Biomass transformation webs provide a unified approach to consumer–resource modelling

Abstract

INTRODUCTION

Current approaches to modelling food webs (Pimm 1982; Winemiller & Polis 1996) come in many guises including Lotka–Volterra community assemblages (Ackland & Gallagher 2004), modified Lotka–Volterra trophic webs (Arditi & Michalski 1995; Getz et al. 2003), information theoretical aspects of trophic flow (Ulanowicz 2004), trophic flow models (Jordán 2000), trophic mass-balance models (Moloney et al. 2005), energy flow models (Jordán 2000), bioenergetic models (Romanuk et al. 2009), ecological networks (Brose 2010; Jörgensen & Fath 2006), nutrient cycling (Allen & Gillooly 2009) and carbon flows (Sandberg et al. 2000; Finlay et al. 2002). Each approach best addresses a specific class of questions such as stability properties (Neutel et al. 2002), effects of web structure on productivity (Carpenter et al. 1985) or biodiversity (Bascompte 2009), transport properties among spatial elements (Polis et al. 1997; Power & Dietrich 2002) or linkage structure (Williams & Martinez 2000). None of these approaches embeds into food webs, as seemlessly as the biomass transformation web (BTW) approach developed here, all possible consumer types, particularly scavengers and parasites.

Many food web studies include one or more detrital components (Moore et al. 2004; Szwabiński et al. 2010). However, BTW takes this a step further by dividing populations into both live and dead biomass components, as well as classifying consumers of plant, animal or particulate organic material based on whether they mine or gather resources. As a result, BTW leads to a natural 10-way classification of basic consumer categories (Fig. 1). Here, miners are idealised as sessile extractors of pooled resources such as phloem-feeding aphids or blood-sucking ticks, and gatherers are idealised as mobile extractors of resource packets such as grasshoppers eating leaves or cats hunting prey. The relevance of this miner–gatherer dichotomy will become clearer in the general modelling section when we consider how resource deficits over various periods of time affect biomass dynamics and may ultimately lead to starvation of individuals.

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Figure 1

Consumer categories (see Appendix S1 and Table S1 for more details).

The general formulation of BTW, presented in the next section, uses metaphysiological concepts of biomass dynamics (Getz 1991, 1993, 2009) to characterise growth in terms of consumption, conversion (digestion and anabolism) and metabolism, as well as mortality in terms of both extraction (i.e. the population in question is a resource for other populations in the food web) and senescence. In the BTW formulation, senescence is regarded as mortality due to all factors other than extraction, and thus includes deaths due to ageing, non-infectious disease (e.g. cancer, organ failure), starvation and infectious disease when the agent of the disease is itself not explicitly modelled within the food web. At the end of this article, although, a BTW model that explicitly includes the pathogenic bacterium, Bacillus anthracis, as a consumer in its own right, and hence affects its hosts through the processes of extraction rather than senescence, is formulated to study the dynamics of a food web centred around the occurrence of anthrax in large mammalian herbivores (primarily zebra and elephant) in Etosha National Park (ENP), Namibia. One of the interesting questions that will be explored is the effect of carcass subsidies from anthrax deaths on the population dynamics of black-backed jackal (Canis mesomelas) scavengers.

Before we can formulate a model that includes elephants (Loxodonta africana), zebra (Equus quagga), B. anthracis, jackals and various small mammal species that are predated by jackals, we need to develop a general approach to modelling such food webs. As no general paradigm currently exists that includes scavengers (jackals in our case) and parasites (B. anthracis in our case), our first task is to develop such a paradigm, which we call BTW because of its focus on biomass transfers among food web components. The novelty of the approach, however, requires that we both clarify the kinds of consumers occurring in BTWs – that is, the 10 categories referred to earlier – and develop details needed to capture differences among consumer types within the context of our unified approach to modelling all types of consumer–resource interactions. Of course, no general formulation can cover the complexities of all consumer–resource interactions. The range covered by the formulation presented here, however, is much broader than other existing formulations, as illustrated in the final section of this article, where a BTW model is presented of an anthrax-centred food web in ENP.

GENERAL BTW FORMULATION

Biomass transformation web is based on a set of principles that specify how the abundances of live, x_i(t), and dead, y_i(t), biomass of the ith (i = 1, …, n) population or functional group, referred to as compartments in a food web, change over time. Biomass can be transferred among compartments as a result of processes of extraction, diversion, conversion, metabolism, live biomass senescence or dead biomass decay back into environmental constituents (nutrients, organic molecules, etc.) (Box 1). The latter for simplicity are represented by the aggregated scalar concentration variable y₀ that can be generalised to a multivariable vector, as needed. Additionally, if individuals in the ith group take in less biomass or resources than are required to meet basal metabolic needs, then they accumulate a feeding deficit stress v_i(t) over time. This deficit can be accommodated by drawing upon an implicit stored live biomass component (e.g. stored in the form of fats or sugars) allowing accommodation to take place over extended periods of time (McCue 2010). This accommodation occurs through organisms adjusting growth and reproduction schedules until resource intake is restored to needed levels or individuals ultimately die from the starvation when critical (i.e. final starvation) levels reached. The appropriate forms of the functions that determine the accumulation and accommodation rates and the way senescence depends on v_i(t) are likely to be influenced by the feeding ecology of species i, with gatherers more likely than miners able to tolerate extensive periods of stress from deficit feeding (i.e. starvation).

From these considerations, the state of a BTW at time t ≥ 0 is represented by the vectors x(t) = (x₁(t),…,x_n(t))′, y(t) = (y₀(t),y₁(t),…,y_n(t))′ and v(t) = (v₁(t),…,v_n(t)) (where ' denotes vector transpose because vectors have column rather than row representations). The equations of BTW are formulated in Box 1 with descriptions of variables, process functions and parameters listed in Table 1. These BTW equations (eqn 2) include the influence of the accumulated deficit stress on live biomass senescence into dead biomass. This senescence happens at an accelerating rate with increasing deficit stress until the rate is infinitely fast when starvation level is reached. Throughout the formulation of the BTW equations, we consider various functions with arguments x, y, v and t. For notational convenience, when a function, say ϕ_i(x(t),y(t),t), is considered purely in terms of time, we use the notation ϕ̃_i(t) = ϕ_i(x(t),y(t), t) to avoid confusion.

In closing, the general formulation presented in Box 1 does not explicitly account for faecal waste or external inputs (Polis et al. 1997) other than y₀. The BTW formulation can easily be extended to include one or more faecal waste components (e.g. in systems where different species of dung beetle exploit the dung of different species, as in Larsen et al. 2006) and other external drivers (e.g. emigration) as needed.

CONSUMER CATEGORIES AND TERMINOLOGY

One of the concepts associated with BTW that requires refinement is how to treat consumers that feed exclusively on live vs. dead material. In particular, terms exist to distinguish between consumers of live and dead flesh but not in the case of plant material. Another concept, which is an area for future research, relates to how we should characterise the effects of deficit stress on senescence in different kinds of consumers, particularly gatherers vs. miners. In particular, we need to develop ways of characterising the deficit stress accumulation-rate functions V_i(_i(t) − ϕ̃_i(t), v_i(t)) and the deficit stress accommodation functions w_i(t) that are consumer-type dependent. To facilitate such refinements in modelling the effects of deficit resource intake on different kinds of consumers, the categorisation scheme presented in Fig. 1 (cf. Appendix S1 and Table S1) unambiguously defines consumer categories that distinguish among consumer types. In particular, the scheme proposes that the words decomposer and detritivore should be reserved for organisms that, respectively, mine and gather bits of organic matter independent of source.

Beyond the 10 primary categories illustrated in Fig. 1, we can also classify the consumer world into various compound categories that are useful to consider when developing the specific structure of the general equations presented in eqn 2 (Box 1). Four such categories, three of which already exist, take on the following rigorous definitions: parasites and croppers are miners and gatherers of live biomass, respectively, whereas saprophages and scavengers are miners and gatherers of dead biomass, respectively.

In the development of these categories and when considering the processes that affect growth and senescence in formulating the basic building blocks presented in the next section, we focus on what we call first-order processes and factors and differentiate between direct and indirect effects as defined by:

SOME BASIC BUILDING BLOCKS

Before developing a model of a particular system and exploring its behaviour, it is useful to consider some specific building block functions for representing the processes of extraction, diversion and mortality transformation.

Consumer–resource interactions

Consumer–resource interactions were first considered in the context of prey–predator, plant–herbivore and host–parasite systems (Murdoch et al. 2003; Turchin 2003) and are the core motif of a food web (Bascompte 2009; Bascompte & Melián 2005). We can dynamically isolate this interaction from a surrounding food web by assuming that:

1. The resource population consumes biomass, nutrients or energy, which is at an abundance or concentration of y₀(t) in the environment, through a recipient-controlled process.
2. The extractive part of consumer mortality (the other part is the senescence process) is determined by an external input to the system (e.g. a constant or donor-controlled harvesting rate).

In our treatment below, we assume y₀ to be an underlying constant or specified time-varying environmental input, x₁ to be a resource population that lives off of y₀ and x₂ to be a consumer that most generally consumes x₁, y₁, y₂ and influences all the rates (Fig. 2), but is itself subject only to senescence mortality. The biomass flows and transformations that generally occur can be categorised as follows, with cannibalism now emerging very naturally because of the live–dead biomass dichotomy:

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Figure 2

A biomass flow diagram of a general resource–consumer system modelled by eqn 6 in which the consumer, but not the resource population, may be subject to cannibalism. For simplicity, the diversion functions θ₁₂ and θ₂₂ are not illustrated.

Resource growth transformation

The total biomass (or nutrient if the population is at the lowest trophic level in a food chain) flow rate f₀₁y₀x₁ is transformed from y₀ into x₁.

Resource death transformation

The total biomass flow rate m₁x₁ is transformed from live resource x₁ into dead resources y₁.

Diversion transformation

A proportion θ₁₂ of the total extracted biomass flow rate f₁₂x₁x₂ is transformed from live resource x₁ into dead resources y₁.

Extracted live biomass transformation

A proportion (1–θ₁₂) of the total biomass flow rate f₁₂x₁x₂ is transformed from live resource x₁ into cropper or parasitic consumer biomass x₂.

Cannibalistic transformation

Biomass flow f_ii from live resource or consumer population x_i is transformed back into consumer biomass x_i.

Extracted dead biomass transformation

Biomass flow g₁₂ from dead resources y₁ transformed into consumer biomass x₂. (As mentioned above, in a more general treatment, we can separate out coprophagy by adding an explicit faecal waste variable z₁ to the resource population.)

Consumer death transformation

Total biomass flow rate m₂x₂ from live consumers x₂ is transformed into dead consumers y₂.

Cannibalistic-scavenger transformation

Biomass flow g_ii from dead resource or consumer population y_i is transformed into consumer biomass x_i.

In the context of the feeding flows f₁₂ and f₂₂, as formulated in our general model given by eqn 2, we need to account for, using our θ_i2 functions (i = 1, 2), the dual transformation processes of live-to-live and live-to-dead flows as a result of consumer feeding activities and, of course, we also need to account for conversion inefficiencies through the conversion functions K_i. With the above focal transformation processes, and for simplicity confining cannibalism to the consumer alone, we obtain the following consumer–resource model of a closed system (if subsidies flow into the web from the outside, then these need to be included, e.g. see Polis et al. 1997) as a special case of the general BTW model (Fig. 2) presented in eqn 2 (Box 1):

(6)

where

and

.

Various special cases arise by allowing different combinations of the extraction rates f₁₂, g₁₂, f₂₂ and g₂₂ to be non-zero (Table 2) along with zero or non-zero diversion functions θ₁₂ and θ₂₂ (e.g. panels a–d in Fig. 3). Additional cases arise when considering harassment and stress-inducing first-order effects that consumers may have on resource individuals. Three of these are as follows:

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Figure 3

Specific cases of the biomass flow diagram illustrated in Fig. 2 (same legend applies), with flow and influence structures detailed in Table 2, for (a) a cropper, with the link that corresponds to the influence of harassment specifically labelled; (b) a parasite, with the overlapping death compartments indicated; (c) scavengers that feed only on dead resources and (d)scavengers that feed both on dead resources and cannibalistically on their own dead biomass.

Table 2

A selection of ideal consumer types as denned by the arguments of BTW functions f, g, θ and m, CN CR, DV, HR, PS and SC

Extracted live resource f21 (•), • =	Diverted live resource θ21 (•), • =	Extracted dead resource g21(•), • =	Cannibalism: Live f22 (•), • =	Diverted live cannibalism θ22(•), • =	Cannibalism: dead g22(•), • =	Resource mortality m1(•), • =	Consumer mortality m2(•), • =	Selection of functional types
x1, x2	0	0	0	0	0	ϕ 1	ϕ 2	CR only (e.g. many frugivores)
x1, x2	0	0	0	0	0		ϕ 2	CR-HR (e.g. leopards*)
x1, x2	0	0	0	0	0			PS (endo- and ectoparasites)
x1, x2, y1	x1, x2, y1	0	0	0	0	ϕ 1	ϕ 2	CR-DV (e.g. some grasshoppers†)
x1, x2, y1	x1, x2, y1	0	0	0	0	ϕ1, μ1(x2)		CR-DV-HR (elephants‡, cheetahs§)
0	0	x2, y1	0	0	0	ϕ 1	ϕ 2	SC Type I (necro-, saprophages and decomposers)
0	0	x2, y1, y2	0	0	x2, y1, y2	ϕ 1	ϕ 2	SC Type II (e.g. some vultures¶)
x1, x2, y1	x1, x2, y1	x1, x2, y1	0	0	0		ϕ 2	HR-CR-SC Type I (e.g. lions**)
x1, x2, y1, y2	x1, x2, y1, y2	x1, x2, y1, y2	0	0	x1, x2, y1, y2	ϕ 1	ϕ 2	CR-SC Type II (possibly wild dog ††)
x1, x2, y1	x1, x2, y1	0	x1, x2, y1	x1, x2, y1	0	ϕ 1	ϕ 2	CR-DV-CN (e.g. some fish‡‡)
x1, x2, y1, y2	x1, x2, y1, y2	x1, x2, y1, y2	x1, x2, y1, y2	x1, x2, y1, y2	x1, x2, y1, y2	ϕ 1	ϕ 2	CR-DV-CN-SC (e.g. some crabs§§)

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CN, cannibal; CR, cropper; DV, diverter; HR, harasser; PS, parasite; SC, scavenger.

^*Leopards generally are able to feed on their kills in trees to protect against diverting dead biomass to scavengers.

^†Gandar 1982.

^‡Skarpe et al. 2004.

^§Marker et al. 2003.

^¶Bearded vulture chicks will sometimes be fed their dead siblings, Margalida et al. 2004.

^**Hopcraft et al. 2006.

^††Scavenging is rare (Creel & Creel 1995) and cannibalism could be related to infanticide and hence more complex (Robbins & McCreery 2001).

^‡‡Smith & Reay 1991.

^§§Mckillup & Mckillup 1996.

Extraction harassment

The per capita rate f₀₁y₀ at which the resource population extracts food (or energy) from the environment is a non-increasing function of consumer density x₂: that is, . A case in point is predator avoidance by elk in Yellowstone has led to elk feeding at higher elevations for longer periods of time in the spring. This has had the ecological knock-on effect of allowing aspen seedlings at lower elevations to survive and stands of aspen trees to recover (Ripple & Beschta 2007).

Exploitation stress effects on growth

The conversion functions and may vary due to the stress that herbivores induce on plants or predators induce on prey. For example, wild dogs reduce the rate at which their prey are able to reproduce (Creel et al. 2009), while some herbivores invoke a defensive response in plants (Karban 2008) that diverts resources that would otherwise have been allocated to growth and reproduction.

Feeding deficit stress effects on senescence

The per capita rate m₁ of the resource population is a non-increasing function of x₂ because consumers may induce a feeding deficit stress response of some kind on individuals in the resource population that leads to increased mortality rates through senescence. The most ubiquitous examples are parasites that are pathogenic to some degree.

ANTHRAX IN ETOSHA

In the development of a model that can address questions relating to both endemic and outbreak dynamics of pathogens in food webs, with specific application to an anthrax-centred food web in ENP, Namibia, we draw upon eqn 2, as well as equations developed in Appendix S3 modelling the interaction of a consumer that is both a cropper and scavenger in a food web. Bacillus anthracis, the agent responsible for anthrax, is a Gram-positive bacterium that persists in a sporulated life stage in patches of suitable soil – referred to here as locally infectious zones (LIZs) – where its ability to infect herbivores decays over time (Hugh-Jones & Blackburn 2009). During the ENP wet season, individual zebra, springbok, wildebeest and oryx ingest lethal doses of B. anthracis spores, contract the anthrax disease and die (Lindeque & Turnbull 1994). On the other hand, individual elephants range widely and are more likely to die of anthrax during the dry season. Diseased carcasses year round are open by various carcasivores (several kinds of vultures) and carnivores (hyenas and lions), but especially black-backed jackals (Canis mesomelas) that are both carcasivores and opportunistic croppers of small mammals (rodents, lagomorphs, newborn springbok), birds, reptiles, invertebrates (e.g. dung beetles), and even seeds and fruit (Kaunda & Skinner 2003).

The area of ENP around Okaukuejo is semi-arid, where outbreaks of anthrax predictably occur each year (the mean annual rainfall at Okaukuejo was 384 mm from 1934–2007 – see Turner et al. 2010) and consequently ecologically less complex than anthrax in wetter savanna systems such as Zimbabwe, where outbreaks are less predictable and can be highly disruptive to the ecoystem (Hugh-Jones & Blackburn 2009). As the dominant anthrax-mediated transformation process of live-to-dead animal biomass each year around Okaukuejo occurs in the zebra (Equus quagga) and elephant populations (Loxodonta africana), a combined population of these two species provides the focal live resource (x₁) and dead (y₁) resource compartments in a model of anthrax in ENP (Fig. 4; Appendix S4), although anthrax does infect many other species in ENP.

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Figure 4

A simplified anthrax-centred biomass transformation web in Etosha National Park, Namibia. See Appendix S4 for equations modelling this system.

Bacillus anthracis spores (abundance x₂) are distributed within LIZs across a several 1000 km² grazing plain in ENP. A simple spatially aggregated BTW model of anthrax dynamics can be developed, as detailed in Appendix S4, using eqn 12 to model the B. anthracis spore–host (elephant/zebra) interaction. By simply varying the growth parameter a₂ in eqn 12, this model nicely replicates both endemic and epidemic disease dynamics (Fig. 5). Further in the latter case, the outbreaks do not cause the host population to collapse to exceptionally low levels, a situation typical of dynamics predicated by Lotka–Volterra type models. More specifically, in panel (a) (Fig. 5), the population converges to an endemic phase that it similarly converges to in panel (b) when the density-dependent mortality factor μ₂₂ is removed (i.e. set to zero). As the factor a₂ controlling the number of spores entering the environment per unit biomass of infected carcass is increased from a₂ = 0.5 (panel b) to a₂ = 0.8 (panel c), regular outbreaks that appear slightly dampened over time occur every 3 years, although the severity dramatically increases and frequency decreases to once every 7–8 years when the spore production rate increases by 50% to a₂ = 1.2. Thus, the relatively simple two-dimensional model represented by eqn 12 is easily able to capture the range of observed endemicity of anthrax in ENP to the subdecadel and decadel outbreaks in places such as Kruger National Park in South Africa and wildland areas in Zimbabwe (Hugh-Jones & Blackburn 2009).

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Figure 5

The zebra/elephant biomass abundance x₁(t) (Scale 1 = 18 000 metric tons) and anthrax spore abundance x₂(t) (Scale 1 = 200 unspecified units) solutions to eqn 12 are plotted over a 30-year period for the parameter values given in Table S2 (Appendix S5), except as noted: (a) a₂ = 0.5 and μ₂₂ = 0.0001, (b) a₂ = 0.5 and μ₂₂ = 0, (c) a₂ = 0.8 and μ₂₂ = 0, and (d) a₂ = 1.2 and μ₂₂ = 0.

An additional compartment in the model is live jackal biomass at abundance x₃, as jackals scavenge both diseased carcasses and those arising from lion and hyena kills. Jackals also scavenge other carcass species and crop a variety of small animals that we structurally represent through live (x₄) and dead (y₄) resource biomass compartments in the model (Fig. 4; Appendix S4). To keep the model simple, lions (Panthera leo) and spotted hyena (Crocuta crocuta) that prey upon the various ungulates are included in the environment as donor-controlled cropper–scavenger extraction processes (Appendix S4).

If we now include inertial variables v₁, v₃ and v₄ (i.e. for all the live biomass compartments except anthrax x₂), then the resulting BTW models contains nine dynamic equations, as detailed in Appendix S4. This model, in lumping together trophic functional groups such as ‘elephants and zebra’ and ‘other small mammals’, and in ignoring spatial and seasonal structure, is obviously crude. Constructing a model that splits apart these functional groups, and includes migratory seasonal movements and other spatio-temporal structures, is a task worthy of several PhD studies. Thus, the analysis here is only meant to illustrate how a model based on BTW principles can be assembled rather than reflect the current state of biological knowledge of the system under consideration. To this end, the parameters derived in Appendix S5 are crude ballpark estimates that allow the model to be used as a tool for suggesting research priorities and directions rather than answering well-posed research questions. This is appropriate given that current abundance estimates are rather crude and, in particular, jackal abundances are not known within a factor of two of real levels, while the distribution of anthrax spores across the landscape is unknown.

For the set of parameter values given in Table S2 (Appendix S5), the model presented in Appendix S4 predicts the equilibrium values (which can be interpreted as long-term averages) (₁,ŷ₁,₂,₃,ŷ₃,₄)′ = (7437,70.0,31.2,2351,3.49,57.6) (units are metric tons except for x₂, which needs further studies to ground the arbitrary units used here). An important approach to exploring models with uncertain parameter values is to carry out some kind of sensitivity analysis (Saltelli et al 2000) as a way to use the model to inform ecologists what parameters in the model are most critical to characterising variables of interest. For example, we see from panel (a) in Fig. 6 that halving the jackal maximum extraction rate parameter a₄ results in the reduction of the predicted equilibrium by 1/3. Also if we ask the question what will happen if jackal were to only scavenge carcasses (i.e. setting wf= 0), then the jackal population falls by three quarters when a₄ = 0.16 or collapses completely when a₄ = 0.08.

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Figure 6

Jackal biomass abundance x₄(t) (Scale 1 = 20 metric tons) solutions to eqn 12 are plotted over a 25-year period for the parameter values given in Table S2 (Appendix S5), except as labelled in (a) for the four illustrated cases. The same applies to (b), except here the deficit stress variable v₄ (Scale 1 = 10⁻¹) is also plotted for the labelled case.

The jackal population in panel (a) (Fig. 6) is modelled under endemic anthrax conditions. If the system is perturbed into outbreak mode by setting μ₂₂ = 0, then under conditions where the jackal population only scavenge, during an outbreak the population may nearly double and then rapidly decline at the end of an outbreak (panel b in Fig. 6). The reason why the decline is rapid is apparent from the graph of v₄ in panel (b) (Fig. 6): at the end of an outbreak, once the excess carcasses have all been consumed, the elevated jackal population begins to starve, as evidenced by the rapid rise in the value of v₄, and the effects of accelerated senescence due to the associated deficit stress now set in.

CONCLUSION

Consumer–resource models can be traced back to the work of Lotka and Volterra in the mid 1920s, with much of the current work on this topic (Murdoch et al. 2003; Turchin 2003) rooted in Lotka and Volterra's original two-dimensional formulation. The Lotka–Volterra model with its extensions to include various types of nonlinear predator-response-to-prey-density functions has been applied to quite different kinds of consumer–resource processes, including plant–herbivore, prey–predator and host–parasite interactions; but also with a notable lack of focus on scavenger–carrion interactions (Nuria & Fotuna 2006). The BTW presented here deals with all these various consumer–resource interactions, but its approach to growth as a function of biomass extraction leads more naturally than Lotka–Volterra-like approaches to distinguishing among different kinds of consumer through incorporation of biomass diversion, scavenging, parasitism and consumer-harassment processes. The price we pay for this refinement is that the general consumer–resource formulation is now four to six rather than two dynamically linked equations, although, as we have seen, the dimension can be reduced to two when focusing on special cases. The gain though is considerable in that our view of the kinds of resource–consumer interactions that can occur (Fig. 1; Appendix S1) is now considerably enlarged. Along with this enlargement comes a whole new set of ecological and evolutionary questions that can be addressed in a quantitatively rigorous framework using methodologies, such as evolutionarily stable strategy theory, that have proved their worth when used in conjunction with Lotka–Volterra type formulations of population interactions (Cressman & Garay 2003). Among these questions are how might we expect the dynamics of the feeding deficit stress variables v_i to reflect the life history dichotomy of miners vs. gatherers. This is an issue that relates to time constants associated with rates of feeding deficit stress accumulation and accommodation, as well as time-to-death under complete starvation and how life histories evolve to deal with variable interresource encounter periods for gatherers. Moreover, for many species, metabolic rates may adaptively decrease when food intake rates do not meet normal metabolic needs [i.e. periods where v_i increases because φ_i – α_i, so that is an adaptive strategy], as is the case of animals that go into hibernation during seasonal resource dearths.

Although the formulation, through the inclusion of the feeding deficit stress variables v_i, deals with the problem that population processes need to include inertial effects, including the well-studied maternal effect (Inchausti & Ginzburg 1998), the approach ultimately needs to be generalised to take into account two forms of heterogeneity that apply to all paradigms and not just BTW. The first form is spatial heterogeneity and requires elaboration of how particular population processes vary over space and how animals move to mitigate gradients in these processes that naturally arise, such as moving to places where feeding rates can be higher or where they are less likely to succumb to being extracted by predators. The second form is individual variation due to both genetic and random processes, in which two individuals in the same place are subject to different rates of food acquisitions, different rates of parasitism and different risks of being consumed by other species. This results in a phenomenon known as buffering (Revilla & Wiegand 2008).

One of the strengths of BTW is that it provides a unified framework in which the approach to modelling populations to first order is independent of the trophic level. Another strength of BTW is that it deals with scavengers just as easily as it does with croppers or parasites. Thus, it has application to a much wider array of food web systems than current methods generally have including systems of importance to disease management or conservation biology. A case in point is evaluating how many dead trees are needed to sustain the white-backed woodpecker that relies on insect larvae that use dead trees as a food source during winter (Gjerde et al. 2005).

The BTW presented here is an outgrowth of the metaphysiological approach to modelling trophic interactions (Getz 1991, 1993). This approach, by taking a biomass flow rather than a birth-death-migration viewpoint of growth, formulates growth in terms of extraction and senescent processes, rather than directly in terms of the state variables themselves. This allows the formulation of equations to be unified across trophic levels, as evidenced by the general model (Box 1). In addition, the structure of the equations is rather transparent and, hence, easily implemented for particular systems, as illustrated in our derivation of the nine-variable anthrax BTW model in Appendix S4.

In terms of food web theory itself, by extending populations to account separately for both live and dead biomass, new topological relationships emerge and the strength of these relationships, as mediated by scavengers, parasites, disease and senescence processes, can now be included. This topological refinement will affect such characterising measures as L, the number of feeding relationships in webs, which has been used in a number of food web studies (e.g. Williams & Martinez 2000; Romanuk et al. 2009). As Lazzaro et al. (2009) recently pointed out: ‘The structure and dynamics of prey populations are shaped by the foraging behaviours of their predators. Yet, there is still little documentation on how distinct predator foraging types control biodiversity, food web architecture and ecosystem functioning’. This statement was made in the context of foraging strategies per se (e.g. visual vs. filter feeders) rather than in the context of biomass type (viz. live vs. dead). The statement applies equally well to the BTW formulation, which places live and dead material on an explicit co-footing. Similar considerations arise in the context of adaptive foraging in food webs with flexible topology (Křivan & Schmitz 2003), particularly in the context of adaptive scavenging. The BTW formulation provides a paradigm for exploring these various questions more thoroughly and systematically than before. Furthermore, once we move beyond homogeneous to spatially structured webs, the need for a miner–gatherer dichotomy takes on additional force when the ecological aspects of movement (Nathan et al. 2008) are introduced.

Supplementary Material

ACKNOWLEDGEMENTS

Footnotes

REFERENCES