For theoretical intuition into the effects of different dietary strategies on herbivore population dynamics, we combine the analysis of two models: one extremely simple discrete-time population model that does not explicitly consider vegetation pools but assumes seasonal variation in forage quality and another that adapts the well-studied Lotka-Volterra consumer-resource model with one herbivore and two logistically growing resource pools, corresponding either to grass and trees or two different grass pools between which herbivores migrate. We are primarily interested in how herbivore population sizes change with respect to the degree of seasonal diet switching by herbivores, which depend on including limitations on plant productivity (capturing the benefit of switching diet, as either forage quality or total availability is depleted seasonally) and herbivore feeding behavior (as intake and digestive efficiency, including possible costs of the switching strategy itself). Despite its simplicity and a long history of attention, the Lotka-Volterratype system (described in further detail below) resists full formal analysis, and we have therefore presented predictions via computation (see Fig. 1 and figs. S2 to S5); it is for this reason that we have included the even simpler model for more complete analytical intuition.
For empirical evaluation, herbivore census data were extracted from the database previously published by Hempson et al. (14). Data were included for protected areas in Eastern and Southern Africa with an area > 500 km2, rainfall between 400 and 1000 mm year1, and good conservation status at the time of the census (see Fig. 2 and fig. S6) (33). These criteria aim to identify environmentally comparable regions with intact wildlife populations and to minimize the intensive management and edge effects in small reserves. Data for migratory populations were more sparse, because these have been heavily depleted through hunting and fragmentation (24). We identified six migratory populations of four species, including (i) wildebeest (Connochaetes taurinus) and zebra (E. quagga) in the Serengeti, (ii) white-eared kob (Kobus kob) and tiang (i.e., topi or tsessebe, Damaliscus lunatus) in the Boma-Jonglei, (iii) wildebeest in Tarangire-Manyara, and (iv) wildebeest in Liuwa.
The percentage C4 grass component of African herbivore diets was estimated by synthesizing data from published sources (8, 9, 11, 34, 35) and averaging across regions and studies for each species (see table S1). This percentage was reflected about the 50% diet composition axis to estimate the degree of dietary mixing via the equation: 50 |50 percent C4 grass|. This has a maximum value of 50% when the diet is half C4 grass and a minimum value of 0% if C4 grass is all or none of the diet.
The main objective of our analyses was to determine whether the level of grass-browse mixing in a herbivores diet has an influence on its abundance. Herbivore abundance was estimated as individual density (the number of individuals km2) or as metabolic biomass density (body mass0.75 individual density km2). These data were then log transformed before fitting linear mixed-effects models with scaled % dietary mixing, species body mass, and their interaction as fixed effects in the full model. Species identity and protected area identity were included as random effects to account for processes including climatic and edaphic limitations on productivity and top-down carnivore effects on herbivore populations. Models were fitted in R (version 3.3.3) using the lme4 package.
First, we consider a population N that experiences alternating wet and dry seasons, with population growth from one wet season to the next described by the equationNw,t+1=wmd12mNw,twhere w corresponds to the wet-season growth rate of the population, d to its dry-season growth rate, m to the length of the wet season in months (this choice of unit of length is arbitrary and not important for model dynamics), and t to the time elapsed in years. If we nondimensionalize by N0 without loss of generality, thenNw(t)=[wmd12m]t(1)
These growth rates can be adapted to describe the case of grazers, browsers, and mixed feeders. Grazers grow at rates wG and dG in the wet and dry season, respectively, and browsers at rates wB and dB.
We assume that mixed feeders graze in the wet season and browse in the dry season, consistent with empirical observations (9). Hence, we also make the crucial assumption that, in the wet season, the potential growth rate of a herbivore is higher on grass, whereas in the dry season, the potential growth rate of a herbivore is higher on browse (i.e., that wG > wB and dB > dG). We also assume that the intrinsic quality of grazing or browsing forage probably does not differ by herbivore type, but that there are potential costs to generalism that might contribute to the relative success of the strategy (36); therefore, we assume that mixed feeders suffer some inefficiency in how they grow on both grass and browse, scaling their wet and dry season growth rates as cGwG and cBdB, with cG and cB < 1. Note that higher c denotes higher mixed-feeder efficiency.
A mixed-feeder population M therefore achieves higher population numbers than a grazer G and a browser population B, respectively, when1[dGdB]12mandcGmcB12m>[wBwG]m(2)
See fig. S1 for a graphical illustration of these conditions. Ecologically, they suggest the intuitive result that mixed-feeder abundances will exceed grazer abundances when dry season browse is sufficiently better than dry season graze to compensate the costs of mixed feeding. By the same token, mixed-feeder abundances will exceed browser abundances when wet-season graze is enough better than wet-season browse to compensate the costs of mixed feeding. In the case where there are no costs to a mixed-feeding strategy, these conditions reduce to our assumption that grass is better forage in the wet season and trees in the dry; but in the case where mixed feeding carries a cost, the success of the strategy is determined by how seasonal trees and grasses are relative to each other. Thus, the relative responses of trees versus grasses to seasonality are fundamental to determining the benefits of mixed feeding. In extreme cases, this is obvious: Diet switching is obviously disadvantageous when inefficiencies are overwhelming or when grass survives but there is nothing to browse in the dry season, as in heavily deciduous systems in the tropics.
This model can also be used to analogize the dynamic of a migratory grazer, with similar results. In that case, cGwG and cBdB correspond to the cost-adjusted growth rate of the migratory grazer on the wet seasonpreferred grass pool and the dry-season forage reservoir, respectively. Thus, a migratory grazer population grows to a larger size than its nonmigratory equivalent when the benefits of switching to the dry-season reservoir outweigh the costs of doing so. In this analogy, the costs of migrating may be energetic, rather than anatomical (as above), since migratory populations often have nonmigratory conspecifics. However, the analogy is limited by the fact that often, the benefit of migrating is that there is more (not better) food at the destination, and so the discrete-time model presented here is a poor analogy; see the coupled consumer-resource model below for a different perspective on this issue.
Note that the results presented in fig. S1 are qualitatively similar when we consider a discrete-time logistic model, especially when growth rates are slow relative to carrying capacity and identical when carrying capacity is taken to be the same across herbivore types (analysis not shown).
We have used a variant of the well-studied Lotka-Volterra consumer-resource model with one herbivore and two resource pools, corresponding either to grass and trees or two different grass pools between which herbivores migrate. For this detailed model description, we describe the two resource pools as grass and tree foliar biomass; all that is required to turn this into a simple model for migration, however, would be to change the names of the resources to, e.g., two different grass pools between which herbivores migrate. Here, grass and tree foliar biomasses (the resources, G and T) accumulate logistically with some growth rate (~carbon assimilation, AG and AT) and carrying capacity (KG and KT). Herbivores eat grass for a fixed fraction of time G and eat trees the rest of the time (T = 1 G), in proportion to their availability at a rate that depends on bite size (i.e., handling efficiency, G and T). Note that for the purposes of analysis, and always occur together and could be considered as one parameter; however, we maintain the distinction between the two to preserve their biological meaning. Foliage is converted to herbivore biomass depending on how nutritious food is and how efficient digestion is (combined into one term, G and T). This yields the following system of equationsdGdT=AGG(1GKG)GGGHdTdt=ATT(1TKT)TTGHdHdt=[GGGG+TTTT]HH(3)where is the mortality rate of the herbivore. For specialist herbivores (with either T or G = 1) in a nonseasonal environment, the equilibria of this system and their stability are well known. Those familiar with this model can skip two paragraphs to .
As a review, taking the example of a specialist grazer, T approaches its carrying capacity KT and does not interact with grass or the herbivore population. We are left with a two-dimensional system with zero isoclines from Eq. 3 atH=AG(1GKG)GGandG=GGG(4)respectively, and equilibria occur where these zero isoclines intersect (as illustrated, e.g., in fig. S2A). Stability is given by the Jacobian evaluated at equilibriumJ=[AGGKGGGGGGGH0](5)for which the trace is always negative and the determinant is always positive, such that, according to Routh-Hurwitzs stability criteria, any equilibrium that exists is also stable for all biologically realistic (i.e., positive) regions of parameter and state space (see also fig. S2, A and B). The example of a specialist grazer is directly analogous to a specialist browser as well (see fig. S2, I and J).
Analysis is slightly more complicated for a mixed feeder (with either T or G = 1) in a nonseasonal environment because the system is three dimensional (see figs. S3 and S4 for examples of trajectories in three-dimensional space) but nonetheless straightforward. In this case, equilibria are well defined by Eq. 3, and again, their stability is this time given by the (now) three-dimensional JacobianJ=[AGGKG0GGG0ATTKTTTTGGGHTTTH0](6)
In this case, Routh-Hurwitzs criteria for stability require that the trace be negative, the determinant also negative, and the determinant greater than the product of the trace and the sum of the determinants of the dominant subminors; here again, it is straightforward to show that any equilibrium that exists is also stable for all biologically realistic (i.e., positive) regions of parameter and state space.
The next key component of the model is seasonal variation: We assume that seasons alternate predictably, with effects on plant productivity (via A) and, depending on herbivory type, on herbivore diet. We assume that grazers graze and browsers browse all year. However, mixed feeders change their diets seasonally (9), switching from wet-season grazing to dry-season browsing when grass resources are exhausted and/or decrease in quality. By analogy, a migratory grazer might change resource pools seasonally from a preferred resource to a forage reservoir in the dry season. This seasonal change in productivity complicates analysis, even when the herbivore is a specialist grazer or browser (see fig. S2, C, G, and K). Although we can be sure that plant and herbivore population trajectories are always moving toward the seasonal stable equilibria described above, there is no guarantee that the system reaches equilibrium within a season (and, in fact, given that ungulates usually live multiple if not many years, reaching equilibrium within a season seems unlikely). Instead, we see the emergence of cycles in plant and herbivore abundance in response to alternating seasons. These seem, for broad ranges of parameter space, to tend toward stable cycles, as the system moves along deterministic trajectories toward (but not always reaching) seasonal equilibria (see figs. S2 to S4).
In the trivial case where mixed feeders perform better in both wet and dry seasons than pure grazers or browsers, analysis would be simpler: Mixed feeders would achieve higher population sizes overall (37). However, we must make assumptions to mirror a reality that directly violates this most trivial case, and mixed feeders may not always perform better overall than grazers or browsers. Although more extensive work has been done on similarly structured aquatic systems that reach equilibrium within a season (38), currently available analytical tools cannot go much further than this. We proceed for further intuition via computation methods below.
In reality, mixed feeders do better than grazers only in the dry season (when grass has run out) and better than browsers only in the wet season (when grass is more abundant and/or easier to eat than browse). The best-case scenario in this is that mixed feeders do exactly as well as grazers when grazing and exactly as well as browsers when browsing. However, mixed-feeder disadvantages may be more severe if mixed feeders, as generalists, are less efficient grazers than grazing specialists and less efficient browsers than browsing specialists. There are two possible ways to include the costs of mixed feeding (and analogous costs of migrating spatially). Mixed feeders may digest foliage less efficiently, payable as a fractional decrease in digestive efficiency (where efficiency = 1 corresponds to no cost; applied multiplicatively to the first two terms of Eq. 3). Alternatively, mixed feeders may have less efficient mouth shapes for grazing and browsing, resulting in a decrease in intake efficiency (applied multiplicatively to the last terms of the first two of Eq. 3 and the first two terms of the last of Eq. 3).
Via computation across a broad range of parameter space, we find that when the costs of mixed feeding are high, mixed feeders do not achieve higher abundances than grazers and browsers (see Fig. 1, A and D, and fig. S5, A and D). However, mixed-feeder advantages are relatively robust to mild decreases in feeding efficiency due to mixed feeding; in fact, both intake and digestive efficiency costs are widely debated in the literature, and recent syntheses suggest that mixed feeders have only slightly lower feeding or digestive efficiencies than grazers or browsers (36). Thus, we should expect mixed feeders to have increasing abundances with increases in the degree of mixed feeding, for realistic efficiency estimates (see Fig. 1, C and F).
To generate computation results, we have used Runge-Kutta fourth-order integration in the package deSolve in R, version 3.2.2. For all results shown herein, wet and dry seasons each last one-half a time step (with one unit of time assumed to be a year), and the transition between the two is abrupt (instead of, e.g., sinusoidal, which would capture a more gradual transition between wet and dry seasons). In the main text (see Fig. 1), we present results assuming AG,wet = 10, AT,wet = 5, AG,dry = AT,dry = 0, KG = KT = 1000, G = 0.02, T = 0.08, G = 0.08, T = 0.05, = 0.8, c = 0.95, and c = 0.8, except where parameters are varied for the parameter sweep, incorporating the assumptions that tree foliage is more nutritious than grass but that taking large bites of grass is easier than selective browsing of trees (see also fig. S3 for trajectories for a subset of those simulations). However, for generality, we also provide another simulation set that makes neutral assumptions about the quality and handling times of grass and trees (AG,wet = AT,wet = 10, KG = KT = 1000, G = T = 0.05, G = T = 0.05, = 0.8, c = 0.95, and c = 0.8; see figs. S4 and S5). Across all simulations, grazers graze and browsers browse all year; we additionally assume that mixed feeders exclusively graze in the wet season but that they switch to browsing in the dry (with G,wet = T,dry = 1, such that diet mixing = 50%). Where we vary the degree of diet mixing for parameter sweeps (in Fig. 1, B, C, E, and F, and fig. S5, B, C, E, and F), we achieve this by varying T,dry.
Note that, here, we consider only the population dynamics of a single herbivore at a time, ignoring the dynamics of the diverse food web, which have been considered in some depth elsewhere (18, 39). However, results using metabarcoding approaches suggest that diverse herbivores in savannas compete minimally, so this simplification may in a narrow sense be realistic; how this niche differentiation arises in a competitive, evolutionary context, especially in view of the advantages of generalism, may be of theoretical interest. Also note that our models consider the dynamics of grass and tree accumulation separately; although these may interact (40), tree-grass coexistence is not the subject of this work, and so we approximate equilibrium competition via limitations on the respective carrying capacities of tree and grass foliar biomass. Elaborations on these themes may be of future interest.
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Seasonal dietary changes increase the abundances of savanna herbivore species - Science Advances
