Unique Feature For Ecosystem Engineering

In this section I propose a specific ecological situation that might well be reserved for the term ecosystem engineering. There is a feedback aspect of the species-environment interaction, as expressed by Solomon (1949), that hasn't garnered much apparent attention by the main proponents of ecosystem engineering, and is completely distinct from the keystone species concept. However, Gurney and Lawton (1996), Wright et al. (2004), and Cuddington and Hastings (2004) address this feedback feature quite well, and it is this feature, or some general representation of it, that I propose be reserved for ecosystem engineering. The feedback idea has also been addressed quite well by Cuddington et al. (unpublished), and my aim here is to present an abbreviated and specific pedagogical version of that latter model.

The caricature of the model for ecosystem engineering sits somewhere between the two concepts pictured in Figure 3.1A and 3.1B, combining species interactions and the environment and feedback between the two, and resulting in the interaction picture of Figure 3.2. In particular, the species "produces" the environmental variable at a per capita rate g > 0, and the environment, in turn, affects either or both of the birth and death rates of the species. The representation used here combines

V / \ / \

\

Births P ^

Population

Deaths\ ^

FIGURE 3.2 The presence of an environment affected by a species, which has demographic rates that are themselves affected by the environment, sets up a situation that, with a positive-feedback loop, can lead to extreme environmental modification and runaway species growth.

FIGURE 3.2 The presence of an environment affected by a species, which has demographic rates that are themselves affected by the environment, sets up a situation that, with a positive-feedback loop, can lead to extreme environmental modification and runaway species growth.

density-independent birth and death factors into a single term, and, likewise, the density-dependent factors; hence one set of interaction parameters, KE and aE, affect both birth and death rates. A mathematical representation takes the logistic growth equation and adds the effects of the coupled environment, giving

In the preceding equations I have taken the liberty of simplifying the species' interactions by making the density-dependent and density-independent coefficients have a linearized environmental dependence, similar to the linearization with density of the logistic growth model and to which umbrage may be taken. I have also assumed that the environment decays to a set point of E=E0 > 0 in the absence of the engineering species, represented by n = 0. The environmental trait, E, is produced proportional to the species density, but production by n is independent of the level of E. Note that g = 0 recovers Equation 1 describing Figure 3.1A, and KE = aE = 0 recovers Equation 2a describing Figure 3.1B. For the sake of reducing the number of parameters, in the remainder of the analysis I will assume that the environmental set point is E0 = 0 in the absence of species n.

Once again, my goal is conceptual clarity, and I follow the approach of Cuddington et al. (unpublished), but for only a single exemplary case with an evolutionary extension. In particular, I assume that the species has positive growth when rare in the absence of environmental feedback, or K0 > 0.

Suppose now, for ease of analysis, that the environmental variable is one that adjusts rapidly compared with the population dynamics of the purported engineering species, such that we can always consider the environment to be in equilibrium with the species n's population density. In other words, whenever n changes a little or a lot, the environment responds immediately, taking on the value that satisfies dE/dt = 0. This is called a "quasiequilibrium" assumption. In the case described by Equation 7, the environment takes on the quasiequilibrium value, E* = gn/p. This value can then be placed into the equation for the population dynamics, under the assumption of instantaneous feedback,

The per capita growth rate as a function of density n is depicted in Figure 3.3 for two useful limiting situations. When the environment has no effect on density-dependent regulation, or aE = o, the isolated effects of the environment on density-independent growth, shown in Figure 3.3A, has three levels of effect. When KE < o, the equilibrium population density is suppressed, and when KE > o it is enhanced, but when KE > pa^lj, the engineering effects of density-independent growth overwhelm the unen-gineered density-dependent regulation, resulting in runaway environmental modification and positive per capita growth for all population densities.

The second limiting case examines effects of the environment on density-dependent regulation given fixed KE < paolg, shown in Figure 3.3B. The result is uneventful when aE > o, and the environment increases density-dependent regulation, depressing the equilibrium population density of species n. However, an interesting scenario occurs for the case aE < o, when interactions with the environment reduce the density-dependent regulation in the system. This case results in two changes. First, the stable equilibrium population density associated with the aE = o situation is enhanced. Second, a new upper equilibrium is introduced, but this equilibrium is unstable. If the population density were, somehow, to find itself above this upper equilibrium, runaway environmental modification and positive population growth would take place.

Thus, the scenario outlined by Solomon (1949) leads to some interesting ecological situations, particularly what he would call "favorable environmental modification." It is only an aside that, clearly, no ecologist, not even a theoretical one, argues for the realism of runaway growth. All that runaway growth means in a model is that, given the ingredients put into it, population regulation has been lost. Fortunately, there are other ecological ingredients that can regulate populations, but such regulating

FIGURE 3.3 Net per capita growth rates under different scenarios for environment-dependent contributions to density-dependent (aE) and density-independent (KE) rates. It is assumed that g > 0 and, thus, the environment takes on positive values. (A) If the environment only affects density-independent rates, mutants with larger values of KE always have positive per capita growth rates when the wild-type, with lower values, are at equilibrium. If KE exceeds pa0/g, a runaway growth situation occurs. (B) Fixing KE for clarity, it is seen that mutants with smaller values of aE have positive per capita growth rates near the wild-type equilibrium. Interestingly, an unstable equilibrium occurs for higher density when aE < 0, meaning that a runaway growth situation can occur if the density ever exceeds this equilibrium.

FIGURE 3.3 Net per capita growth rates under different scenarios for environment-dependent contributions to density-dependent (aE) and density-independent (KE) rates. It is assumed that g > 0 and, thus, the environment takes on positive values. (A) If the environment only affects density-independent rates, mutants with larger values of KE always have positive per capita growth rates when the wild-type, with lower values, are at equilibrium. If KE exceeds pa0/g, a runaway growth situation occurs. (B) Fixing KE for clarity, it is seen that mutants with smaller values of aE have positive per capita growth rates near the wild-type equilibrium. Interestingly, an unstable equilibrium occurs for higher density when aE < 0, meaning that a runaway growth situation can occur if the density ever exceeds this equilibrium.

ingredients are irrelevant to the question at hand. However, the conditions for runaway growth indeed indicate an interesting outcome worthy of understanding.

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