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description= An approach to modelling food web biomass flows among live and dead compartments within and among species is formulated using metaphysiological principles that characterise population growth in terms of basal metabolism, feeding, senescence and ...;
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new classification scheme involving 10 primary categories of consumer types these types in various combinations rigorously distinguish scavenger from parasite herbivory from phytophagy and detritivore from decomposer application of the approach to particular consumer resource interactions is demonstrated culminating in the construction of an anthrax centred food web model with parameters applicable to etosha national park namibia where deaths of elephants and zebra from the bacterial pathogen bacillus anthracis provide significant subsidies to jackals vultures and other scavengers keywords anthrax bacillus anthracis etosha national park food web models host parasite jackals metaphysiological models plant herbivore prey predator scavengers 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 figure 1 open in a new tab 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 i th 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 y 0 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 v i t v i s 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 box 1 general formulation biomass extraction extraction of live and dead biomass from j to live i is at rates f ji x y t x j x i and g ji x y t y j x i respectively these functions are 0 if a trophic relationship is absent live biomass diversion only a proportion θ ji x y t of extracted live biomass j actually flows into i the remainder flows to dead biomass j live and dead biomass conversion biomass flowing from live and dead j into i is converted with efficiencies k ji f x y t and k ji g x y t respectively biomass incorporation the per capita rate at which biomass is incorporated from all sources into live i is from the above omitting functional arguments ϕ i σ j 1 n k ji f θ ji f ji x j k ji g g ji y j 1 biomass metabolisation the per capita rate at which biomass is metabolised into water and other by products is α̃ t α i x y t dependence on x and y is likely to be weak deficit stress accumulation and accommodation whenever ϕ̃ i t α̃ i t 0 for extended periods of time a deficit stress variable v i t monitors this deficit via a deficit stress accumulation rate function v i that depends on both current storage deficit stress v i t and current net converted biomass rate ϕ̃ i t α̃ i t a discounting function w i t s that approaches zero as time s 0 recedes into the past is used to account for accommodation of this feeding deficit stress through reductions in growth and reproduction rates live biomass senescence each unit in the population is subject to a per capita senescence rate m i that includes all sources of mortality other than extraction where m i has a minimum background rate that increases with increasing v i t such that m i v i as v i v i s where v i s is a death by starvation level dead biomass decay for population i dead biomass decays back into the environment at a per capita rate δ i x y t any dependence on x and y is likely to be weak accounting for all these processes in a model that averages out spatial structure produces the dynamic model omitting process functional arguments except in the last equation where they are needed for clarity dx i dt ϕ i x i σ j 1 n f ij x j x i m i x i α i x i recall m i depends on v i dy i dt m i x i σ j 1 n 1 θ ji f ij x j x i σ j 1 n g ij x j y i δ i y i v i t t w i t s v i α i s ϕ i s v i s ds 2 thelast integral equation can be transformed into a differential equation as discussed in appendix s2 from these considerations the state of a btw at time t 0 is represented by the vectors x t x 1 t x n t y t y 0 t y 1 t y n t and v t v 1 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 v i s 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 table 1 a summary of functions and description of parameters appearing in the model symbol description equations t time the independent variable state variables x i live biomass of population i equation 2 y i dead biomass of population i equation 2 v i v deficit stress in population i equation 2 eqn 3 z x live consumer biomass x feeding on resource input z equations 3 and 4 processes f̃ t f x t y t t representing f purely as a function of time for all functions below f ij f live i biomass extraction to j equations 1 and 2 eqn 4 g ij dead i biomass extraction to j equations 1 and 2 θ ij θ live i to dead j biomass diversion equations 1 and 2 eqn 4 k ij f and k ij g κ conversion of live and dead i respectively to j equation 1 eqn 4 φ i φ per unit i biomass incorporation equations 1 and 11 eqn 4 α i α per unit i metabolism equation 2 eqn 3 m i m per unit i senescence mortality other than extraction equations 2 and 10 eqn 5 δ i per unit i dead biomass decay equation 2 v i deficit stress accumulation rate equation 2 w i deficit stress accommodation over time equation 2 μ x density dependent component of m x v equation 5 parameters a a 1 maximum extraction rate in f in f 01 equation 4 eqn 9 a 2 rate at which diseased carcass produces spores equation 11 b b 1 resource density at which extract rate is a 2 in f in f 01 equation 4 eqn 9 c c 1 c 12 weakness inverse of strength of competition in f in f 01 equation 4 eqn 9 γ abruptness in onset of competition in f equation 4 v s v 1 s starvation level for deficit stress variable v v 1 equation 5 eqn 7 μ i μ ij parameters in μ and μ i i 0 1 2 j 1 2 see text and equation 10 a i b i c i v i s γ i w f w g h index are specific to anthrax model in appendix equation 21 open in a new tab 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 y 0 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 α̃ 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 order effect of processes and factors the sensitivity of the output of a model to perturbations of model parameters either singly or in combination has various but precise mathematical definitions saltelli et al 2000 using any appropriate analytical method if the sensitivity of some output to a process or factor represented through parameter perturbations is an order of magnitude i e 1 unit on a log 10 scale greater than another then the effect of the process or factor on the measure can be said to be of order one higher than the other with the highest being first order in our formulations we focus only on first order processes and fact...
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