Accommodating environmental variation in population models: metaphysiological biomass loss accounting

Authors


Correspondence author. E-mail: norman.owen-smith@wits.ac.za

Summary

1. There is a pressing need for population models that can reliably predict responses to changing environmental conditions and diagnose the causes of variation in abundance in space as well as through time. In this ‘how to’ article, it is outlined how standard population models can be modified to accommodate environmental variation in a heuristically conducive way. This approach is based on metaphysiological modelling concepts linking populations within food web contexts and underlying behaviour governing resource selection. Using population biomass as the currency, population changes can be considered at fine temporal scales taking into account seasonal variation. Density feedbacks are generated through the seasonal depression of resources even in the absence of interference competition.

2. Examples described include (i) metaphysiological modifications of Lotka–Volterra equations for coupled consumer-resource dynamics, accommodating seasonal variation in resource quality as well as availability, resource-dependent mortality and additive predation, (ii) spatial variation in habitat suitability evident from the population abundance attained, taking into account resource heterogeneity and consumer choice using empirical data, (iii) accommodating population structure through the variable sensitivity of life-history stages to resource deficiencies, affecting susceptibility to oscillatory dynamics and (iv) expansion of density-dependent equations to accommodate various biomass losses reducing population growth rate below its potential, including reductions in reproductive outputs. Supporting computational code and parameter values are provided.

3. The essential features of metaphysiological population models include (i) the biomass currency enabling within-year dynamics to be represented appropriately, (ii) distinguishing various processes reducing population growth below its potential, (iii) structural consistency in the representation of interacting populations and (iv) capacity to accommodate environmental variation in space as well as through time. Biomass dynamics provide a common currency linking behavioural, population and food web ecology.

4. Metaphysiological biomass loss accounting provides a conceptual framework more conducive for projecting and interpreting the population consequences of climatic shifts and human transformations of habitats than standard modelling approaches.

Introduction

There is a pressing need for models that can reliably predict the consequences of human-modified habitats, climates and ecosystem contexts for animal and plant populations (Norris 2004; Coreau et al. 2009). Theoretical population ecology has developed increasingly sophisticated models, but these models can prove inadequate for diagnosing the causes of increasing or decreasing populations (Caughley 1994). According to Getz (1998), ‘the art of modelling in population ecology may have more to do with fixing what is most glaringly wrong with models than in finding the right model’. Models encapsulate, in heuristically simplified form, what we currently know and understand about the system being modelled. Yet, a gulf still exists between elegant theoretical models and the real-world biological processes that they supposedly represent. Nevertheless, judging models merely by their fit to data is misdirected; simplicity and generality are equally important (Ginzburg & Jensen 2004). Clearly, some reconciliation is needed between the theoretical models that have become standard in the literature and the biological and environmental processes that generate changes in populations (Owen-Smith 2010a). The manifold influences of climatic variation on the population dynamics of large mammalian herbivores are especially well documented (Owen-Smith 2010b). In this article, I describe how various aspects of environmental variation can be accommodated within population models in ways that have the potential to be heuristically productive, illustrated using working examples.

Models of population dynamics have taken various forms. Single-species population models such as logistic growth equations and variants thereof represent density-dependent feedbacks regulating populations around some environmental carrying capacity, but not what determines this capacity. Time series elaborations allow for the perturbing effects of stochastic environmental variation on the population density level manifested, but still assume some constant ‘attractor’ about which population abundance varies (Royama 1992; Denis & Taper 1994; Bjornstad & Grenfell 2001). Assumptions of a stable attractor become increasingly untenable the longer populations are observed (Pimm & Redfearn 1988; Owen-Smith & Marshal 2010), and spatial variation modifies the population abundance manifested (Brown, Mehlman & Stevens 1995; Hobbs & Gordon 2010). Moreover, the discrete, generally annual time steps usually adopted in these models obscure the seasonal variation in resources and conditions that contributes to the population dynamics generated (Owen-Smith 2002a). Models of interacting populations based on the Lotka–Volterra equations emphasize the propensity to oscillations of predators and their prey, or more generally consumers and resources, but do not represent the environmental contexts that may promote or suppress oscillatory dynamics (Turchin 2003; Owen-Smith 2002b; Gross, Gordon & Owen-Smith 2010). Furthermore, for most consumers, the resource base is actually constituted by multiple populations with distinct dynamics and other properties affecting choices among them. Demographically structured models highlight how distinctions in rates of survival and reproduction among age or stage classes modify the dynamics generated, but only superficially represent the environmental influences affecting these vital rates (Caswell 2001; Lande, Engen & Saether 2003). Individual-based models incorporate unlimited contextual detail in principle, but in practice pose considerable challenges for parameterization as well as conceptual interpretation of the sources of the dynamics generated (Grimm & Railsback 2005).

In this ‘how to’ article, I describe how these standard models can be modified to represent intrinsic and extrinsic processes generating population dynamics in ways that are more conducive to diagnosing the causes of population change. This approach draws on metaphysiological modelling concepts developed originally by Getz (1991, 1993) to embed interacting populations within food web contexts and expanded by me to link population dynamics to the behavioural ecology of resource selection in variable environments (Owen-Smith 2002a). Parallels exist with the dynamic energy balance models formulated by Kooijman et al. (2008); Kooijman, van der Hoeven & van der Werf (1989); Nisbet et al. (2000) and De Roos & Persson (2001), but these authors place their emphasis on intrinsic physiological mechanisms rather than on the extrinsic environmental influences. Moreover, material resources needed for constructing biomass can restrict population performance as well as the supply of energizing substrates.

The structure of this article will be as follows. First, I will describe how the Lotka–Volterra equations, and modifications thereof, can be elaborated to accommodate environmental influences potentially affecting population dynamics in a general conceptual way. Second, I will show how this approach can be used to explain spatial variation in habitat capacity, as reflected by the population abundance supported, using real-world data. Third, I will illustrate how population structure can be incorporated into these models, using a classical case study as an example. Last, I will suggest how simple single-species models could be modified to reconcile them with the metaphysiological perspective through a general framework for biomass loss accounting. To assist readers in developing and applying this approach, I have supplied the computational code that was developed for these specific examples in Appendices S1–S5, together with listings and some explanation of the parameter values used, provided in these appendices. This approach has proved its heuristic value in my postgraduate teaching (Owen-Smith 2007).

Metaphysiological population modelling

The foundation of the metaphysiological modelling approach rests on the Lotka–Volterra equations of coupled predator–prey (or consumer-resource) dynamics. Expressed phenomenologically,

image(eqn 1)
image(eqn 2)

where X represents the prey, or resource, population, Y the predator, or consumer, population, R the resource production function, U the function governing uptake or extraction of resources by consumers, G the functional gain in consumer abundance as a result of this uptake and M the function governing intrinsic mortality or other losses from the consumer population. In practice, consumer dynamics is usually represented by coupling a hyperbolically saturating extraction function (equivalent to a Holling Type II ‘functional response’) with a constant per capita mortality loss:

image(eqn 3)

where c is the conversion coefficient from resources into consumers, umax is the maximum rate of resource extraction, x1/2 is the resource abundance level at which the rate of resource extraction reaches half of its maximum and m represents the constant proportional mortality loss. Somewhat weirdly, this formulation suggests that consumers die at a constant rate in the absence of food, ameliorated by food gained. Rather than being constant, the mortality loss would be expected to rise as the food supply becomes increasingly deficient. Nevertheless, even with superabundant food, some minimal mortality would be expected through terminal senescence (dying of old age). Moreover, resource uptake must be offset against intrinsic metabolic attrition.

Equation 2 can be expanded into more mechanistic form as follows, with all rate functions expressed on a per capita basis:

image(eqn 4)

where MP,Y represents metabolic attrition, MQ the resource-dependent mortality and MZ the additive mortality dependent on predator or parasite abundance Z (see Fig. 1 for a pictorial representation). We have now separated three loss components: (i) metabolic dissipation, (ii) mortality predisposed by resource deficiencies (e.g. through starvation, potentially amplified by other agents) and (iii) additional mortality imposed by predation and parasitism. Note that the above mortality functions subsume deaths that would otherwise have occurred through senescence, because organisms die sooner when the food supply becomes inadequate or when predators kill them. Note further that eqns 1–4 are expressed in differential calculus form, implying that changes in consumer and resource populations occur effectively continuously. This requires that the currency for expressing population changes becomes aggregate biomass, rather than numbers, so that per capita rates become relative rates per unit of biomass. Populations get bigger and their demand on resources greater when the individual organisms expand in size as well as numerically. There is no term for a birth rate in eqn 4 nor in Lotka–Volterra equations in general; how the aggregate population biomass is partitioned among individuals varying in size and age is a secondary consideration.

Figure 1.

 Diagrammatic outline of the biomass accounting model, showing the relative biomass growth rate dB/Bdt dependent on the difference between biomass gained from resources consumed and biomass lost through metabolic attrition plus mortality from various causes (from Owen-Smith 2002a).

Eq. 4 was mnemonically labelled the ‘GMM’ model in Owen-Smith (2002a), to suggestively capture the dependence of population growth on resource Gains relative to Metabolic and Mortality losses (as opposed to Births, Immigration, Deaths and Emigration in classical BIDE models). The latter emphasize the demographic mechanisms generating population changes. In contrast, the GMM model focuses attention on the extrinsic influences contributing to population change: food resources, or more specifically the rate at which these can be captured and converted into consumer biomass; metabolic attrition, including both a maintenance component and additional expenditures that may be incurred in searching for food and from coping with weather conditions; and mortality risks, generated partly by resource deficiencies relative to metabolic requirements and partly from the abundance of predators (or parasites) and their food-seeking activities.

To make eqn 4 operational, the forms of the functional relationships represented by G, MP, MQ and MZ (omitting the ys indicating the consumer population) need to be specified. Let us assume a mechanistic Holling Type II formulation for the extraction response, and resource-dependent mortality inversely dependent on the ratio of resource gains to metabolic requirements, while leaving the metabolic rate constant and additive predation absent:

image(eqn 5)

where = area searched per unit time, = fraction of the resource available for consumption, = handling time per unit of food and = a proportional constant. Note that G represents the specific form of the resource gain function on the left. The maximum growth rate of the consumer population may need to be capped under highly favourable environmental conditions, to ensure that the annual increase does not exceed realistic expectations.

Before this model can be exercised, we must also specify functional forms for the resource dynamics represented by eqn 1. If the resource is vegetation being consumed by herbivores, a logistic production function can reasonably be assumed, i.e. the plants grow until their leaf canopy saturates the available ground surface for capturing sunlight. Note that this formulation represents the seasonal growth dynamics of vegetation biomass, not the change in the plant population. The predation term MZ becomes replaced by the extraction function U(X,Y) expressed in Holling Type II form. Note further that plant growth occurs only when conditions are sufficiently warm or wet – plants become dormant and show no growth during most of the winter or dry season. Hence eqn 1 becomes

image(eqn 6)

where rx = intrinsic growth rate of vegetation and Xmax its maximum biomass, while g most simply switches between values of 1 (summer or wet season) and 0 (winter or dry season).

The output of the coupled eqns 5 and 6, parameterized to represent a large mammalian herbivore interacting with a homogeneous vegetation resource, illustrates how plant biomass gets reduced progressively during the dormant season, effectively through consumption (Fig. 2). This seasonal reduction in food availability causes the biomass trend of the herbivore population to oscillate seasonally between positive and negative, once its abundance becomes sufficient to substantially deplete the vegetation resource during the dormant season. In this seasonal environment, the herbivore biomass does not attain any equilibrium with resource supply rates. Its potential abundance would be much higher if the wet season conditions persisted and zero if the dry season conditions lasted long enough. Nevertheless, inertial constraints limit the biomass oscillations within a circumscribed range, and annually censused abundance levels can appear to be constant if conditions do not change between years. A density feedback arises indirectly through the seasonal resource depression, greater when the herbivore population is higher.

Figure 2.

 Output of seasonal biomass accounting model, showing coupled herbivore (thick lines) and plant (thin lines) biomass dynamics. Grey lines represent within-year dynamics, while black lines connect abundance levels expressed at the same stage annually (modified from Owen-Smith 2002a).

In this model, seasonality is stabilizing, by restricting the abundance that the herbivore population can attain and hence its impact on vegetation resources. For the plant population to persist from 1 year to the next, it must have sufficient underground (or otherwise ungrazable) biomass to regenerate foliage at the start of each growing season. If the herbivore population becomes sufficiently great, it could restrict the build-up of these stored reserves, or damage the tissues containing them, such that plants die and the plant population generating the annual forage production shrinks. Representing plant population processes would require further elaboration of eqn 6. One simple way of doing this is described in Chapter 13 of Owen-Smith (1988) (repeated in Chapter 11 of Owen-Smith 2007). However, even if plant populations are protected from such ‘over-grazing’ through inaccessible biomass components, the herbivore population can exhibit oscillatory dynamics if food quality (governed by the conversion coefficient c) is so high so that the herbivore population trend does not become negative until very little vegetation remains towards the end of the dormant season. Any variation in plant production from 1 year to the next, as might be governed by rainfall variation, contributes towards precipitating periodic crashes in herbivore population biomass under these conditions.

This model represents the herbivore population as being limited solely by the amount of vegetation remaining towards the end of the dormant season. In practice, diminishing food quality during the dormant season may be the major limitation. The can be represented by making the conversion coefficient c a function of the declining vegetation biomass over the dormant season. Note that this pattern is distinct from the ‘forage maturation’ concept (Fryxell 1991), whereby the nutritional value of vegetation decreases as biomass increases and tissues mature during the course of the growing season. If through no other mechanism, forage quality will decrease over the course of the dormant season through selective consumption of the more nutritious plant parts and species.

A simple way of representing declining food quality is to fall back on the Michaelis–Menton formulation of the resource uptake response represented in eqn 3, but make the half-saturation parameter x1/2 somewhat greater in the consumer gain function G than in the resource uptake function U in eqns 1 and 2. The effect of seasonally declining food quality represented in this way is to stabilize the dynamics of the herbivore population for parameter values that would otherwise tend to promote oscillatory dynamics.

The simple equations specified earlier can easily be incorporated into a spreadsheet model (see Appendix S2) or coded in some computer programming language (Appendix S3). These simplified models can be used to explore the range in parameter values and specific functional forms that allow relatively stable dynamics of the herbivore population to be generated, in contrast with the irruptive dynamics emphasized by Caughley (1976) and Forsyth & Caley (2006). See Gross, Gordon & Owen-Smith (2010) for a more thorough assessment of the irruptive potential of herbivore populations.

The time step in the model should be made sufficiently short so that spurious lagged effects are not generated. A daily time step would surely be sufficient, but a weekly one is generally adequate for larger organisms. This raises a new issue, arising from mismatches in the time-scales for different processes. Susceptibility to mortality depends not immediately on the daily (or weekly) resource gains, but rather on resources accumulated over some extended period back in time, because of the carryover of stored body reserves. In the extreme, the likelihood of mortality during the dormant season depends on how good conditions were during the preceding growing season, enabling the build-up of fat reserves by animals (and starch reserves by plants; see Getz & Owen-Smith 1999). Just how far back in time the influence on resource-dependent mortality should embrace the daily or weekly gain function G is unclear. In earlier modelling (Owen-Smith 2002a), I simply estimated the annual mortality rate that would apply if the resource conditions existing at each time step remained constant over a year, then averaged these daily or weekly mortality rates to derive the annual mortality that would be effective, aided by appropriate tuning of the proportional constant q. This issue needs further exploration.

Of course, stable consumer dynamics can be expected only if the intrinsic resource dynamics remain unchanged between years. Variable rainfall and hence plant growth could be incorporated into the model by making the switching parameter g in eqn 6 shift between zero and one during the course of the growing and dormant seasons according to some regime, perhaps stochastically. For temperature-driven systems, the intrinsic growth rate rx of the plants could be made variably dependent on daily or weekly conditions.

Alternative functional forms and different parameter values will be necessary if the consumers are not large ungulates consuming herbage. Smaller herbivores must potentially capture resources at a faster rate than larger ones, to govern their higher potential population growth rate. They can do this by consuming higher-quality plant parts, aided for invertebrates by lower metabolic costs. Many small herbivores undergo dormancy during winter, thereby restricting metabolic shrinkage as well as predation when resources are vastly inadequate. Some insects show annual turnover in their populations, with new biomass generated from the tiny carryover incorporated in eggs. Large predators may find their prey easier to catch and kill during the winter or dry season, when herbivores become weakened and perhaps hampered by snow accumulation. Hence, the effective availability of such prey, as governed by a in eqn 5 rather than resource quality c, can vary seasonally. Seasonal switching by consumers among different resource types will be considered in the next section below.

The concepts embodied in the metaphysiological GMM model have the potential to be applied to any animal population, with suitable modifications. I believe that more difficult challenges will come in applying them to plant dynamics, for several reasons: (i) the resource base for plants is constituted by very different sources – CO2 from the atmosphere, various mineral nutrients from the soil, with extraction of both promoted by capture of light photons and enabled by soil moisture, (ii) seasonal growth and decay of plant biomass occur in addition to changes in the plant populations (ramets and genets, Harper 1977) generating this biomass, (iii) much of what might be measured as biomass consists of structural, metabolically inert tissues, and (iv) population growth takes place largely through a lottery for dispersal into vacant sites for establishment. Nevertheless, the seasonal phasing of plant biomass dynamics must be recognized, at least, because of its ramifying influences on the dynamics of all higher trophic levels.

Spatial variation in the population abundance supported: heterogeneous resources

Next, I describe a specific application of the GMM model to address a feature not explained by standard models of population dynamics: regional variation in habitat suitability as indicated by the population abundance attained. The simple model developed above represented the vegetation resource as a single population, but in fact this resource is constituted by a diverse set of plant types differing in their growth characteristics and nutritional value. The model I will now develop is based on assessing the relative contributions of different vegetation components to supporting a herbivore population through the seasonal cycle. It also illustrates how empirical data can be incorporated into the model.

The herbivore is represented by a large browsing antelope, the greater kudu (Tragelaphus strepsiceros). Kudus attain densities of around two animals per km2 in typical savanna vegetation, but are largely absent from regions where fine-leaf umbrella thorn (Acacia tortilis) savanna predominates. In succulent thicket vegetation, kudu densities exceed 10 animals per km2. Data were obtained from observations on the seasonal diet selection of kudus in the Nylsvley Nature Reserve (Owen-Smith & Cooper 1987, 1989; Owen-Smith 1994). Supporting information on available leaf biomass and chemical contents was provided by other contributors to the South African Savanna Biome study (Scholes & Walker 1993). Broad-leaf savanna characterized by wild seringa (Burkea africana) and other deciduous trees predominated in the study area, but patches of fine-leaf savanna where spinescent acacias, including A. tortilis, prevailed were also present and exploited seasonally by the kudus. To make the model tractable, the woody plant species eaten by kudus were grouped into five palatability categories based on seasonal patterns of selection for them shown by kudus and other browsers (Owen-Smith & Cooper 1987). Forbs (herbs apart from grasses) were amalgamated into a single category. The available forage biomass that broad-leaf and fine-leaf savanna presented during the growing season was very similar (Table S1 in Supporting Information). By the late dry season, only a few evergreen species retained much foliage. Evergreen trees or shrubs formed a small proportion of the vegetation in the broad-leaf savanna, but were absent from the fine-leaf savanna. Because vegetation measurements were unavailable for succulent thicket, I assumed that its peak forage biomass was identical to that in the savanna habitats, but with evergreen shrubs prevalent.

For this model, the Holling Type II equation representing the resource gain function, as incorporated in eqn 5, was modified to allow for multiple food types as follows:

image(eqn 7)

where = search rate in area covered per unit time, = acceptance fraction of the food encountered that is consumed, = conversion coefficient from food consumed into herbivore biomass, = available food biomass per unit area and = food uptake (or eating) rate, which is the inverse of the handling time per unit biomass (Owen-Smith & Novellie 1982). The overall rate of food gain was summed over the range r of food types i considered. Adaptive selection among these food types was allowed at each time step, based on maximizing the rate of biomass gain, affecting the acceptance coefficients ai assigned. This results in the progressive incorporation of lower-value food types into the diet as the availability of higher-quality plant types becomes depleted, both through consumption and through intrinsic attrition. The biomass conversion coefficients ci take into account both the relative digestibility of the food types and the conversion from plant dry mass to animal live mass. Over a daily time frame, the gain function needs to be multiplied by the proportion of the day spent foraging, relative to resting or other activities not associated with feeding, which was assumed to be constant.

The resource-dependent growth potential (RGP) of the herbivore population, excluding mortality, was estimated at each weekly time step from the extent to which biomass gains (G) exceeded metabolic maintenance requirements (MP), relative to the basal metabolic rate (P0), all consistently expressed as biomass fluxes:

image

Next, the weekly susceptibility of the herbivores to mortality as a result of the widening resource deficits incurred during the dry season needs to be estimated. The projected annual mortality rate if the current conditions persisted was assumed to be linearly but inversely dependent on the ratio between resource gains and metabolic requirements:

image

where q0 is the intersection and q the slope coefficient for the linear relationship. Weekly estimates were averaged to obtain the effective mortality loss over the course of the dry season. The mortality rate for the first week of the dry season, before resources became depleted, was assumed to represent the constant low mortality rate during the wet season. The annual mortality rate was the average of the wet season and dry season mortality estimates. The annual population growth rate was determined by subtracting this mortality loss from the maximum population growth potential. Values assumed for animal parameters, as a population average, are listed in Table S2 in Supporting Information. All of them were based on either empirical observations or literature values, except for the mortality intercept and slope coefficient. To assign credible estimates for the latter, I assumed that the annual biomass loss through resource-dependent mortality would be 30% per year when the biomass gain barely met maintenance metabolic requirements, diminishing to zero when the nutritional gain was 25% above requirements.

The last step entailed estimation of the annual biomass growth rate attained by the kudu population over a range of fixed stocking densities. Greater population densities lead to more rapid depletion of preferred vegetation components during the course of the dry season, and hence greater metabolic deficits towards the end of this season, amplifying mortality (Fig. 3a). The density response curves projected zero population growth intercepts on the X-axis that closely resembled observed kudu densities supported in the three vegetation types (Fig. 3b). This correspondence was perhaps a fortuitous outcome of appropriate parameter values assumed for the mortality loss function. While different values would affect the specific zero growth densities, they would have little influence on the relative abundance levels projected for these distinct habitat types.

Figure 3.

 Output from the habitat suitability model for a browsing ungulate. (a) Relative growth potential through the dry season months associated with zero population growth. (b) Population growth potential relative to the population density, comparing three habitat types: broad-leaf savanna, fine-leaf savanna and succulent thicket. (c) Effect on population performance of adjustments to the broad-leaf savanna vegetation entailing (i) halving the abundance of forbs constituting the highest value resource, (ii) halving the abundance of palatable deciduous plants serving as the staple food resource, (iii) eliminating palatable but spinescent species, (iv) eliminating relatively palatable evergreen species or (v) eliminating unpalatable deciduous species (From Owen-Smith 2002a).

The salient feature of this model is its counterbalancing of resource gains during the wet season against losses incurred during the dry season, when most savanna trees shed their leaves. As a further extension of the model, the relative contributions made by particular resource components towards supporting the kudu population may be evaluated by reducing or eliminating their availability in the model (Fig. 3c). The model output suggests that the early production of new foliage by unpalatable deciduous trees in the transition period between the dry season and the onset of the rains is more influential than the small evergreen constituent for supporting the kudu population in the broad-leaf savanna. The contribution by this otherwise neglected resource component bridges a time when the kudus would have quite severely starved for 2–3 weeks in its absence. This finding suggests that the key resources concept, advocated by Illius & O’Connor (1999) to explain the disproportionate contribution of dry season resources to supporting large herbivore populations, needs to be expanded to encompass distinct resource contributions at different stages of the annual cycle (Owen-Smith 2002a; Chapter 11). The code for this model, written as a program in TrueBASIC, is appended (Appendix S3). A counterpart model described in Owen-Smith (2002a,b) assesses how range condition as indexed by grassland composition affects the stocking density of grazing ungulates supported.

Accommodating population structure

The habitat capacity model developed above considered only the aggregate biomass density of the herbivore population. For some situations, the structuring of the population biomass into age or stage classes makes a big difference to the population dynamics produced. One specific example concerns the dynamics of the Soay sheep population inhabiting the Island of Hirta in the Outer Hebrides. The periodic die-offs manifested by these sheep are predisposed by the population structure interacting with variability in late winter conditions, with lambs and adult males much more susceptible to mortality than adult females (Clutton-Brock et al. 1991; Coulson et al. 2001). The higher proportion of adult females surviving facilitates rapid recovery following these population crashes.

Conceptually, the GMM equation can be applied to any population segment as well as to the aggregate population biomass. Only the parameter values vary, with younger and hence smaller individuals having higher relative metabolic costs, and accordingly greater susceptibility to mortality for similar resource gains (Table S3 in Supporting Information). Why adult males are more vulnerable to mortality is not immediately obvious. The challenge is how to model the progression of individuals through the stage classes from birth to adulthood.

From a biomass accounting perspective, births are represented by the transfer of some fraction of adult female biomass to constitute the newly born segment at a certain stage in the seasonal cycle. Thereafter, juvenile growth is subsidised by maternal transfers up to the age at when such provisioning ceases (age at weaning for mammals). Adult females recover their mass lost to reproduction over the course of the year, while adult males utilize resources but make no material contribution to biomass increase. The population composition in terms of the proportions contributed by these growth stages both restricts the overall population growth rate that can be manifested and underlies the propensity of the population to crash when adverse conditions occur.

Whether the model developed replicated the oscillatory dynamics manifested by the Soay sheep population depended very much on assumptions made about the nutritional quality of the herbage consumed by the herbivores (Fig. 4a). Only when vegetation quality is relatively high does the modelled herbivore population reduce the remaining forage biomass to a sufficiently low level by late winter to precipitate a population crash. The modelling exercise suggested that the inherent instability exhibited by this sheep population, in comparison with the relative stability of the red deer population inhabiting a very similar island environment on Rum (Clutton-Brock et al. 1991), could be largely a consequence of this extrinsic environmental feature. Lowering the vegetation quality uniformly in the model suppressed the oscillations, by reducing the growth rate achieved by the herbivore population (Fig. 4b). The program code for this model is appended in Appendix S5. Further details about the model are given in Chapter 13 of Owen-Smith (2002a,b).

Figure 4.

 Output from the structured population model parameterized to represent Soay sheep dynamics, showing (a) the persistent oscillations generated by high food quality, compared with the observed dynamics, and (b) suppression of the oscillations brought about by lowering the effective quality of the reserve forage exploited during winter (modified from Owen-Smith 2002a).

This model does not incorporate the effects of weather variation, as represented by the occurrence of wet and windy conditions at the end of winter in the case of Soay sheep, on the projected herbivore dynamics. Because these extreme weather conditions impose thermal stress, they would affect the value of the metabolic expenditure MP in eqn 4. However, it is also possible that animals could avoid such expenditures by seeking sheltered sites, thereby entailing costs in terms of curtailed foraging time.

Note that mortality in the juvenile stage makes a relatively lesser contribution to the overall biomass dynamics than mortality in the adult stage, simply as a consequence of the individual weight difference. A further elaboration could represent both biomass and numerical dynamics within stage classes in parallel, thereby enabling changes in individual weight-for-age to be extracted. This would allow the risk of mortality to be related to the current state dependent on past resource gains, rather than on gain functions extending back over some uncertain period, as was suggested above. Again, such modifications are for others to explore.

Demographically structured models conventionally represent only the female segment of the population, provided a shortage of males does not restrict the reproductive output of these females. The metaphysiological approach exposes shortcomings of this simplification, because the males do affect the resource supply remaining through the seasonal cycle, and hence the reproductive success of these females as well as well as the growth rates achieved by immature animals.

Parameterization

An advantage of the metaphysiological approach is that parameter values can be derived mostly from biological process rates, in particular allometric scaling relationships. Power relationships governing the dependence of basal or resting metabolic rates on body mass have become quite firmly established, despite some variation among taxa (Capellini, Venditi & Barton 2010; Glazier 2010). Additional expenditures incurred in particular activities or for thermoregulation can be represented as multiples of the resting rate (Garland 1983; Porter et al. 2002). The maximum biomass growth rate that can potentially be attained is underlain by the maximum metabolic scope, also expressed as a multiple of the basal or resting rate (Hammond & Diamond 1997; Speakman & Krol 2010). Metabolic expenditures, generally expressed in energetic units, need to be converted into their biomass equivalents, as shown in Table S2 (Supporting Information). The minimal mortality loss to senescence is simply the inverse of the maximum potential life span.

Applying these estimates at the population level is complicated by the variable size structure of the individuals constituting the population. For aggregate biomass, an averaged body mass can be used as an approximation. However, organisms with determinate growth, such as mammals and birds, have no individual growth potential once they have attained adult size, apart from the contribution made to offspring growth via body secretions or provisioning. Hence, the potential biomass increase over the annual cycle by a natural population is rather less than the growth that could be attained by a farmer raising a batch of young animals towards maturity. Moreover, the growth of young animals in the wild is restricted by nutritional shortfalls experienced during the adverse season (Parker, Barboza & Gillingham 2009).

The forms of the functional relationships need to be derived empirically or incorporated using logically defendable approximations. The annual biomass increase depends multiplicatively on the biomass increments or decrements incurred at each step or stage during the year. If seasonal resource shortfalls are such that death almost certainly results before conditions improve, the population cannot persist no matter how favourable conditions are at other times of the year.

Biomass loss accounting

Single-species models represent the population growth rate as being reduced below its potential maximum by the ratio between the current abundance and the abundance level at which the net growth rate becomes zero (or carrying capacity), e.g. the logistic equation:

image

where N represents the variable abundance, usually in numerical terms, rmax is the maximum growth rate and K is the carrying capacity. Time series elaborations employ discrete (generally annual) time steps and incorporate lagged density feedbacks as well as external influences reducing or elevating population growth, interpreted as stochastic perturbations:

image

where Nt represents population abundance at time t, βi are coefficients governing the strength of the density feedbacks at increasing time-lags and βW governs the perturbing influence of an external factor W operating at time t-1. Note that this equation can be expanded to incorporate longer lags and additional perturbing influences, as indicated earlier. Furthermore, for statistical reasons, the population abundance is generally log-transformed.

From a metaphysiological perspective, the maximum biomass growth rate depends on the maximum rate at which nutrients can be extracted and converted into biomass, relative to maintenance metabolic requirements. This must equal the maximum numerical growth rate, allowing for the restrictions imposed by population structure. From eqns 3 and 4,

image

where m0 represents combined biomass losses from maintenance metabolism plus mortality through senescence. Substituting this into eqn 5, ignoring additional metabolic costs and additive predation, we obtain

image

which expresses resource-limited growth in biomass Y restricted by the ratio between the metabolic demand and the biomass gained from the material resources consumed.

A structural resemblance with the logistic equation is apparent, but there are some important distinctions. The numerical density variable has been replaced by the metabolic demand on resources associated with the biomass density MP. The carrying capacity constant has been replaced by the variable rate of resource capture G. More fundamentally, a need to resort to a discrete time approximation is obviated through the use of a biomass currency. More generally, the biomass loss accounting needs to be expanded to incorporate additive losses owing to predation and parasitism, variable metabolic costs per unit of biomass imposed by ambient weather conditions and reductions in the growth potential brought about through reproductive failures (delays in the onset of reproduction, lengthened intervals between reproduction and reduced litter or clutch sizes). Hence, in general terms

image(eqn 8)

where MF,Y expresses the losses in biomass that might otherwise have been generated owing to reproductive shortfalls, and other terms are as in eqn 4. Reproductive losses are likely to depend on conditions earlier in time than those affecting mortality losses. Note that all of the terms on the right entail subtractions if rmax is indeed the maximum potential biomass growth rate.

In practice, all simulations entail discrete time steps – the question is how fine-scaled to make the iterations, noting that this defines the minimum time-lag recognized. Annual steps are too coarse to expose seasonal processes, but weekly iterations may seem unnecessarily detailed. Minimally, the year could be partitioned between the ‘fat’ season when most reproduction, and hence population expansion, takes place and the ‘lean’ season when most mortality occurs. Models incorporating such a bipartite division have been formulated for small mammals (Stenseth et al. 2002) and an ungulate (Coulson et al. 2008). However, finer temporal divisions could prove more insightful, especially to capture brief periods when extreme conditions make a disproportionate contribution to mortality (Wilmers & Getz 2004; Chan et al. 2005; Hone & Clutton-Brock 2007). Their effect on population biomass dynamics depends not only on the severity of the conditions but also on the duration over which these conditions persist, the stage in the annual cycle when they occur, and earlier circumstances affecting the body reserves carried forward (Getz & Owen-Smith 1999; Owen-Smith 2000; Hallett et al. 2004). The periods distinguished can be chained in terms of their contribution to annual biomass change, recognizing that a nonpositive outcome at any stage eliminates the population:

image(eqn 9)

where the numbers refer to the sequential periods over which the aggregate mortality loss MX is accounted for multiplicatively. Notably, mortality losses during various stages of the year are likely to depend on different predictors with reference to resource dependencies and environmental conditions (Payne & Wilson 1999).

Biomass loss accounting as expressed above could be applied similarly to any population segment, after having identified the maximum reproductive potential and minimal mortality rate expected for each segment. The latter should be close to zero until the maximum longevity is approached, except perhaps for newly born or hatched offspring. For example, I related the age-class-specific survival or mortality rates of a large herbivore to the annual rainfall total, governing resource production, relative to the population biomass density (Owen-Smith 1990, 2000) and subsequently distinguished the contributions of seasonal rainfall components (Owen-Smith, Mason & Ogutu 2005).

Discussion

In this ‘how to’ article, I have illustrated how metaphysiological modelling concepts can be applied at different levels of resolution, from quite abstract in the first and last examples to empirically fairly detailed in the second and third ones. As with all models, the level of detail required depends on the purpose for which the model is being developed. More abstract models need incorporate only the functional forms of the relationships that emerge from the underlying mechanisms. Units of population biomass (quantity) alone may be inadequate, and a second state variable representing the effective quality of this biomass, as governed by population structure or body reserves carried forwards from past conditions, could capture these effects in a simple way (Getz & Owen-Smith 2011). More detailed models allowing for variable resource selection could draw on principles of dynamic optimization (Clark & Mangel 1999) and the state-dependent perspective upon which it is based. I applied this approach to assess the optimal investments of resources gained towards growing bigger, growing fatter, or growing offspring during sequential stages of the annual cycle, distinguishing life-history stages (Chapter 7 in Owen-Smith 2002a). This illustrated why maximizing resource gains is not always optimal, especially when resources are plentiful and storing additional fat is detrimental for survival prospects.

The simplest models can be formulated within spreadsheets. Models taking into account resource heterogeneity and the adaptive choices among resource components or distinct habitats, or population structuring into life-history stages, require the additional capabilities provided by a programming language. The essential features of the metaphysiological approach are (i) subannual time steps, using a biomass currency, (ii) distinctions among processes contributing to reductions in population growth below its potential, (iii) structural consistency in the representation of population dynamics in interacting populations and (iv) capacity to accommodate environmental variation across space as well as through time. Furthermore, biomass dynamics provides a consistent measure of utility relating behavioural to population ecology, potentially conveyed upwards into food web analyses.

The examples of biomass accounting that I have used were based on large mammalian herbivores, because it is for this trophic guild that the effects of seasonality in plant growth on consumer dynamics are most strikingly evident. Seasonal variation in prey selection has been documented for large mammalian carnivores (Owen-Smith 2008), but with the adverse season tending to be the opposite of that for their herbivore prey. For small mammalian and insect herbivores, the timing of life-history events in the context of seasonal plant dynamics can be especially important for the annual abundance level attained. Even for large herbivores, a mismatch in the timing of reproduction can have substantial consequences for population dynamics (Post & Forchhammer 2008). For most plants, a biomass rather than numerical currency is obviously necessary, but what fraction of their tissues is effectively biomass, rather than merely providing structural support, becomes an issue. A further complication is that plants derive energizing and material resources from different sources and via different tissues. Moreover, soil moisture operates somewhat differently through enabling resource capture, rather than contributing materially. Among animals, nesting sites for birds can be interpreted as having a functionally comparable enabling role. Spatial variation in the risk of predation can have an overriding influence on where populations do or do not exist (Chirima et al. unpublished).

Metaphysiological biomass accounting represents population dynamics as essentially disequilibrial, alternating between growth towards the extremely high abundance that might be attained if conditions remained constantly benign and the extirpation that would result if the adverse season conditions prevailed for long enough. This approach provides the conceptual framework most conducive for addressing the disruptive effects of shifting climatic regimes and human transformations of habitat conditions on population dynamics and ultimately for the persistence of these populations and the species they represent.

Acknowledgements

The stimulus for this review originated with the working group on ‘Dynamics of large herbivore populations in changing environments’ established in the National Center for Ecological Analysis and Synthesis at the University of California Santa Barbara. Its final shape was improved by critical feedback received from Wayne Getz, Lev Ginzburg and Eloy Revilla enabled our joint support by the Stellenbosch Institute for Advanced Study. I am indebted to Tim Coulson for his constructive criticisms of previous drafts.

Ancillary

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