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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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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 factors that dominate demographic times scales recognising that second and lower order processes and factors may have importance on ecological succession and evolutionary time scales direct vs indirect extraction effects in the context of btw only we define factors parameters process descriptions that have direct extraction effects to be those that alter the values of the biomass extraction rates f ij x y and g ij x y for any given state x y all other types of factors are said to have indirect extraction effects i e they ultimately affect extraction through their influence on the state x y 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 isolated population growth as a first step in developing our understanding of btw dynamics we consider the canonical example of a single population described by biomass variables x t and y t growing in a constant environmental resource matrix at level z t which we use instead of y 0 to reduce the use of subscripts furthermore as we are now dealing with only one population we can drop all subscripts under the assumption that the population is isolated from extraction by other populations assuming that the metabolic maintenance rate α is constant and as discussed in appendix s2 that the deficit stress accumulation function is simply ṽ s max 0 α̃ ϕ̃ t it follows from eqn 2 box 1 that the equations for the variables x t and v t satisfy the closed system dx dt ϕ z x m x v α x dv dt max 0 α ϕ t max 0 α ϕ t t s 3 once x t and v t have been found then the dead compartment can be generated from the equation dy dt m x v α x δ y equation 3 constitute a two variable description of inertial population growth that represents a mechanistic alternative to the more axiomatic approach taken by ginzburg and colyvan 2004 and the more phenomenological quantity quality approach taken by getz owen smith in preparation in particularising these equations we note from eqn 1 for constant conversion and diversion proportions k and the θ respectively that the form of biomass incorporation function φ z x kφf z x z depends on f z x the form we select is that of a holling type ii where the half maximum extraction parameter b is modified as discussed in getz 1993 to incorporate the effects of abrupt intraspecific competition getz 1996 thus our extraction and biomass incorporation functions respectively are f z x a b 1 x c γ z and ϕ z x k θ az b 1 x c γ z where a 0 is the maximum extraction rate b 0 is a resource level that reduces the intake rate to half its maximum when interspecific competition is absent formally as x 0 and c 0 is a parameter that determines the intensity of interspecific competition for a given abruptness parameter γ 1 the key difference between eqn 4 and a holling type ii functional response is the dependence of the denominator on the consumer biomass abundance x this dependence is required when extractive rates are reduced under conditions of diminishing availability of resources per unit consumer getz 1984 abrams ginzburg 2000 in this case competition is directly experienced through interference abrams 1988 rather than emerging indirectly through reductions in future levels of resource moreover for simplicity we assume that the per capita senescence function m x v is separable in the variables x and v and can be expressed in the form m x v μ x v s v s v 5 this form implies m x v as v v s with the result that the population plunges to zero when the starvation deficit limit v s is reached a general form for μ x that allows us to account for density related effects that either reduce or increase mortality with increasing density is μ x μ 0 μ 1 x 1 μ 2 x where we assume that μ i 0 i 1 2 or 3 and μ x 0 for all x 0 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 the resource population consumes biomass nutrients or energy which is at an abundance or concentration of y 0 t in the environment through a recipient controlled process 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 y 0 to be an underlying constant or specified time varying environmental input x 1 to be a resource population that lives off of y 0 and x 2 to be a consumer that most generally consumes x 1 y 1 y 2 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 figure 2 open in a new tab 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 θ 12 and θ 22 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 f 01 y 0 x 1 is transformed from y 0 into x 1 resource death transformation the total biomass flow rate m 1 x 1 is transformed from live resource x 1 into dead resources y 1 diversion transformation a proportion θ 12 of the total extracted biomass flow rate f 12 x 1 x 2 is transformed from live resource x 1 into dead resources y 1 extracted live biomass transformation a proportion 1 θ 12 of the total biomass flow rate f 12 x 1 x 2 is transformed from live resource x 1 into cropper or parasitic consumer biomass x 2 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 g 12 from dead resources y 1 transformed into consumer biomass x 2 as mentioned above in a more general treatment we can separate out coprophagy by adding an explicit faecal waste variable z 1 to the resource population consumer death transformation total biomass flow rate m 2 x 2 from live consumers x 2 is transformed into dead consumers y 2 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 f 12 and f 22 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 dx i dt ϕ i α i m i f i 2 x 2 x i dy i dt α i m i x i 1 θ i 2 f i 2 x i x 2 g i 2 x 2 δ i y i i 1 2 v i t t w i t s v i α i s ϕ i s v i s ds 6 where ϕ i k 01 f f 01 y 0 and ϕ 2 k 12 f θ 12 f 12 x 1 k 22 f θ 22 f 22 x 2 k 12 g g 12 y 1 k 22 g g 22 y 2 various special cases arise by allowing different combinations of the extraction rates f 12 g 12 f 22 and g 22 to be non zero table 2 along with zero or non zero diversion functions θ 12 and θ 22 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 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 f 21 diverted live resource θ 21 extracted dead resource g 21 cannibalism live f 22 diverted live cannibalism θ 22 cannibalism dead g 22 resource mortality m 1 consumer mortality m 2 selection of functional types x 1 x 2 0 0 0 0 0 ϕ 1 ϕ 2 cr only e g many frugivores x 1 x 2 0 0 0 0 0 ϕ 1 μ 1 x 2 d μ 1 dx 2 0 ϕ 2 cr hr e g leopards x 1 x 2 0 0 0 0 0 ϕ 1 μ 1 x 2 d μ 1 dx 2 0 ϕ 2 m 1 dm 2 dm 1 0 ...
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