So far, we’ve modeled scale disparities of up to 4 orders of magnitude between predators and prey, and it could be interesting to go further! In our simulations, the prey and the environment are always at the same scale, and the prey tracks the environment pretty closely (see the upper right graph). Predators only ‘perceive’ the environment via prey abundances. So when the entire range of environmental variation (in this case, the full sine wave) fits within a single cell of the predator lattice, predators only respond to the average value of the environment, as our example output suggests (center figure). Increasing the scale disparity between predator and environment shouldn’t change this result.
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That said, I’d also like to point out that we make the simplifying assumption that juveniles disperse, while adults do not; and that each time step encompasses one generation of each species. So our results say more about differences in home range size than about migration possibilities. To look at migrations specifically, we’d want to use a smaller time step and allow for adult dispersal. I think that we could deal with large-magnitude spatial scale disparities in the same way we do now, but we’d have to be thoughtful in choosing an appropriate temporal scale to report on.

To answer your question, I’m going to concentrate on the prey-predator scale disparity here, because in all runs we’ve done so far, prey populations are at the same scale as the environmental variation. It would be interesting to change that in the future, because there are undoubtedly cases where a prey species can’t perceive all variation in its environment. I would expect the result to be similar to what we see for the predator under a scale disparity.
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It’s possible to get a rough estimate of whether our scale disparities are realistic, using home-range data for predator-prey pairs. For example, the average home range size of bobcats in central Florida is 1450 Ha for females and 2550 Ha for males (Wassmer et al. 1998). The average home range size of one of their main food sources, the marsh rabbit, is 3.96 Ha (Forys & Humphrey 1996). These numbers give spatial scale disparities of 1:366 and 1:643, which is comfortably within the range of disparities that we modeled (1:1 to 1:10,000).
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There is also some history in the literature of experimental manipulation in spatial scale disparities. Kawata and Agawa (1999) imposed scale disparities of 1:49, 1:196 and 1:441 in an algae-snail system by changing the size of the algae’s substrate plates. Interestingly, in the latter two cases, the snails responded to the algae as if it was homogeneously distributed.

Thanks! It’s certainly a lot of material to communicate.
To address your question: The previous work that we’ve seen tends to either model predators and prey on the same lattice (similar to our S1 simulation runs) or as a mean-field source of prey mortality (similar to our S100). Much of this literature also does not look at species interactions in the context of a spatially varying environment, as we do. Our work is novel in that it provides a way to study and compare the same-scale and mean-field extremes of scale disparity, as well as the entire range between them.

Thanks! You make a lot of good points, so I’m afraid you’re in for a long response.
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What we hope to get in the end is an understanding of how scale disparity in species interactions affects the patterns we see on the landscape.
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In our model, we assume that predators don’t respond to spatial pattern at any finer grain than that of their own lattice. So we expect that increasing the disparity between predators and their prey/environment would cause the distribution of predator populations to be increasingly spatially averaged compared to the prey/environment pattern. That’s what we see when we decrease the environmental pattern size with respect to the extent of the predator populations (center figure)- the differences between large and small predator populations decrease. We generally see the same thing as we increase the prey-predator scale disparity (bottom graph), although there’s a region where the 1:25 predators do worse than the 1:100 predators. We’ve got standard error on there (and it’s tiny!), so we’re still mulling over what this could mean.
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As for the prey, we expect all prey populations to converge to their carrying capacity in the absence of a predator, which we saw (result not shown). In the presence of a predator at the same scale as the prey, we expected the prey to be displaced from the ‘best’ lattice cells, because predators have access to all of the information about the prey’s spatial distribution in this case. We expected that this displacement would decrease with increasing scale disparity, because predation pressure would be spread over more populations. What we saw instead was that scale disparity didn’t have much of an effect at all (upper graph)! Changing the environmental pattern did have some effect even though the prey and environment are at the same spatial scale, because dispersal distances were constant.
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What would a cool result look like? It would be really interesting to find situations where predators track the environment more closely than their prey, despite not having direct access to environmental pattern information. We’ve had hints of this so far, both in our work and in the work of another team member who is applying this model to social evolution. It’s something we’re still exploring.
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For us, success would be a model that meets our expectations in known test cases (e.g. without predation, without dispersal, with minimal/maximal scale disparity), and in intermediate cases,
tells us something new about how scale disparities between interacting species affect patterns we see in the natural world.

I’m glad you enjoyed the video, and hope you find it useful! All we mean by ‘prey fidelity’ is how well the distribution of prey populations tracks environmental variation. Our environmental pattern is represented by values defining the prey’s carrying capacity in every cell. So any process that reduces the prey populations in each lattice cell from that carrying capacity will reduce the fidelity of the prey pattern to the environmental pattern.

Glad you liked the video!

Because of the huge variety in scale-disparate processes that occur in ecological systems (and beyond), we set out to make our model as general as possible. For example, our code allows the user to define the order in which the population processes are executed. We also coded for the possibility that the prey is at the larger scale- which would be the case if you were studying, say, carnivorous plants.

We’re currently collaborating with another student who is modifying this model to study questions in social evolution. Right now, she’s looking at the consequences of disparities between the size of villages and the governments that oversee them.

To answer your first question, the minimum for our metric is 0.75 (when the environment is made up of a single sine wave, as it is here). We used the 2D Fourier transform in our analysis because it explicitly pinpoints the different scales that make up spatial patterns. Because our environment is sinusoidal, its transform only has peaks in the center and at n steps away from the center, where n is the number of periods in the region. All patterns can be expressed as a sum of sine waves, so any reduction in the ability of our simulated species to track the environment shows up in the Fourier transform as the appearance of peaks at other frequencies. Our metric compares the location and size of peaks in the Fourier transforms of the environment and species lattices.
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So far, we’ve just stuck with simple environments, but I can certainly see the draw of a fractal environment. We use continuous boundary conditions, so one advantage to the sinusoid is that we can avoid discontinuities at the ‘edges’. As for the dynamics, we wanted to start with something as linear as possible in order to get a good understanding of our model before complicating it. Our future choices of dynamics will definitely depend on what our next questions are!