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. 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 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 consists of M resources whose abundances are R 1 , … , R M {\displaystyle R_{1},\dots ,R_{M}} and S consumer species whose populations are N 1 , … , N S {\displaystyle N_{1},\dots ,N_{S}} . A general consumer-resource model is described by the system of coupled ordinary differential equations, d N i d t = N i g i ( R 1 , … , R M ) , i = 1 , … , S , d R α d t = f α ( R 1 , … , R M , N 1 , … , N S ) , α = 1 , … , M {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=N_{i}g_{i}(R_{1},\dots ,R_{M}),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=f_{\alpha }(R_{1},\dots ,R_{M},N_{1},\dots ,N_{S}),&&\qquad \alpha =1,\dots ,M\end{aligned}}}
where g i {\displaystyle g_{i}} , depending only on resource abundances, is the per-capita growth rate of species i {\displaystyle i} , and f α {\displaystyle f_{\alpha }} is the growth rate of resource α {\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 are emergent inter-species interactions. Originally introduced by Robert H. MacArthur and Richard Levins, consumer-resource models have found success in formalizing ecological principles and modeling experiments involving microbial ecosystems.
Models
Niche models Niche models are a notable class of CRMs which are described by the system of coupled ordinary differential equations,
d N i d t = N i g i ( R ) , i = 1 , … , S , d R α d t = h α ( R ) + ∑ i = 1 S N i q i α ( R ) , α = 1 , … , M , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=N_{i}g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=h_{\alpha }(\mathbf {R} )+\sum _{i=1}^{S}N_{i}q_{i\alpha }(\mathbf {R} ),&&\qquad \alpha =1,\dots ,M,\end{aligned}}}
where R ≡ ( R 1 , … , R M ) {\displaystyle \mathbf {R} \equiv (R_{1},\dots ,R_{M})} is a vector abbreviation for resource abundances, g i {\displaystyle g_{i}} is the per-capita growth rate of species i {\displaystyle i} , h α {\displaystyle h_{\alpha }} is the growth rate of species α {\displaystyle \alpha } in the absence of consumption, and − q i α {\displaystyle -q_{i\alpha }} is the rate per unit species population that species i {\displaystyle i} depletes the abundance of resource α {\displaystyle \alpha } through consumption. In this class of CRMs, consumer species' impacts on resources are not explicitly coordinated; however, there are implicit interactions.
MacArthur consumer-resource model (MCRM) The MacArthur consumer-resource model (MCRM), named after Robert H. MacArthur, is a foundational CRM for the development of niche and coexistence theories. The MCRM is given by the following set of coupled ordinary differential equations: d N i d t = τ i − 1 N i ( ∑ α = 1 M w α c i α R α − m i ) , i = 1 , … , S , d R α d t = r α K α ( K α − R α ) R α − ∑ i = 1 S N i c i α R α , α = 1 , … , M , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&={\frac {r_{\alpha }}{K_{\alpha }}}\left(K_{\alpha }-R_{\alpha }\right)R_{\alpha }-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}} where c i α {\displaystyle c_{i\alpha }} is the relative preference of species i {\displaystyle i} for resource α {\displaystyle \alpha } and also the relative amount by which resource α {\displaystyle \alpha } is depleted by the consumption of consumer species i {\displaystyle i} ; K α {\displaystyle K_{\alpha }} is the steady-state carrying capacity of resource α {\displaystyle \alpha } in absence of consumption (i.e., when c i α {\displaystyle c_{i\alpha }} is zero); τ i {\displaystyle \tau _{i}} and r α − 1 {\displaystyle r_{\alpha }^{-1}} are time-scales for species and resource dynamics, respectively; w α {\displaystyle w_{\alpha }} is the quality of resource α {\displaystyle \alpha } ; and m i {\displaystyle m_{i}} is the natural mortality rate of species i {\displaystyle i} . This model is said to have self-replenishing resource dynamics because when c i α = 0 {\displaystyle c_{i\alpha }=0} , each resource exhibits independent logistic growth. 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). Steady states of the MCRM satisfy the competitive 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 occupiable ecological niches is equal to the number of non-depleted resources.
Externally supplied resources model 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: d N i d t = τ i − 1 N i ( ∑ α = 1 M w α c i α R α − m i ) , i = 1 , … , S , d R α d t = r α ( κ α − R α ) − ∑ i = 1 S N i c i α R α , α = 1 , … , M , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=r_{\alpha }(\kappa _{\alpha }-R_{\alpha })-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}} where all the parameters shared with the MCRM are the same, and κ α {\displaystyle \kappa _{\alpha }} is the rate at which resource α {\displaystyle \alpha } is supplied to the ecosystem. In the eCRM, in the absence of consumption, R α {\displaystyle R_{\alpha }} decays to κ α {\displaystyle \kappa _{\alpha }} exponentially with timescale r α − 1 {\displaystyle r_{\alpha }^{-1}} . This model is also known as a chemostat model.
Tilman consumer-resource model (TCRM) The Tilman consumer-resource model (TCRM), named after G. 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's R* rule. It is described by the following set of coupled ordinary differential equations: d N i d t = τ i − 1 N i ( ∑ α = 1 M w α c i α R α − m i ) , i = 1 , … , S , d R α d t = r α ( K α − R α ) − ∑ i = 1 S N i c i α , α = 1 , … , M , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=r_{\alpha }(K_{\alpha }-R_{\alpha })-\sum _{i=1}^{S}N_{i}c_{i\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}} where all parameters are shared with the MCRM. In the TCRM, resource abundances can become nonphysically negative.
Microbial consumer-resource model (MiCRM) The microbial consumer resource model describes a microbial ecosystem with externally supplied resources where consumption can produce metabolic byproducts, leading to potential cross-feeding. It is described by the following set of coupled ODEs: d N i d t = τ i − 1 N i ( ∑ α = 1 M ( 1 − l α ) w α c i α R α − m i ) , i = 1 , … , S , d R α d t = κ α − r R α − ∑ i = 1 S N i c i α R α + ∑ i = 1 S ∑ β = 1 M N i D α β l β w β w α c i β R β , α = 1 , … , M , {\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}(1-l_{\alpha })w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=\kappa _{\alpha }-rR_{\alpha }-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha }+\sum _{i=1}^{S}\sum _{\beta =1}^{M}N_{i}D_{\alpha \beta }l_{\beta }{\frac {w_{\beta }}{w_{\alpha }}}c_{i\beta }R_{\beta },&&\qquad \alpha =1,\dots ,M,\end{aligned}}} where all parameters shared with the MCRM have similar interpretations; D α β {\displaystyle D_{\alpha \beta }} is the fraction of the byproducts due to consumption of resource β {\displaystyle \beta } which are converted to resource α {\displaystyle \alpha } and l α {\displaystyle l_{\alpha }} is the "leakage fraction" of resource α {\displaystyle \alpha } governing how much of the resource is released into the environment as metabolic byproducts.
Symmetric interactions and optimization
MacArthur's Minimization Principle 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., r α ≫ 1 {\displaystyle r_{\alpha }\gg 1} ) that they become entrained to species dynamics and are constantly at steady state (i.e., d R α / d t = 0 {\displaystyle {\mathrm {d} }R_{\alpha }/{\mathrm {d} }t=0} ) so that R α {\displaystyle R_{\alpha }} is expressed as a function of N i {\displaystyle N_{i}} . With this assumption, one can express species dynamics as,
d N i d t = τ i − 1 N i [ ∑ α ∈ M ∗ r α − 1 K α w α c i α ( r α − ∑ j = 1 S N j c j α ) − m i ] , {\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 ∑ α ∈ M ∗ {\displaystyle \sum _{\alpha \in M^{\ast }}} denotes a sum over resource abundances which satisfy R α = r α − ∑ j = 1 S N j c j α