The concept of sustainability is based on the qualitative or quantitative assessment of environmental interactions between a system and its environment at large, and implies the identification of a certain number of natural—and possibly finite—resources so that the dynamic interplay among species in a given ecological niche can be modeled to study the time evolution of the system. A first problem is thus that of “measuring” the resources, i.e., of finding a quantifier that is at the same time useful, convenient and general enough to represent material and immaterial fluxes. A second problem is posed by the empirical observation that most existing species display a multitude of “modes” of natural interrelation, like indifference, competition, cooperation, antagonism, adaptation, etc. A third problem is the “flexible attitude” exhibited by most species towards the substitutability of certain resources [1,2,3].

Since the very first attempts to construct behavioral and evolutionary models of living organisms, two substantially different approaches have been taken:

Nature’s intertwined trophic chains are far too complex a system to be amenable to a tractable comprehensive mathematical approach: for example, it is impossible to classify within general schemes voluntary or instinctive behaviors that, especially in superior species, can be treated only by recurring to species-specific, and therefore non-general, assumptions. Reasoning in an ethologic sense, though, it is legitimate to posit [7] the dependence of the dynamics of a certain species on the available resources (measured by extended exergy), formulate a proper mathematical model on its basis, and apply the model to predict the dynamics of a single species (local level) or of an entire ecological niche (global level): since a large amount of experimental data are available on the dynamics of certain populations, the results can subsequently be validated and verified. It was shown in two previous papers [4,5] that such an analysis of the sustainability concept leads to credible results. In the model adopted in those papers, “competition” was among the species and for the available resources. The underlying assumptions are quite simple:

The resulting equation for a single population (for the complete derivation, see [4,5]) couples the population size (its numerosity N) with the exergy rate Ė(t) of the global exergy resources exploited by the population in its ecological niche:

where the dot denotes the time derivative, the constant r is the limit growth rate of the population (in the case of infinitely abundant flow of resources) and μ is the intrinsic mortality rate of the species (calculated in the case in which the availability of resources drops to zero). It is useful to briefly discuss the behavior of the solutions of this equation in a very simple but realistic situation. Suppose that the incoming exergy rate is constant in time: this means the population N(t) can avail itself of the same exergy inflow at every instant of time. What one expects is then a “logistic” behavior, in the sense that, after an initial transient, the solution N(t) approaches a limit depending on the parameters of the model and on the available exergy flow. Indeed in this simple case the equation possesses an implicit solution given by:

where N₀ = N(t = t₀) and . It is clear that as t → ∞, N(t) → N^*. The behavior is indeed similar to that of a logistic-type equation under a carrying capacity constraint: here however the specific exergy consumption rate and the parameters of the model exactly define the carrying capacity. Note that in this case, under a constant influx condition (and therefore an infinite amount of cumulative exergy) at its disposal, the numerosity can sustain itself to the value N^* for an infinite time. Obviously this is no longer true when describes a finite amount of cumulative exergy (see [4,5]). Notice also that the carrying capacity N^* is proportional to Ė but also inversely proportional to c, a model constant representing the specific exergy consumption rate corresponding to“minimum survival” of the population; indeed if in a given interval of time the total energy consumption rate Ė(t) of the population is much higher than cN(t) then the population increases in that interval, while in the opposite situation the population would decrease: this means that the curve N(t) “follows” in some sense the curve Ė(t). A last remark: in the previous simple situation when Ė(t) is constant we can have two qualitative different behavior of the solution N(t) (see e.g. [4]): if at t = t₀ the initial population size is larger than N^*, then the population will monotonically (and exponentially) decrease to the value N^*, while if the population size at t = t₀ is smaller than N^*, then the population will increase to the value N^*. Some plots of the transient N(t) are provided in Figure 1: the gray curves represent the first case, the black ones the latter.

In the case of Z populations the previous equation generalizes to:

where again the r_j, j=1…Z, are the limit growth rates of each population and the μ_j are the intrinsic mortality rate of each species. Now the “source term” Ė_j(t) represents the exergy rate allocated to population j and, as such, it will depend on the strength of the mutualism and competition between populations i and j. Once the characteristics of each species have been assigned (r_j, μ_j and Ė_j(t) above), a “scenario” is completely identified by specifying the evolution of the incoming total exergy rate: for each scenario, equations (3) describe the evolution of the Z populations. For a detailed analysis on the dynamics of competing populations governed by Equation (3), we refer readers to [4]. The relative strength of the interactions between populations i and j, that we describe by means of a set of parameters characteristic of the species and of their ecological niche, determines the relative amount of the global incoming exergy allocated to each population.

In this paper we show that the system (3) is comprehensive and general enough to encompass the three following experimentally observed modes of species interaction [8,9]:

Consider two species that share a single renewable resource and who “feed” on their respective discharges (Figure 2). Notice that the relative strength of the interactions between populations i and j, that we describe by means of a set of parameters characteristic of the species and of their ecological niche, determines the relative amount of the global incoming exergy allocated to each population. Our model assumes that, at each instant of time, the rate of exergy accumulation or depletion (Ė₁ and Ė₂ here) is the force that drives the increase (or decrease) in the respective numerosities: it will be therefore regarded as “useful input”.

The exergy “balances” read:

Notice that, since exergy is not a conserved quantity, in this paper the expression “exergy balance” is adopted to denote the closure of the exergy equation obtained by introducing the exergy destruction E_δ.

We assume that the allocation of the exergy flows Ė₁ and Ė₂ increases with the numerosity of the respective population. So we set: and , γ₁ and γ₂ (both in J/individual/day) being the specific exergy consumptions of the two species (notice that β₁+β₂=1 and that as requested by the imposed boundary conditions). In most real situations, the parameters γ_i could depend also on time: in this paper, for simplicity, we posit that they are constant. Note that the parameter a_nk is the amount of the discharge of species k “recycled” by species n: if a_nk=a_kn=0, the two populations are indifferent to each other, i.e., neither one exploits the discharges of the other as its own exergy input). Since the instantaneous rate of discharge and destruction of the exergy flow can be only a fraction of the incoming one, we set and where w_i and δ_i are two dimensionless species-specific constants measuring the portion of incoming flow being discharged and destroyed by population N_i: In the model presented here, these “discharges” are NOT equivalent to the detritus (which is a pure material flow rate): they include also any form of exergy released by species j (thermal, chemical, etc.)

It is clear that this interpretation for the constants w_i and δ_i implies the constraints w_i + δ_i ≤ 1. Equations (4) and (5) then result in:

Due to the constraints w_i + δ_i ≤ 1 we see that the above expressions are indeed, at every instant, non-negative, as requested by their physical interpretation. It is convenient to set p_i = 1 − w_i − δ_i: by substituting (6) and (7) into (3), we obtain:

Where for simplicity we set and . The strong coupling between the numerosity of each species and the time evolution of ALL the participating species is apparent. Notice that in this case, Z = 2, but the formulae can be easily extended to account for whatever number Z of interacting populations, see [1]. We shall assume in the following that the exergy influx Ė_in is constant and that the model coefficients γ, w, δ, r, μ and c are known for all species considered. The type and degree of interaction between the two species is represented by the terms _k and γ_k that respectively contain the “β_j” (that express the exergy voracity of species j) and the “α_ij,” that quantify the degree of mutualism. Let us examine, on the same line as proposed in [8], the three possible general behaviors. Notice that Jørgensen [8] uses “cooperation” and “parasitism” to indicate “mutualism” and “commensalism” respectively.

The first conclusion that can be derived from the arguments developed in this paper is that the axiom that is at the foundation of our reasoning, namely that resource consumption (material and immaterial) can be completely and unambiguously quantified in terms of exergy flows, is a convenient and fruitful working hypothesis. The model of population dynamics derived from such axiom results in strongly coupled and intrinsically non-linear evolution equations. In this paper, the time evolution patterns of two species sharing an ecological niche have been studied under three different scenarios: indifference, commensalism and mutualism. Our results indicate that:

In all cases, the sustainability of the point in state-space of the surviving population(s) prescribes a certain well-identified numerosity.

In spite of the apparent success of our model, caution ought to be exercised in generalizing its results: nature is far more complex than any complicated and non-linear model makes it appear, and the results presented here are but a basis—and an incentive!—for further study. In particular, the following problems need to be carefully addressed if this type of approach is brought down from a descriptive to an applicative or predictive level: