Consistency, competitive exclusion and coexistence in complex plankton ecosystem models

We synthesise the generic properties of ecologically realistic multi-trophic level models and define criteria for ecological realism. We define three simple properties that all sensible ecosystem models should have: 1. Independence of scale: the functions that describe the change in the size of the population (x(i)) are independent of the scale at which we consider the population, that is: 1/x(i) dx(i)/dt = f(i)(x(1),x(2),...,x(n);N) for i = 1,2...,n. (1) 2. Conservation of mass: we assume that mass of a key limiting nutrient (N) is conserved, with implicit remineralisation of detritus into inorganic nutrient slaved to the ecosystem, that is: x(1) + x(2) + + x(n) + N = N-T double left right arrow (x) over dot(1) + (x) over dot(2) + + (x) over dot(n) = -(N)over dot. (2) We scale the system so that the total nutrient N-T = 1, with 0 <= x(i) <= 1 and. 0 < x(1) + x(2) + + x(n) < 1 and use equation (2) to eliminate N from equation (1). The lowest trophic level (x(1) at least) therefore grows on inorganic nutrient and must be an autotroph. 3. Resource limitation: we stipulate that every population must be explicitly limited by a finite resource (R-i), that is, the rate at which the population can grow decreases as availability of its limiting resource decreases. This may be expressed formally by the resource ray gradient condition: (r) under tilde (i).del f(i) equivalent to r(1)(i) partial derivative f(i)/partial derivative r(1)(i) + r(2)(i) partial derivative f(i)/partial derivative r(2)(i) + + r(n)(i)partial derivative f(i)/partial derivative r(n)(i) equivalent to x(1) partial derivative f(i)/partial derivative x(1) + x(2)partial derivative f(i)/partial derivative x(2)+ +(1-x(j))partial derivative fi/partial derivative(1-x(j))+ + x(n)partial derivative f(i)/partial derivative x(n)<0, (3) where <(r)under tilde>(i) = (x(1),...,1-x(j),...,x(n)) is a ray spanning the resource space for species x(i) that feeds on species x(j). To be a sensible ecology, the ray sign conditions f(i)(R-i = 1) > 0 > f(i)(R-i = 0) where R-i is the amount of resource i must also be met. This defines an "ecospace" in which all ecologically realistic dynamics are confined, and construct "resource rays" that define the resources available to each species at every point in the ecospace. Resource rays for a species are lines from a vertex of maximum resource to the opposite boundary where no resources are available. The growth functions of all biota normally decrease along their resource rays, and change sign from positive to negative. This property prescribes that each species must have a zero isosurface that divides the ecospace, and provides a simple test for ecological consistency. We use the properties of our consistent ecologies to develop heuristics that illuminate the key mechanisms that allow the coexistence of explicit competitors in these systems. Our approach unifies many theoretical and applied models in a common biogeochemical framework, providing a useful tool to generate new insights into the properties of complex ecosystems.