In theoretical ecology and nonlinear dynamics, consumer-resource models (CRMs) are a class of ecological models in which a community of consumer species compete for a common pool of resources.
Consumer-resource models have served as fundamental tools in the quantitative development of theories of niche construction, coexistence, and biological diversity.
These models can be interpreted as a quantitative description of a single trophic level.
A general consumer-resource model is described by the system of coupled ordinary differential equations,
, depending only on resource abundances, is the per-capita growth rate of species
An essential feature of CRMs is that species growth rates and populations are mediated through resources and there are no explicit species-species interactions.
Originally introduced by Robert H. MacArthur[3] and Richard Levins,[4] consumer-resource models have found success in formalizing ecological principles and modeling experiments involving microbial ecosystems.
[5][6] Niche models are a notable class of CRMs which are described by the system of coupled ordinary differential equations,[7][8] where
In this class of CRMs, consumer species' impacts on resources are not explicitly coordinated; however, there are implicit interactions.
[9][10] The MCRM is given by the following set of coupled ordinary differential equations:[11][12][8]
Given positive parameters and initial conditions, this model approaches a unique uninvadable steady state (i.e., a steady state in which the re-introduction of a species which has been driven to extinction or a resource which has been depleted leads to the re-introduced species or resource dying out again).
It is described by the following set of coupled ordinary differential equations:[11][8]
It is described by the following set of coupled ordinary differential equations:[8]
[7] The microbial consumer resource model describes a microbial ecosystem with externally supplied resources where consumption can produce metabolic byproducts, leading to potential cross-feeding.
[13][8] For the MacArthur consumer resource model (MCRM), MacArthur introduced an optimization principle to identify the uninvadable steady state of the model (i.e., the steady state so that if any species with zero population is re-introduced, it will fail to invade, meaning the ecosystem will return to said steady state).
To derive the optimization principle, one assumes resource dynamics become sufficiently fast (i.e.,
[14][15] MacArthur's Minimization Principle has been extended to the more general Minimum Environmental Perturbation Principle (MEPP) which maps certain niche CRM models to constrained optimization problems.
When this symmetry condition is satisfied, it can be shown that there exists a function
, the steady-state uninvadable resource abundances and species populations are the solution to the constrained optimization problem:
can be interpreted as a distance by defining the point in the state space of resource abundances at which it is zero,
Each of these conditions specifies a region in the space of possible steady-state resource abundances, and the realized steady-state resource abundance is restricted to the intersection of these regions.
The structure and locations of the intersections of the ZNGIs thus determine what species and feasibly coexist; the realized steady-state community is dependent on the supply of resources and can be analyzed by examining coexistence cones.
is defined to be the set of possible resource supply vectors which will lead to a community containing precisely the species
To see the cone structure, consider that in the MacArthur or Tilman models, the steady-state non-depleted resource abundances must satisfy,
As the surviving species are exactly those with positive abundances, the sum term becomes a sum only over surviving species, and the right-hand side resembles the expression for a convex cone with apex
In an ecosystem with many species and resources, the behavior of consumer-resource models can be analyzed using tools from statistical physics, particularly mean-field theory and the cavity method.
These distributions of steady-state abundances can then be determined by deriving mean-field equations for random variables representing the steady-state abundances of a randomly selected species and resource.
In the MCRM, the model parameters can be taken to be random variables with means and variances:
), the steady-state resource and species abundances are modeled as a random variable,
This mean-field framework can determine the moments and exact form of the abundance distribution, the average susceptibilities, and the fraction of species and resources that survive at a steady state.