Intheoretical ecology andnonlinear dynamics,consumer-resource models (CRMs) are a class ofecological models in which acommunity of consumer species compete for a common pool of resources. Instead of species interacting directly, all species-species interactions are mediated through resource dynamics. Consumer-resource models have served as fundamental tools in the quantitative development of theories ofniche construction,coexistence, andbiological diversity. These models can be interpreted as a quantitative description of a singletrophic level.^[1][2]

A general consumer-resource model consists ofM resources whose abundances are${\displaystyle R_1,\dots ,R_M}$ andS consumer species whose populations are${\displaystyle N_1,\dots ,N_S}$. A general consumer-resource model is described by the system of coupled ordinary differential equations,where${\displaystyle g_i}$, depending only on resource abundances, is the per-capita growth rate of species${\displaystyle i}$, and${\displaystyle f_{\alpha }}$ is the growth rate of resource${\displaystyle \alpha }$. An essential feature of CRMs is that species growth rates and populations are mediated through resources and there are no explicit species-species interactions. Through resource interactions, there areemergent inter-species interactions.

Originally introduced byRobert H. MacArthur^[3] andRichard 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${\displaystyle \mathbf{R}\equiv (R_1,\dots ,R_M)}$ is a vector abbreviation for resource abundances,${\displaystyle g_i}$ is the per-capita growth rate of species${\displaystyle i}$,${\displaystyle h_{\alpha }}$ is the growth rate of species${\displaystyle \alpha }$ in the absence of consumption, and${\displaystyle -q_{i\alpha }}$ is the rate per unit species population that species${\displaystyle i}$ depletes the abundance of resource${\displaystyle \alpha }$ through consumption. In this class of CRMs, consumer species' impacts on resources are not explicitly coordinated; however, there are implicit interactions.

The MacArthur consumer-resource model (MCRM), named afterRobert H. MacArthur, is a foundational CRM for the development of niche and coexistence theories.^[9][10] The MCRM is given by the following set of coupled ordinary differential equations:^[11][12][8]where${\displaystyle c_{i\alpha }}$ is the relative preference of species${\displaystyle i}$ for resource${\displaystyle \alpha }$ and also the relative amount by which resource${\displaystyle \alpha }$ is depleted by the consumption of consumer species${\displaystyle i}$;${\displaystyle K_{\alpha }}$ is the steady-state carrying capacity of resource${\displaystyle \alpha }$ in absence of consumption (i.e., when${\displaystyle c_{i\alpha }}$ is zero);${\displaystyle {\tau }_i}$ and${\displaystyle r_{\alpha }^{-1}}$ are time-scales for species and resource dynamics, respectively;${\displaystyle w_{\alpha }}$ is the quality of resource${\displaystyle \alpha }$; and${\displaystyle m_i}$ is the natural mortality rate of species${\displaystyle i}$. This model is said to have self-replenishing resource dynamics because when${\displaystyle c_{i\alpha }=0}$, each resource exhibits independentlogistic growth. Given positive parameters and initial conditions, this model approaches a uniqueuninvadablesteady 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).^[7] Steady states of the MCRM satisfy thecompetitive exclusion principle: the number of coexisting species is less than or equal to the number of non-depleted resources. In other words, the number of simultaneously occupiableecological niches is equal to the number of non-depleted resources.

The externally supplied resource model is similar to the MCRM except the resources are provided at a constant rate from an external source instead of being self-replenished. This model is also sometimes called the linear resource dynamics model. It is described by the following set of coupled ordinary differential equations:^[11][8]where all the parameters shared with the MCRM are the same, and${\displaystyle {\kappa }_{\alpha }}$ is the rate at which resource${\displaystyle \alpha }$ is supplied to the ecosystem. In the eCRM, in the absence of consumption,${\displaystyle R_{\alpha }}$ decays to${\displaystyle {\kappa }_{\alpha }}$ exponentially with timescale${\displaystyle r_{\alpha }^{-1}}$. This model is also known as a chemostat model.

The Tilman consumer-resource model (TCRM), named afterG. David Tilman, is similar to the externally supplied resources model except the rate at which a species depletes a resource is no longer proportional to the present abundance of the resource. The TCRM is the foundational model for Tilman'sR* rule. It is described by the following set of coupled ordinary differential equations:^[8]where all parameters are shared with the MCRM. In the TCRM, resource abundances can become nonphysically negative.^[7]

The microbial consumer resource model describes a microbial ecosystem with externally supplied resources where consumption can produce metabolic byproducts, leading to potentialcross-feeding. It is described by the following set of coupled ODEs:where all parameters shared with the MCRM have similar interpretations;${\displaystyle D_{\alpha \beta }}$ is the fraction of the byproducts due to consumption of resource${\displaystyle \beta }$ which are converted to resource${\displaystyle \alpha }$ and${\displaystyle l_{\alpha }}$ is the "leakage fraction" of resource${\displaystyle \alpha }$ governing how much of the resource is released into the environment as metabolic byproducts.^[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.,${\displaystyle r_{\alpha }\gg 1}$) that they become entrained to species dynamics and are constantly at steady state (i.e.,${\displaystyle \mathrm{d}{R}_{\alpha }/\mathrm{d}t=0}$) so that${\displaystyle R_{\alpha }}$ is expressed as a function of${\displaystyle N_i}$. With this assumption, one can express species dynamics as,$${\displaystyle \frac{\mathrm{d}{N}_i}{\mathrm{d}t}={\tau }_i^{-1}{N}_i\left[\sum _{\alpha \in M^{\ast }}{r}_{\alpha }^{-1}{K}_{\alpha }{w}_{\alpha }{c}_{i\alpha }\left(r_{\alpha }-\sum _{j=1}^S{N}_j{c}_{j\alpha }\right)-m_i\right],}$$where${\displaystyle \sum _{\alpha \in M^{\ast }}}$ denotes a sum over resource abundances which satisfy${\displaystyle R_{\alpha }=r_{\alpha }-\sum _{j=1}^S{N}_j{c}_{j\alpha }\ge 0}$. The above expression can be written as${\displaystyle \mathrm{d}{N}_i/\mathrm{d}t=-{\tau }_i^{-1}{N}_i\mathrm{\partial }Q/\mathrm{\partial }{N}_i}$, where,$${\displaystyle Q(\{N_i\})=\frac{1}{2}\sum _{\alpha \in M^{\ast }}{r}_{\alpha }^{-1}{K}_{\alpha }{w}_{\alpha }{\left(r_{\alpha }-\sum _{j=1}^S{c}_{j\alpha }{N}_j\right)}^2+\sum _{i=1}^S{m}_i{N}_i.}$$

At un-invadable steady state${\displaystyle \mathrm{\partial }Q/\mathrm{\partial }{N}_i=0}$ for all surviving species${\displaystyle i}$ and${\displaystyle \mathrm{\partial }Q/\mathrm{\partial }{N}_i>0}$ for all extinct species${\displaystyle i}$.^[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 the population growth conferred upon a species by consuming a resource is related to the impact the species' consumption has on the resource's abundance through the equation,$${\displaystyle q_{i\alpha }(\mathbf{R})=-a_i(\mathbf{R})b_{\alpha }(\mathbf{R})\frac{\mathrm{\partial }{g}_i}{\mathrm{\partial }{R}_{\alpha }},}$$ species-resource interactions are said to besymmetric. In the above equation${\displaystyle a_i}$ and${\displaystyle b_{\alpha }}$ are arbitrary functions of resource abundances. When this symmetry condition is satisfied, it can be shown that there exists a function${\displaystyle d(\mathbf{R})}$ such that:^[7]$${\displaystyle \frac{\mathrm{\partial }d}{\mathrm{\partial }{R}_{\alpha }}=-\frac{h_{\alpha }(\mathbf{R})}{b_{\alpha }(\mathbf{R})}.}$$After determining this function${\displaystyle d}$, the steady-state uninvadable resource abundances and species populations are the solution to theconstrained optimization problem:The species populations are theLagrange multipliers for the constraints on the second line. This can be seen by looking at theKKT conditions, taking${\displaystyle N_i}$ to be the Lagrange multipliers:Lines 1, 3, and 4 are the statements of feasibility and uninvadability: if${\displaystyle {\overline{N}}_i>0}$, then${\displaystyle g_i(\mathbf{R})}$ must be zero otherwise the system would not be at steady state, and if${\displaystyle {\overline{N}}_i=0}$, then${\displaystyle g_i(\mathbf{R})}$ must be non-positive otherwise species${\displaystyle i}$ would be able to invade. Line 2 is the stationarity condition and the steady-state condition for the resources in nice CRMs. The function${\displaystyle d(\mathbf{R})}$ can be interpreted as a distance by defining the point in the state space of resource abundances at which it is zero,${\displaystyle {\mathbf{R}}_0}$, to be its minimum. TheLagrangian for thedual problem which leads to the above KKT conditions is,$${\displaystyle L(\mathbf{R},\{N_i\})=d(\mathbf{R})-\sum _{i=1}^S{N}_i{g}_i(\mathbf{R}).}$$ In this picture, the unconstrained value of${\displaystyle \mathbf{R}}$ that minimizes${\displaystyle d(\mathbf{R})}$ (i.e., the steady-state resource abundances in the absence of any consumers) is known as the resource supply vector.

The steady states of consumer resource models can be analyzed using geometric means in the space of resource abundances.^[16][17][8]

For a community to satisfy the uninvisibility and steady-state conditions, the steady-state resource abundances (denoted${\displaystyle {\mathbf{R}}^{\star }}$) must satisfy,$${\displaystyle g_i({\mathbf{R}}^{\star })\le 0,}$$for all species${\displaystyle i}$. The inequality is saturated if and only if species${\displaystyle i}$ survives. 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 boundaries of these regions, specified by${\displaystyle g_i({\mathbf{R}}^{\star })=0}$, are known as the zero net-growthisoclines (ZNGIs). If species${\displaystyle i=1,\dots ,S^{\star }}$survive, then the steady-state resource abundances must satisfy,${\displaystyle g_1({\mathbf{R}}^{\star }),\dots ,g_{S^{\star }}({\mathbf{R}}^{\star })=0}$. 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.

The structure of ZNGI intersections determines what species can feasibly coexist but does not determine what set of coexisting species will be realized. Coexistence cones determine what species determine what species will survive in an ecosystem given a resource supply vector. A coexistence cone generated by a set of species${\displaystyle i=1,\dots ,S^{\star }}$ is defined to be the set of possible resource supply vectors which will lead to a community containing precisely the species${\displaystyle i=1,\dots ,S^{\star }}$.

To see the cone structure, consider that in the MacArthur or Tilman models, the steady-state non-depleted resource abundances must satisfy,$${\displaystyle \mathbf{K}={\mathbf{R}}^{\star }+\sum _{i=1}^S{N}_i{\mathbf{C}}_i,}$$ where${\displaystyle \mathbf{K}}$ is a vector containing the carrying capacities/supply rates, and${\displaystyle {\mathbf{C}}_i=(c_{i1},\dots ,c_{i{M}^{\star }})}$ is the${\displaystyle i}$th row of the consumption matrix${\displaystyle c_{i\alpha }}$, considered as a vector. 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 aconvex cone with apex${\displaystyle {\mathbf{R}}^{\star }}$ and whose generating vectors are the${\displaystyle {\mathbf{C}}_i}$ for the surviving species${\displaystyle i}$.

In an ecosystem with many species and resources, the behavior of consumer-resource models can be analyzed using tools from statistical physics, particularlymean-field theory and thecavity method.^[18][19][20] In the large ecosystem limit, there is an explosion of the number of parameters. For example, in the MacArthur model,${\displaystyle O(SM)}$ parameters are needed. In this limit, parameters may be considered to be drawn from some distribution which leads to a distribution of steady-state abundances. 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:${\displaystyle ⟨c_{i\alpha }⟩=\mu /M,\mathrm{var}(c_{i\alpha })={\sigma }^2/M,⟨m_i⟩=m,\mathrm{var}(m_i)={\sigma }_m^2,⟨K_{\alpha }⟩=K,\mathrm{var}(K_{\alpha })={\sigma }_K^2.}$

With this parameterization, in the thermodynamic limit (i.e.,${\displaystyle M,S\to \mathrm{\infty }}$ with${\displaystyle S/M=\mathrm{\Theta }(1)}$), the steady-state resource and species abundances are modeled as a random variable,${\displaystyle N,R}$, which satisfy the self-consistent mean-field equations,^[18] where${\displaystyle ⟨N⟩,⟨N^2⟩,⟨R⟩,⟩R^2⟩}$ are all moments which are determined self-consistently,${\displaystyle Z_R,Z_N}$ are independentstandard normal random variables, and${\displaystyle \nu =⟨\mathrm{\partial }N/\mathrm{\partial }m⟩}$ and${\displaystyle \chi =⟨\mathrm{\partial }R/\mathrm{\partial }K⟩}$ are average susceptibilities which are also determined self-consistently.

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.

Similar mean-field analyses have been performed for the externally supplied resources model,^[11] the Tilman model,^[12] and the microbial consumer-resource model.^[20] These techniques were first developed to analyze therandom generalized Lotka–Volterra model.

• Theoretical ecology
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• Lotka–Volterra equations
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• Cui, Wenping; Robert Marsland III; Mehta, Pankaj (2024). "Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics".arXiv:2403.05497 [q-bio.PE].
• Stefano Allesina's Community Ecology course lecture notes:https://stefanoallesina.github.io/Theoretical_Community_Ecology/

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