Accommodating environmental variation in population models: metaphysiological biomass loss accounting
2011; Wiley; Volume: 80; Issue: 4 Linguagem: Inglês
10.1111/j.1365-2656.2011.01820.x
ISSN1365-2656
Autores Tópico(s)Animal Ecology and Behavior Studies
ResumoJournal of Animal EcologyVolume 80, Issue 4 p. 731-741 'HOW TO…' PAPERFree Access Accommodating environmental variation in population models: metaphysiological biomass loss accounting Norman Owen-Smith, Corresponding Author Norman Owen-Smith Correspondence author. E-mail: norman.owen-smith@wits.ac.zaSearch for more papers by this author Norman Owen-Smith, Corresponding Author Norman Owen-Smith Correspondence author. E-mail: norman.owen-smith@wits.ac.zaSearch for more papers by this author First published: 24 February 2011 https://doi.org/10.1111/j.1365-2656.2011.01820.xCitations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat 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, (eqn 1) (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: (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: (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 1Open in figure viewerPowerPoint 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: (eqn 5) where s = area searched per unit time, a = fraction of the resource available for consumption, h = handling time per unit of food and q = 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 (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 2Open in figure viewerPowerPoint 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: (eqn 7) where s = search rate in area covered per unit time, a = acceptance fraction of the food encountered that is consumed, c = conversion coefficient from food consumed into herbivore biomass, F = available food biomass per unit area and u = 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: 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: 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 an
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