Ideally, we would like to eradicate them, but well established invasive predators are very hard to eradicate – particularly because Australia is so big. To limit the damage that predators, like feral cats and foxes, do we control their numbers in important locations – such as in national parks. When we remove predators from a region, it is not permanent. The space is soon taken up by other predators moving in to fill the space. Hence, to keep invasive predator densities low, control efforts need to go on continuously – making the process very expensive. We would like to limit the costs as much as possible. Mike’s and my new paper addresses how control efforts should vary spatially in this situation.

Cats are one of the introduced predators which have spread across the entire continent.

Feral cats are one of the introduced species which have spread across the entire continent. (source)

Qualitatively, the answer is quite straightforward. If you are trying to protect a location, you should focus your control efforts close by and taper the effort off further away. Although it seems kind of obvious, this doesn’t happen in practice. Managers tend to control in a buffer zone around the location. Determining a good buffer zone is also hard to do, especially if you are trying to do it experimentally. This paper is a first step in tackling the problem from a mathematical modelling perspective. The spatial distributions that we have presented are probably too intricate to reproduce in practice. We are now working on finding distributions which can be used in practice, which are almost as good as the optimal solution.

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Quantifying Randomenss

Last week I wrote about the use of random walks and the diffusion equation in ecology. I’m going to continue on with this theme and tell you about the first research problem that I worked on. This was back at the start of 2011, when I was doing a summer vacation research project.

The idea is: how do you tell if some collection of objects is spatially random. I’m going to illustrate this idea by considering a clowder of cats. If somebody gives you a photo with the positions of cats in their backyard, how can you tell if they are just at random positions or not? It might seem like a funny question, but in the context of my previous post, you require the cats to be in random positions if you want to model their movement with the diffusion equation.

To answer the question, we need to know what spatial randomness looks like. It’s a slightly strange thing to think about. It’s quite easy to imagine what random looks like, but it’s very hard to actually write down a definition. However, it’s quite easy to state some things that aren’t  random. For example, if all the cats were in one corner

backyard_corner

or if they were spread out evenly

backyard_even

we would say that they are not in random positions. So, what does randomness look like then? Luckily, we have a measure for this. We call it I, which stands for index. I’m sure we could have come up with something better, but this is all a bit obscure anyway, so it doesn’t really matter. To calculate the I, we divide the region up in to boxes, count how many cats are in each box and calculate the variance. You then divide the variance by the largest possible variance for the system. Which, in our case, is when all the cats are a single box. This means that the index ranges from 0 (the cats are uniformly spread out) to 1 (the cats are really not at all spread out).

We actually know what the value of the index should be if the objects are spread out randomly. This is called the index of complete spatial randomness, Icsr , which is one divided by the number of objects. In our example this is equal to 0.125. The closest that we can get to this is by having one or two boxes with two cats. It looks something like this:

backyard_csr1 backyard_csr2

If you want to know all the gory details (such as how do we know what the value of Icsr  is), the paper is available here.

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Random Walks and Diffusion in Ecology

Last week I had my very first paper accepted! It’s some work that came out of my masters, which I did in the mathematical biology group at Melbourne University. To mark the occasion, I’m going to write a few posts about random walks and how they relate to biology and ecology.

One of the most common models for animal movement in ecology is diffusion. It has been used by people like Ronald Fisher and James Murray, and it continues to be used extensively today. However, the reason why diffusion is a good model of animal movement is not obvious. Particularly when you consider that the diffusion equation also describes heat flow.

The diffusion equation is derived assuming that movement is random. The concept of a random walk is quite simple. Imagine a piece on an (infinitely large) chessboard. If you move the piece to a random adjacent square every second, then the piece performing a random walk. Getting from this simple idea to the diffusion equation takes a little sweat, which I won’t put you through. However, the basic idea is that if you make the squares on your chessboard really small and start moving your piece really often, then, out pops the diffusion equation.

Diffusion Equation

You can read all about the derivation here. Visually, the process of shrinking the lattice looks like this:

Random walk with large steps

Random walk with large steps

Random walk with small steps

Random walk with small steps

Random walk with tiny steps

Random walk with tiny steps

The diffusion equation gives average behavior: it will give you the probability that a particle on a random walk will be at a specific site at a given time. It will give you the expected behaviour of a large group, but will not be able to tell you where specific individual is.

Of course you can’t use the diffusion equation to model anything. You need the animals to be moving fairly randomly, without strong interactions. For example, you could not model individuals in a school of fish. This actually correspond to negative diffusivity, which is a different kettle of fish.

The diffusion equation actually works quite well for some systems which violate the assumptions. We assumed that the step sizes were infinitely small, but the equation works perfectly well when the step sizes are relatively large. There are all kinds of variations on the basic random walk. You can always derive a nonlinear diffusion equation for a random walk, but the standard diffusion will often suffice. For example, the diffusion equation will do just fine if you are modelling animals which are on a persistent random walk.

The diffusion equation is an exceptionally useful equation. However, it is important to understand the assumptions that were made in the derivation when you use it. Even if you’re going to ignore the assumptions and use it anyway.

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Six Months Down

I’m quickly approaching the six month stage of my PhD. It has certainly been quite an exciting time and a big change from last year. I was in one of the largest research groups in the maths department. It turns out that a group of 9 is actually quite small.

As well as the massive size of the group, there is also a really big range of skills in QAEco. Much of the group’s work is centred on using mathematics for either modelling or improved decision-making in an ecological setting. Therefore there are people here with an ecological background who haven’t done maths since high school, people like me with a background in mathematics and statistics, and everyone in-between. Working so closely with people with different backgrounds is a huge advantage. For me this is that I can easily ask ecologists what they think of my models, which means I am more aware of potential issues than I would be otherwise.

I’ve managed to fit in a couple of trips to Brisbane in an attempt to avoid as much of the cold Melbourne weather as I can. I spent two weeks at the University of Queensland attending the Australian Mathematical Sciences Institute winter school, and in about a week I will be heading back for the annual CEED conference.Me with Ian Walker (Minister for Science, Information Technology, Innovation and the Arts) and Andrew Cramer at the winter school opening ceremony.

Me with Ian Walker (Minister for Science, Information Technology, Innovation and the Arts) and Andrew Cramer at the winter school opening ceremony.

Although not everything at the winter school was particularly relevant to me, it was worth the trip. I got to meet people from EDG, spend way too much time playing ‘killer bunnies’ with mathematicians and be constantly surprised and amazed by how much bigger the birds are in Brisbane.

The week after the winter school was the AMSI run Mathematics for Planet Earth – The Conference, held in Melbourne. I managed to throw a last minute poster together for the conference during some of the less interesting winter school lectures and my flight home to present at the conference. Later that week I finished my first paper from PhD work (on optimal spatial baiting), which is still in review. Hopefully everything goes well!

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