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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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n the resource population that leads to increased mortality rates through senescence the most ubiquitous examples are parasites that are pathogenic to some degree croppers diverters and harassers croppers defined above to be primarily gatherers of live animal or plant biomass may or may not divert a significant flow of resource biomass to the dead resource compartment herbivorous grasshoppers for example divert up to 40 of what they eat gandar 1982 while carnivorous cheetahs divert up to 15 of what they hunt marker et al 2003 they may also harass individuals in the resource population yellowstone wolves for example fit into the cropper diverter harasser category and play a critical role in stabilising populations that scavenge on wolf produced carrion by producing elk carcasses year round and reducing the strong pulse of elk carcass towards the end of winter wilmers et al 2003 wilmers getz 2005 by definition since croppers strictly do not scavenge and if they are not cannibals then the only non zero extraction function in eqn 6 is f 12 which consequently in this case does not depend to first order on the dead biomass components y 1 and y 2 table 2 furthermore in the simplest case we assume that at least to first order the per capita senescence rates m i i 1 2 of individuals in the resource and cropper populations depend only on the deficit stress variables v i thus applying eqn 5 to each population for constant background senescence rates μ i 0 we obtain m i v i μ i v i s v i s v i in this case the differential equations in resource and cropper abundance x 1 and x 2 respectively are independent of variables y 1 and y 2 under these assumptions we can write down the following four variable inertial model as a special case of eqn 6 for croppers that also may harass the resource dx 1 dt ϕ 1 α 1 μ 1 v 1 s v 1 s v 1 f 12 x 2 x 1 dx 2 dt ϕ 2 α 2 μ 2 v 2 s v 2 s v 2 x 2 v i t t w i t s v i α i s ϕ i s v i s ds i 1 2 7 where ϕ 1 k 1 θ 1 f 01 y 0 and ϕ 2 k 2 f 12 x 1 8 the effects of harassment of resource individuals can be incorporated by generalising the feeding f 01 to be a monotonically decreasing function of x 2 in this case the simple extension of eqn 4 to f 01 y 0 x 1 x 2 a 1 b 1 1 x 1 c 1 x 2 c 12 γ y 0 9 sufficesfor some constant c 12 0 parasites and pathogens all types of organisms in food webs can be parasites and their hosts can be any kind of organism marcogliese cone 1997 lafferty et al 2008 parasites as defined in fig 1 are miners of live biomass thus on the gather miner spectrum the opposite of parasites are croppers while on the live dead spectrum the opposite of parasites are saprophages parasites are not obligate killers although some groups are such as insect parasitoids godfray 1994 macro endoparasites that consume a significant proportion of their host s biomass with up to 39 of the soft tissue body mass being accounted for by trematodes in one host snail species hechinger et al 2009 have a direct effect on their hosts through biomass extraction in addition such parasites usually have an indirect effect through increasing their host s senescence rate microparasites fungi protozoans bacteria and viruses are much less likely to have direct effects on the biomass of their hosts but those that are pathogenic may have considerable indirect effects on senescence some microparasites have evolved to become symbiotic example being flagellate protozoan parasites of the termite gut that help termites digest cellulose an important issue relating to host parasite dynamics is the question of average parasite load at the population level vs actual parasite load in individual hosts this problem can and has been addressed in several ways bearing in mind that a curvilinear but monotonically increasing relationship between host densities and parasite abundance is likely to occur as has been demonstrated in the context of gastrointestinal strongylid nematodes across 19 mammalian host species arneberg et al 1998 when parasite infections are widespread among all individuals in a host population then a simple approach to modelling parasite population abundance is to assume that all dynamics can be characterised in terms of an average infection intensity u in the host population in this case if the host and parasite populations have live biomass abundances x 1 and x 2 respectively then the proportion of host biomass infected is p inf x 2 ux 1 this approach of using an average intensity of parasitism across all hosts is likely to be more applicable to macroparasite infections such as those by nematodes cestodes and trematodes than bacterial or viral infections where individuals are regarded as either infected or not infected moreover in the latter case dose of infection may be a factor in determining whether an individual host succumbs to an associated disease claridge et al 2002 the use of average intensities of infection may also be applicable to microscopic parasites such as coccidia found in the gastrointestinal tracts of almost all vertebrate ruminants or even malarial plasmodia where abundance in hosts mosquito vectors in this case see dawes et al 2009 is important in determining the death rate additional complexities arise such as parasites increasing the vulnerability of their hosts to predators hudson et al 1992 but we leave such complexities to future studies parasite death rates themselves occur both independent of host deaths that is when the parasites die within the host or during the process of transmission from one host to another and with the death of host individuals fig 2 in some cases parasites may cause disease but then jump from one host to another before the death of the host this is particularly true of ectoparasites such as ticks that vector various diseases including rocky mountain spotted fever and lyme disease in humans in this case however three rather than two species are involved in our btw model the process of transmission itself is not explicitly considered but assumed to occur at finer temporal and possibly spatial scales than the spatio temporal scale of the btw paradigm to obtain this finer level of resolution requires that we divide the population further into susceptible and infected individuals with transmission assumed to follow either a mass action principle a frequency dependent transmission principle or more generally a saturating rate for which mass action and frequency dependent transmission are special cases mccallum et al 2001 with these various points in mind one approach is to assume the existence of a background host mortality rate that is enhanced by the presence of the parasites x 2 to yield the expression m 1 μ 1 μ 12 x 2 v 1 s v 1 s v 1 10 for constants μ 1 0 and μ 12 0 one might also assume that the parasite has a background mortality rate as the parasite is cleared from hosts by mechanisms that relate to the host immune system and parasite senescence additionally in many cases parasites might die along with hosts at a rate proportional to the parasite induced host death rate where this factor of proportionality depends on the ratio x 2 x 1 under these assumptions it follows that m 2 x 2 μ 2 v 1 s v 1 s v 1 x 2 μ 21 x 2 x 1 m 1 x 1 μ 2 μ 21 μ 1 μ 21 μ 12 x 2 v 1 s x 2 v 1 s v 1 where we note that μ 21 has a value that takes account of the factor 1 u mentioned above in calculating the proportion p inf discussed above viral and in most cases bacterial pathogens need to be treated differently than parasites whose primary effect on the host is the resources extracted from the host population in the case of microbial pathogens that either kill their hosts or go on to be defeated by the host s immune system the first order effect is the increase in the mortality rate of hosts rather than a decrease in the host biomass in this case the pathogen s abundance x 2 may best be measured as the number of pathogens per unit biomass of host or per unit volume of the host s blood phloem tissue or cytoplasm as discussed in the next section for microparasites such as b anthracis the infectious agent are spores that enter the environment soon after the death of a host in this case all other factors equal the rate of growth of the number of b anthracis spores in the environment will be proportional to the pathogen induced mortality μ 12 x 2 v 1 s v 1 s v 1 thus it follows that ϕ 2 a 2 μ 1 μ 12 x 2 v 1 s v 1 s v 1 11 where a 2 0 the rate at which a diseased carcass contributes spores to the environment may itself depend on many other factors such as the presence of scavengers to open the carcass moreover the background mortality rate m 2 of spores x 2 in the environment may decrease with increasing density x 2 of spores if spores at infectious sites are degraded at a greater per capita rate when present at high than at low densities in this case to first order we might assume m 2 μ 2 μ 22 x 2 if we now ignore the inertial component by forcing v t 0 for all t then the host pathogen equations that arise are dx 1 dt k 1 a 1 y 01 b 1 1 x 1 c 1 γ 1 y 01 α 1 μ 1 μ 12 x 2 x 1 dx 2 dt a 2 μ 12 x 1 α 2 μ 2 μ 22 x 2 x 2 12 which provides a novel alternative formulation to numerous existing approaches e g murdoch et al 2003 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 x 1 and dead y 1 resource compartments in a model of anthrax in enp fig 4 appendix s4 although anthrax does infect many other species in enp figure 4 open in a new tab a simplified anthrax centred biomass transformation web in etosha national park namibia see appendix s4 for equations modelling this system bacillus anthracis spores abundance x 2 are distributed within lizs across a several 1000 km 2 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 a 2 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 μ 22 is removed i e set to zero as the factor a 2 controlling the number of spores entering the environment per unit biomass of infected carcass is increased from a 2 0 5 panel b to a 2 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 a 2 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 figure 5 open in a new tab the zebra elephant biomass abundance x 1 t scale 1 18 000 metric tons and anthrax spore abundance x 2 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 a 2 0 5 and μ 22 0 0001 b a 2 0 5 and μ 22 0 c a 2 0 8 and μ 22 0 and d a 2 1 2 and μ 22 0 an additional compartment in the model is live jackal biomass at abundance x 3 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 x 4 and dead y 4 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 v 1 v 3 and v 4 i e for all the live biomass compartments except anthrax x 2 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...
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