^{1}

^{2}

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What models and statistical tools can best help us assess how ecosystems respond to the impact of multiple factors, such as disease, predation, fire, and rain?

The anthrax pathogen,

Mathematical models are used to explore questions regarding what factors tip the balance in favor of top-down or bottom-up control. The most versatile models from a trophic point of view are those that take a consumer-resource perspective, irrespective of the particular trophic level under consideration (_{2}. Further, and more specifically (as illustrated in Holdo et al.'s Figure 4B), they assess the relative importance of the human-elephant-tree cascade (influenced by poaching) compared with the rinderpest-wildebeest-fire-tree cascade that has resulted from the near eradication of rinderpest—a highly contagious measles-like virus the infects cattle and buffalo, giraffe, kudu, wildebeest, and other artiodactyls—through the vaccination of cattle in east Africa in the early 1960s.

In the simplest trophic foodweb models, each component is represented by a single variable, be it the biomass density, population density, or some other appropriate measure of abundance or amount. Consider the case where _{i}_{i}_{−1} is the amount of resources available to individuals in component _{i}_{+1} is the amount of component

the amount that each unit in component

the per unit (e.g., per capita) growth rate of the component _{i}_{i}_{−1} (see _{i}

component _{i}_{+1}_{i}_{+1}(_{i}_{i}_{+1}) (from condition 1 above the total amount transferred from _{i}_{+1}_{i}_{0}(_{1} = _{0} =

the per unit extraction rate is given by

the growth rate is the hyperbolic (i.e., metaphysiological) function

the population itself is free from predation, then the governing equation reduces to the ubiquitous logistic growth equation

Extending this case to a two-trophic case of an idealized population _{2} exploiting a second idealized population _{1} = _{0} =

This 2-D consumer-resource equation, unlike the usual Lotka-Volterra (L-V) equation, has the internal logic of applying the same growth and extraction principles at both trophic levels. In contrast, the L-V equation arises by assuming that the second trophic level either consumes the first at a rate _{2}(_{1},_{2}) ≡ _{2}(_{1}) = δ_{2}_{1} (mass action) or _{2}(_{2}) = _{2}−

Holdo et al.

The first class of processes are episodic perturbations such as outbreaks of fires and diseases that “burn” their way through systems, or environmental switches such as those driven by ENSO (El Niño-Southern Oscillation); although we need to bear in mind that what might be episodic and intense at one spatial scale appears as more regular and less intense at a larger spatial scale. Irregular episodic events (best characterized in terms of probability distributions describing the frequencies and intensities of occurrences) can determine the dominant state of ecosystems even when such events are infrequent

The second class of processes relates to species that extract resources from underlying trophic levels at relatively steady rates (

All biological systems constitute a mixture of idealized continuous processes such as expectations of time of death, and idealized discrete processes—typically diurnally or seasonally cued, such as the timing of reproduction in many temperate species. Data, however, are invariably discrete and relate either to counting events occurring within demarcated intervals of time or space, to measuring traits categorized into discrete classes, or to rounding measurements to a given number of significant digits. A statistically oriented approach is to fit the parameters of discrete time equations to sets of time-series data {_{i}_{i}_{i}_{i}_{i}_{i}_{i}^{2}) with mean and variance parameters μ_{i}_{i}^{2} that can also be estimated during the data fitting process. Adding a stochastic component allows process specification error and environmental stochasticity to be included

A common, but problematic way to generate _{i}_{i}_{i}_{i}_{−1},_{i}_{i}_{+1},ν) = log[_{i}_{i}_{−1},_{i}_{i}_{+1},ν)]. Thus, if we now take the logarithm on both sides of Equation (2), we obtain the relationship

Various forms for _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{j}_{j}_{j}_{j}

Whatever the particular functional form of the _{i}_{j}

Holdo et al.

The unusual aspect of Holdo et al.'s study

Holdo et al.

The effect of elephants is through regular browsing and coppicing of trees, fire through episodic burns linked to fuel load, wildebeest after being released from the suppressing effects of endemic rinderpest (a morbillivirus of artiodactyls), and rain through its connections to all system components. Holdo et al.

_{−1}, _{i}_{i}_{i}_{i}_{i}_{−1}, _{i}_{2}, red) at the second trophic level, except now _{3}, black) at the third trophic level, except now _{4} identically 0).

Rapid land-use change in Africa is creating ever-smaller islands of wild savanna habitat in a sea of engineered landscapes. Coupled with climate change and the threat of emerging diseases

I would like to thank Paul Cross, James Lloyd-Smith, Leo Polansky, Norman Owen-Smith, Wendy Turner, and George Wittemyer for comments that have greatly improved this manuscript.