## D x xTdxdt dxdtTx xTJ JTx

where J is the Jacobian matrix. It is obvious that the dynamics of the perturbation are governed by the term (J + JT)/2, which is another matrix called the Hermitian part of J, or H(J). The reactivity is simply the dominant eigenvalue of H(J). Assuming that the eigenvalues of J have negative real parts, if the dominant eigenvalue of H( J) is positive, then a perturbation will initially grow before converging asymptotically back to equilibrium, but if the dominant eigenvalue of H( J) is negative, then the perturbation will converge monotonically back to equilibrium.

If we return to the Jacobian described by Eq. (12.5), we can easily calculate H( J), for which the dominant eigenvalue is:

. l2kd - Iul + ^]ll2k2d2 - 4Iulkd + 2I2u2 + 3l3kd + 2l4 + 2l3 - 3Iul2 + l2kd

1 2l

Litter return and coexistence of two species

When the numerator is positive, then the system is reactive and perturbations are initially amplified by the flux and circulation of nutrients. Now, this is a rather formidable looking beast, but we can get lucky and it turns out that by setting the numerator equal to 0 and solving for one parameter in terms of the others, many of the terms cancel out. This is easily done using software such as Matlab or Mathematica. Let's find a critical value for the input, I, in terms of all other parameters that separates a stable but reactive system from one which is stable but nonreactive. Perturbations to this system will be initially amplified when:

A large input of nutrients will therefore cause a perturbation to grow before it decays, and the larger the input the faster the growth and the greater the magnitude of growth. Thus, fertilizer input to a lake from runoff could result in very large departures from equilibrium even if the system remains stable and eventually converges asymptotically to equilibrium. Similarly, we can also solve for a critical decay rate that also determines when perturbations will be amplified:

A large value for k, or a fast decay rate of dead biomass, will also increase the chance that a perturbation will initially grow. Similar calculations also show that a high uptake rate also has the same effect.

A high input rate, a high uptake rate, or a fast decay rate each increase the amount of a nutrient cycling through the system. In addition, a high input or uptake rate or a fast decay rate increase productivity because the flux of nutrients into the resource pool (and therefore available for uptake into biomass) is thereby magnified. Therefore, perturbations to productive systems are more likely to be initially amplified compared with perturbations to unproductive systems, even though both systems may be asymptotically stable to perturbations in the long run.

Systems in which a perturbation is initially amplified but then return to equilibrium are said to be resilient - as we have seen, systems with rapid uptake and decay rates are prone to be resilient. Systems in which the initial perturbation is damped monotonically are said to be resistant.

In the previous chapter, we saw that two species could not stably coexist if they each have Lotka-Volterra uptake forms. But in that model, the nutrient content of a species was exported from the system upon death. What if we allow the nutrient contents of both species to be recycled into R after they die (Fig. 12-6)? This gives the other species another chance at getting the nutrient. Will litter return allow stable coexistence with Lotka-Volterra uptake forms?

To examine this, we introduce equations for two species, each with their own characteristic uptake, death, and litter decay rates:

Fig. 12-6 A two-species nutrient cycling model with litter return modeled by Eq. 12.8.

dP1 ~dt dP2

The equilibria are:

As with the Tilman model discussed in the previous chapter, we still don't have a fixed-point coexistence equilibrium. This is rather surprising, since recirculation of the nutrient should allow both species to "share" the nutrient.

Moreover, the decay rate puts strict limits on the stability of either monoculture. To see this, let's look at the eigenvalues of the Jacobian evaluated at the P1 monoculture (Eq. 12.9b). Two of the eigenvalues are:

Note that when the numerator of X2 is negative, P* is automatically positive and stable. We can solve X2 for k1 as before to obtain an analogous k1 critical that separates the P1 monoculture equilibrium (Eq. 12.9b) from the empty resource equilibrium (Eq. 12.9a) by a transcritical bifurcation. The third eigenvalue is

The sign of this eigenvalue separates the P1 monoculture (Eq. 12.9b) from the P2 monoculture (Eq. 12.9c). The sign of A3 depends on the numerator. Let's solve for a critical value of k1 that separates negative values of A3 (and hence stability for the P1 monoculture) from positive values by setting A3 = 0 and solving for k1, yielding:

For stability of the P1 monoculture, k1 must be less than both critical values. Species 1 can achieve this in several ways: (i) by keeping k1, its litter decay rate, as small as possible; (ii) by keeping d1, the proportion of species 1 organic matter in litter, as small as possible; and/or (iii) by keeping the difference between its uptake rate, u1, and that of species 2 as large as possible. All of these conditions result in nutrients being sequestered in live or dead biomass of species 1 and separate a stable P1 monoculture (Eq. 12.9b) from an unstable P2 monoculture (Eq. 12.9c). But by symmetry, we can make the same arguments for the stability of the P2 monoculture. Therefore, the two monoculture solutions are also separated by a transcritical bifurcation which depends on critical combinations of the two species' uptake rates, proportion in dead organic matter, and decay rates.

Therefore, to maintain a stable monoculture, there are critical values not only for uptake and death rates of a species but also for its decay rate. The decay rate of a species' litter is as important a life history trait as is its uptake rates or its death rates. As before with a single species model, the dead material (litter) is still a part of the population - it is merely detached from the live part of the population and subject to different rules. Only now with a two-species model, the decay rate of a species' litter also determines the stability of the species' monoculture against invasion by a competitor. A species can be a better competitor not only by having greater nutrient uptake rates, which is the usual way we think of competition, but also by having slow decay rates for its dead biomass.

You can think of the three equilibria as existing in a state space of axes R*, P*, and P*. These three equilibria are analogous to three ecosystems on a natural landscape: (i) an ecosystem consisting only of R with an input and an output, or a "lake" empty of organisms; (ii) an ecosystem which is a monoculture of P: and R; (iii) an ecosystem of a monoculture of P2 and R. The transcritical bifurcations are functions separating the regions of the state space occupied by the three equilibrium solutions, or ecosystems, at stability. These transcritical bifurcations depend in turn on parameter values. If these parameter values change as one moves across a landscape (because of climatic or other gradients that affect them, for example), then the bifurcations represent real boundaries between different ecosystems.

### Dividing the resource pool and species coexistence

The fact that a model of explicit resource competition which also includes litter return does not allow for stable coexistence of two or more species is surprising for two reasons. First, allowing the resource to recycle through the available pool should allow

Fig. 12-7 A nutrient cycling model with S species, litter return, and partial segregation of the resource pool, redrawn from Loreau (1998).

each species to have a "second chance" at obtaining some of it and thereby avoid extinction from a superior competitor. Second, uptake of nutrients by primary producers and their return through litter and its decay accounts for greater than 90% of the flux of nutrients through most ecosystems, the amount flowing to higher trophic levels being no more than, and usually much less than, 10% of nutrients contained in plant biomass. Is the coexistence of species independent of the mass balance of nutrients cycling through an ecosystem? Clearly this is not the case, as numerous field studies have shown. One way to have stability is, of course, to have the uptake curves be Michaelis-Menten functions which cross, as we saw in the previous chapter. That model, the Armstrong and McGehee model, was able to produce coexistence with limit cycles but without litter return. Is the enormous flux of nutrients in litter and through soil and sediments not as important as the balancing flux in uptake? This asymmetry in the importance of uptake vs. the balance required in litter return and decay seems implausible. Alternatively, perhaps the underlying assumption of a homogeneously mixed resource pool needs to be relaxed to depict the proper mass balance of resource fluxes.

Loreau (1998) presents a very elegant solution to this problem in a model which is an elaboration of an earlier model by Huston and DeAngelis (1994). Loreau separates the nutrient resource into a regional, well-mixed pool (Rr) interacting with small local pools (Rt), one for each of S species, through diffusion from Rr to Ri at a diffusion rate 8 (Fig. 12-7). The transfer of nutrients from live plant species (Pj) to their litter/detritus (Li) and then through decay back to the local and regional resource pools (at rates kj and rj, respectively) are also made explicit. Each species i of S species acquires its nutrients from the local nutrient resource, analogous to local rooting zones, through Lotka-Volterra uptake. Loreau also separates the local resource pools into two components: the concentration of the resource per unit volume and the volume of space (o) containing each local resource pool, analogous to a local rooting zone. Nutrients are exported from the regional pool and the detritus at proportional rate l as with the previous models. The mass balance of nutrient fluxes is described by the following set of coupled differential equations:

dRR ~dt dR

dL dt

Segregating uptake of nutrient inputs from uptake of the inorganic resource pool

The equilibrium solution for the regional resource pool is I + SjuSc,r*

where c, and R*are the average rooting volumes and equilibrium local resource pools across all of i species, respectively and u = 1/(B, + ยก). An interesting consequence of this equilibrium is that stable coexistence of each species with all others requires

The first two terms represent input to the ecosystem (I) plus nutrient input to the regional pool from all local resource pools, which is proportional to SS(R* - R*). This inequality can be restated as:

or that R* < inputs to R*, which is analogous to what we have seen in the previous models. Loreau (1998) further shows that this inequality determines the maximum stable species richness imposed by the mass balance of nutrient flux through the system and the effect of each species on it through attributes which determine per capita flux rates. This is a very remarkable contribution to the current debate on biodiversity and ecosystem properties because it establishes rigorous and testable criteria by which we can determine how biodiversity, species life history traits, and nutrient input-output budgets determine the rate of nutrient cycling and by implication productivity of an ecosystem, a contribution which perhaps has not had the attention it deserves. Species richness (S) and the total nutrient cycling rate

X u,R,P, or X k,L, can both be predicted a priori from the constraints of mass balance and species life history traits.

Pastor et al. (2002) developed and analyzed a model of species coexistence and nutrient flux in peatlands in which mosses and vascular plants draw on different pathways and pools of nutrient fluxes (Fig. 12-8). Ombrotrophic peatlands, otherwise known as bogs, are portions of peatlands in which the accumulation of several meters of peat have raised the rooting zone above the regional water table (Wright et al. 1992). The only inputs of nutrients to bogs are in precipitation, which is effectively

Fig. 12-8 Nutrient cycling in a peatland model, after Pastor et al (2002), for Eq. 12.12.

absorbed by mosses (PM, Urban et al. 1988) and eventually transferred to peat through moss litter. In contrast to mosses, the growth of the vascular plant nutrient pool is determined by the vascular plant nutrient capital itself (PV) with an intrinsic uptake rate u (uptake per unit R and per unit PV) and by R. Live moss and vascular plant nutrient capitals are transferred to litter (LM and LV) at specific rates d1 and d2, respectively. Litter decays at rates intrinsic to each species (k3 and k4 for moss and vascular plant litters, respectively) and eventually becomes peat (LP). The peat in turn decays and releases nutrients into the inorganic nutrient pool at rate k5. Nutrients leave the system from the inorganic nutrient pool at a proportional rate l.

Because mosses and vascular plants do not draw nutrients from a common pool, they do not directly compete for nutrients (although mosses delay the availability of incoming nutrients to vascular plants). However, mosses and vascular plants reduce the growth of each other's nutrient capital through competition coefficients (aVM = effect of mosses on vascular plants and aMV = effect of vascular plants on mosses) representing competition for other resources, most likely competition for light through the shading of mosses by vascular plants or the engulfing of meristemic tissues of vascular plants by vertical growth of the moss carpet (Crum 1992).

The differential equations are:

 dPM dt = 1 d1PM aPM Pv PMPV dPV dt = uPVR - d2Pv - aPvPMPMPV dLM dt = dP - k L dLV dt = d2 Pv - k2Lv k1LM h k2Lv - k3LP dR dt = k3LP - uPVR - There are two equilibria, a moss monoculture: P*m = 1, Lm = 1, P* = L* = 0, L* = 1, R* = 1 (12.13a)