Econstudentlog

How Species Interact

There are multiple reasons why I have not covered Arditi and Ginzburg’s book before, but none of them are related to the quality of the book’s coverage. It’s a really nice book. However the coverage is somewhat technical and model-focused, which makes it harder to blog than other kinds of books. Also, the version of the book I read was a hardcover ‘paper book’ version, and ‘paper books’ take a lot more work for me to cover than do e-books.

I should probably get it out of the way here at the start of the post that if you’re interested in ecology, predator-prey dynamics, etc., this book is a book you would be well advised to read; or, if you don’t read the book, you should at least familiarize yourself with the ideas therein e.g. through having a look at some of Arditi & Ginzburg’s articles on these topics. I should however note that I don’t actually think skipping the book and having a look at some articles instead will necessarily be a labour-saving strategy; the book is not particularly long and it’s to the point, so although it’s not a particularly easy read their case for ratio dependence is actually somewhat easy to follow – if you take the effort – in the sense that I believe how different related ideas and observations are linked is quite likely better expounded upon in the book than they might have been in their articles. The presumably wrote the book precisely in order to provide a concise yet coherent overview.

I have had some trouble figuring out how to cover this book, and I’m still not quite sure what might be/have been the best approach; when covering technical books I’ll often skip a lot of detail and math and try to stick to what might be termed ‘the main ideas’ when quoting from such books, but there’s a clear limit as to how many of the technical details included in a book like this it is possible to skip if you still want to actually talk about the stuff covered in the work, and this sometimes make blogging such books awkward. These authors spend a lot of effort talking about how different ecological models work and which sort of conclusions these different models may lead to in different contexts, and this kind of stuff is a very big part of the book. I’m not sure if you strictly need to have read an ecology textbook or two before you read this one in order to be able to follow the coverage, but I know that I personally derived some benefit from having read Gurney & Nisbet’s ecology text in the past and I did look up stuff in that book a few times along the way, e.g. when reminding myself what a Holling type 2 functional response is and how models with such a functional response pattern behave. ‘In theory’ I assume one might argue that you could theoretically look up all the relevant concepts along the way without any background knowledge of ecology – assuming you have a decent understanding of basic calculus/differential equations, linear algebra, equilibrium dynamics, etc. (…systems analysis? It’s hard for me to know and outline exactly which sources I’ve read in the past which helped make this book easier to read than it otherwise would have been, but suffice it to say that if you look at the page count and think that this will be an quick/easy read, it will be that only if you’ve read more than a few books on ‘related topics’, broadly defined, in the past), but I wouldn’t advise reading the book if all you know is high school math – the book will be incomprehensible to you, and you won’t make it. I ended up concluding that it would simply be too much work to try to make this post ‘easy’ to read for people who are unfamiliar with these topics and have not read the book, so although I’ve hardly gone out of my way to make the coverage hard to follow, the blog coverage that is to follow is mainly for my own benefit.

First a few relevant links, then some quotes and comments.

Lotka–Volterra equations.
Ecosystem model.
Arditi–Ginzburg equations. (Yep, these equations are named after the authors of this book).
Nicholson–Bailey model.
Functional response.
Monod equation.
Rosenzweig-MacArthur predator-prey model.
Trophic cascade.
Underestimation of mutual interference of predators.
Coupling in predator-prey dynamics: Ratio Dependence.
Michaelis–Menten kinetics.
Trophic level.
Advection–diffusion equation.
Paradox of enrichment. [Two quotes from the book: “actual systems do not behave as Rosensweig’s model predict” + “When ecologists have looked for evidence of the paradox of enrichment in natural and laboratory systems, they often find none and typically present arguments about why it was not observed”]
Predator interference emerging from trophotaxis in predator–prey systems: An individual-based approach.
Directed movement of predators and the emergence of density dependence in predator-prey models.

“Ratio-dependent predation is now covered in major textbooks as an alternative to the standard prey-dependent view […]. One of this book’s messages is that the two simple extreme theories, prey dependence and ratio dependence, are not the only alternatives: they are the ends of a spectrum. There are ecological domains in which one view works better than the other, with an intermediate view also being a possible case. […] Our years of work spent on the subject have led us to the conclusion that, although prey dependence might conceivably be obtained in laboratory settings, the common case occurring in nature lies close to the ratio-dependent end. We believe that the latter, instead of the prey-dependent end, can be viewed as the “null model of predation.” […] we propose the gradual interference model, a specific form of predator-dependent functional response that is approximately prey dependent (as in the standard theory) at low consumer abundances and approximately ratio dependent at high abundances. […] When density is low, consumers do not interfere and prey dependence works (as in the standard theory). When consumers density is sufficiently high, interference causes ratio dependence to emerge. In the intermediate densities, predator-dependent models describe partial interference.”

“Studies of food chains are on the edge of two domains of ecology: population and community ecology. The properties of food chains are determined by the nature of their basic link, the interaction of two species, a consumer and its resource, a predator and its prey.1 The study of this basic link of the chain is part of population ecology while the more complex food webs belong to community ecology. This is one of the main reasons why understanding the dynamics of predation is important for many ecologists working at different scales.”

“We have named predator-dependent the functional responses of the form g = g(N,P), where the predator density P acts (in addition to N [prey abundance, US]) as an independent variable to determine the per capita kill rate […] predator-dependent functional response models have one more parameter than the prey-dependent or the ratio-dependent models. […] The main interest that we see in these intermediate models is that the additional parameter can provide a way to quantify the position of a specific predator-prey pair of species along a spectrum with prey dependence at one end and ratio dependence at the other end:

g(N) <- g(N,P) -> g(N/P) (1.21)

In the Hassell-Varley and Arditi-Akçakaya models […] the mutual interference parameter m plays the role of a cursor along this spectrum, from m = 0 for prey dependence to m = 1 for ratio dependence. Note that this theory does not exclude that strong interference goes “beyond ratio dependence,” with m > 1.2 This is also called overcompensation. […] In this book, rather than being interested in the interference parameters per se, we use predator-dependent models to determine, either parametrically or nonparametrically, which of the ends of the spectrum (1.21) better describes predator-prey systems in general.”

“[T]he fundamental problem of the Lotka-Volterra and the Rosensweig-MacArthur dynamic models lies in the functional response and in the fact that this mathematical function is assumed not to depend on consumer density. Since this function measures the number of prey captured per consumer per unit time, it is a quantity that should be accessible to observation. This variable could be apprehended either on the fast behavioral time scale or on the slow demographic time scale. These two approaches need not necessarily reveal the same properties: […] a given species could display a prey-dependent response on the fast scale and a predator-dependent response on the slow scale. The reason is that, on a very short scale, each predator individually may “feel” virtually alone in the environment and react only to the prey that it encounters. On the long scale, the predators are more likely to be affected by the presence of conspecifics, even without direct encounters. In the demographic context of this book, it is the long time scale that is relevant. […] if predator dependence is detected on the fast scale, then it can be inferred that it must be present on the slow scale; if predator dependence is not detected on the fast scale, it cannot be inferred that it is absent on the slow scale.”

Some related thoughts. A different way to think about this – which they don’t mention in the book, but which sprang to mind to me as I was reading it – is to think about this stuff in terms of a formal predator territorial overlap model and then asking yourself this question: Assume there’s zero territorial overlap – does this fact mean that the existence of conspecifics does not matter? The answer is of course no. The sizes of the individual patches/territories may be greatly influenced by the predator density even in such a context. Also, the territorial area available to potential offspring (certainly a fitness-relevant parameter) may be greatly influenced by the number of competitors inhabiting the surrounding territories. In relation to the last part of the quote it’s easy to see that in a model with significant territorial overlap you don’t need direct behavioural interaction among predators for the overlap to be relevant; even if two bears never meet, if one of them eats a fawn the other one would have come across two days later, well, such indirect influences may be important for prey availability. Of course as prey tend to be mobile, even if predator territories are static and non-overlapping in a geographic sense, they might not be in a functional sense. Moving on…

“In [chapter 2 we] attempted to assess the presence and the intensity of interference in all functional response data sets that we could gather in the literature. Each set must be trivariate, with estimates of the prey consumed at different values of prey density and different values of predator densities. Such data sets are not very abundant because most functional response experiments present in the literature are simply bivariate, with variations of the prey density only, often with a single predator individual, ignoring the fact that predator density can have an influence. This results from the usual presentation of functional responses in textbooks, which […] focus only on the influence of prey density.
Among the data sets that we analyzed, we did not find a single one in which the predator density did not have a significant effect. This is a powerful empirical argument against prey dependence. Most systems lie somewhere on the continuum between prey dependence (m=0) and ratio dependence (m=1). However, they do not appear to be equally distributed. The empirical evidence provided in this chapter suggests that they tend to accumulate closer to the ratio-dependent end than to the prey-dependent end.”

“Equilibrium properties result from the balanced predator-prey equations and contain elements of the underlying dynamic model. For this reason, the response of equilibria to a change in model parameters can inform us about the structure of the underlying equations. To check the appropriateness of the ratio-dependent versus prey-dependent views, we consider the theoretical equilibrium consequences of the two contrasting assumptions and compare them with the evidence from nature. […] According to the standard prey-dependent theory, in reference to [an] increase in primary production, the responses of the populations strongly depend on their level and on the total number of trophic levels. The last, top level always responds proportionally to F [primary input]. The next to the last level always remains constant: it is insensitive to enrichment at the bottom because it is perfectly controled [sic] by the last level. The first, primary producer level increases if the chain length has an odd number of levels, but declines (or stays constant with a Lotka-Volterra model) in the case of an even number of levels. According to the ratio-dependent theory, all levels increase proportionally, independently of how many levels are present. The present purpose of this chapter is to show that the second alternative is confirmed by natural data and that the strange predictions of the prey-dependent theory are unsupported.”

“If top predators are eliminated or reduced in abundance, models predict that the sequential lower trophic levels must respond by changes of alternating signs. For example, in a three-level system of plants-herbivores-predators, the reduction of predators leads to the increase of herbivores and the consequential reduction in plant abundance. This response is commonly called the trophic cascade. In a four-level system, the bottom level will increase in response to harvesting at the top. These predicted responses are quite intuitive and are, in fact, true for both short-term and long-term responses, irrespective of the theory one employs. […] A number of excellent reviews have summarized and meta-analyzed large amounts of data on trophic cascades in food chains […] In general, the cascading reaction is strongest in lakes, followed by marine systems, and weakest in terrestrial systems. […] Any theory that claims to describe the trophic chain equilibria has to produce such cascading when top predators are reduced or eliminated. It is well known that the standard prey-dependent theory supports this view of top-down cascading. It is not widely appreciated that top-down cascading is likewise a property of ratio-dependent trophic chains. […] It is [only] for equilibrial responses to enrichment at the bottom that predictions are strikingly different according to the two theories”.

As the book does spend a little time on this I should perhaps briefly interject here that the above paragraph should not be taken to indicate that the two types of models provide identical predictions in the top-down cascading context in all cases; both predict cascading, but there are even so some subtle differences between the models here as well. Some of these differences are however quite hard to test.

“[T]he traditional Lotka-Volterra interaction term […] is nothing other than the law of mass action of chemistry. It assumes that predator and prey individuals encounter each other randomly in the same way that molecules interact in a chemical solution. Other prey-dependent models, like Holling’s, derive from the same idea. […] an ecological system can only be described by such a model if conspecifics do not interfere with each other and if the system is sufficiently homogeneous […] we will demonstrate that spatial heterogeneity, be it in the form of a prey refuge or in the form of predator clusters, leads to emergence of gradual interference or of ratio dependence when the functional response is observed at the population level. […] We present two mechanistic individual-based models that illustrate how, with gradually increasing predator density and gradually increasing predator clustering, interference can become gradually stronger. Thus, a given biological system, prey dependent at low predator density, can gradually become ratio dependent at high predator density. […] ratio dependence is a simple way of summarizing the effects induced by spatial heterogeneity, while the prey dependent [models] (e.g., Lotka-Volterra) is more appropriate in homogeneous environments.”

“[W]e consider that a good model of interacting species must be fundamentally invariant to a proportional change of all abundances in the system. […] Allowing interacting populations to expand in balanced exponential growth makes the laws of ecology invariant with respect to multiplying interacting abundances by the same constant, so that only ratios matter. […] scaling invariance is required if we wish to preserve the possibility of joint exponential growth of an interacting pair. […] a ratio-dependent model allows for joint exponential growth. […] Neither the standard prey-dependent models nor the more general predator-dependent models allow for balanced growth. […] In our view, communities must be expected to expand exponentially in the presence of unlimited resources. Of course, limiting factors ultimately stop this expansion just as they do for a single species. With our view, it is the limiting resources that stop the joint expansion of the interacting populations; it is not directly due to the interactions themselves. This partitioning of the causes is a major simplification that traditional theory implies only in the case of a single species.”

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August 1, 2017 - Posted by | Biology, Books, Chemistry, Ecology, Mathematics, Studies

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