The work builds on the signal processing approach first introduced by Roughgarden in the 1970's. I believe that Roughgarden was mostly interested in teasing out the fundamental rhythms of nature (beyond its merely seasonal ones) but of course there are many ways that spectral analysis can be combined with empirical observation to produce fruitful insights.
In this paper, the authors want to know what form of density dependence regulates coral reef fish populations. (The fact that some such mechanism must be at work is a priori deducible from the fact that populations neither vanish nor explode.) They propose three possibilities: pelagic control (larva die in the plankton), top-down control (predators eat lots of juveniles), or bottom-up control (fish compete for food.) At the bottom of all this is the idea of "recruitment", i.e. the number of youngbloods that get drawn to the reef. Apparently, recruitment levels are observable, but variable, and since fish populations don't directly mimic recruitment levels, deciphering the root regulatory mechanism is difficult.
The point of departure is a discrete model for how both fish count and fish biomass change in the time interval [t,t+1]. Considering juveniles and adults separately, the biomass equations are
$\begin{eqnarray} r_{t+1} = \lambda_{r}(R_t,t) + r_t(1-\beta P_t - \delta)\\ R_{t+1} = \delta r_t + C(R_t) + R_t(1-f-\mu_R)\end{eqnarray}$
while the equations giving population count are
$\begin{eqnarray}N_{r,t+1} = r_{t+1}/m_r\\N_{R,t+1} = \delta N_{r,t} + N_{R,t}(1-\mu_R)\end{eqnarray}$
Here is a list of what the constituent functions and parameters mean:
Functions:
$\lambda_t =$ rate at which fish enter the juvenile category
$C =$ rate of somatic growth
$P_t =$ predator mass
Parameters:
$\beta =$ rate at which juveniles are lost to predation
$\delta$ probability of a juvenile maturing into an adult in a given timestep
$f =$ fecundity rate
$\mu_R =$ adult mortality rate
Parameters that make an appearance at some point:
$\gamma =$ predator food assimilation efficiency
$\mu =$ predator mortality rate
$m_r =$ mass of juveniles (average)
$c =$ rate at which adults consume food
$\phi =$ rate of guild consumption (without competition)
$z =$ ratio of recruiting juvenile to egg mass
$\overline{s} =$ average pelagic survivorship for each gamete
$\overline{R} = $global average adult mass across patches
$\overline{P} =$ global average predator mass across patches
The last group of parameters are used to define the three functions $\lambda$, $C$, and $P_t$ for the three different regulatory regimes. For example, in the top-down case, $P_t$ is given by
$P_{t+1} \equiv P_t(1 + \gamma \beta r_t - \mu)$
while for both the pelagic and bottom-up models, $P_t$ is set equal to a constant $\overline{P}$ (average value.) For the bottom-up model, the function $C$ is a constant, while for both the top-down and pelagic models, $C = cR_t$. Finally, $\lambda$ can be a range of things, but without loss of generality it can be taken to be a constant in all three regimes.
The upshot is that these different equations induce different "transfer" function for the biomass "signals", and thus the relationship between, e.g., the variance in $\lambda$ and the variance in $R$ will be different in the three regimes. The whole point of the paper is to establish qualitative relations along these lines. A representative conclusion: as $\lambda$ increases (in mean, I suppose), the ratio of the variance in $R$ to the variance in $\lambda$ increases, decreases, or stays the same depending on whether the regulation is top-down, bottom-up, or pelagic, respectively. The grand conclusion is a table that shows the qualitative responses in all three regimes to shifting values of three particular parameters ($c$, $\overline{P}$, and $\overline{s}$) where the responses involve arcane quantities like the ratio of the variance of the biomass to the variance of the recruitment rate, etc.
Questions:
1. To what extent are the four quantities used in these qualitative "simple tests" actually measurable with field data?
2. How do they calculate these variances analytically? (There is some talk of linearization; even so.... I suspect a visit to Cox and Miller would be in order.)
3. Would there be some reason to argue that the four quantities they chose were ideal, or ridiculous, or somewhere in between?
4. Is there a continuous time analogue?
5. How much field data is out there? Any in the public domain?
