Consumer-resource model

In theoretical ecology and nonlinear dynamics, consumer-resource models (CRMs), also known as resource-competition models, are a class of ordinary differential equations-based 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 (or multiple) trophic level(s).

A general consumer-resource model consists of M resources whose abundances are $R_{1},\dots ,R_{M}$ and S consumer species whose populations are $N_{1},\dots ,N_{S}$. A general consumer-resource model is described by the system of coupled ordinary differential equations,$${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=N_{i}g_{i}(R_{1},\dots ,R_{M}),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=f_{\alpha }(R_{1},\dots ,R_{M},N_{1},\dots ,N_{S}),&&\qquad \alpha =1,\dots ,M\end{aligned}}$$where $g_{i}$, depending only on resource abundances, is the per-capita growth rate of species $i$, and $f_{\alpha }$ is the growth rate of resource $\alpha$. An essential feature of CRMs is that species growth rates and populations are mediated through resources and there are no explicit species-species interactions. Despite this, there remains implicit dependence due to the interacting nature of the system.

History

Add a bunch... Developed and introduced significantly by Robert H. MacArthur and Richard Levins.

Models

Niche models

Niche models are a notable class of CRMs which are described by the system of coupled ordinary differential equations,

${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=N_{i}g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=h_{\alpha }(\mathbf {R} )+\sum _{i=1}^{S}N_{i}q_{i\alpha }(\mathbf {R} ),&&\qquad \alpha =1,\dots ,M,\end{aligned}}$

where $\mathbf {R} \equiv (R_{1},\dots ,R_{M})$ is a vector abbreviation for resource abundances, $g_{i}$ is the per-capita growth rate of species $i$, $h_{\alpha }$ is the growth rate of species $\alpha$ in the absence of consumption, and $-q_{i\alpha }$ is the rate per unit species population that species $i$ depletes the abundance of resource $\alpha$ 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.[citation needed] The MCRM is given by the following set of coupled ordinary differential equations:$${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&={\frac {r_{\alpha }}{K_{\alpha }}}\left(K_{\alpha }-R_{\alpha }\right)R_{\alpha }-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}$$where $c_{i\alpha }$ is the relative preference of species $i$ for resource $\alpha$ and also the relative amount by which resource $\alpha$ is depleted by the consumption of consumer species $i$; $K_{\alpha }$ is the steady-state carrying capacity of resource $\alpha$ in absence of consumption (i.e., when $c_{i\alpha }$ is zero); $\tau _{i}$ and $r_{\alpha }^{-1}$ are time-scales for species and resource dynamics, respectively; $w_{\alpha }$ is the quality of resource $\alpha$; and $m_{i}$ is the natural mortality rate of species $i$. This model is said to have self-replenishing resource dynamics because when $c_{i\alpha }=0$, 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.

Occasionally, the MacArthur consumer resource model is extended to the asymmetric MacArthur consumer resource model (aMCRM) by changing the relative weights by which resource abundances are depleted through consumption:$${\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}={\frac {r_{\alpha }}{K_{\alpha }}}\left(K_{\alpha }-R_{\alpha }\right)R_{\alpha }-\sum _{i=1}^{S}N_{i}e_{i\alpha }R_{\alpha }$$Restricting $e_{i\alpha }=c_{i\alpha }$ recovers the original MCRM.

Externally-supplied resources model (eCRM)

The externally-supplied resource model (eCRM) is similar to the MCRM except the resources are provided at a constant rate from an external source instead of being self-replenished. It is described by the following set of coupled ordinary differential equations:$${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=r_{\alpha }(\kappa _{\alpha }-R_{\alpha })-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}$$where all the parameters shared with the MCRM are the same, and $\kappa _{\alpha }$ is the rate at which resource $\alpha$ is supplied to the ecosystem. In the eCRM, in the absence of consumption, $R_{\alpha }$ decays to $\kappa _{\alpha }$ exponentially with timescale $r_{\alpha }^{-1}$.

Tilman consumer resource model (TCRM)

The Tilman consumer-resource model (TCRM), named after G. David Tilman, is similar to the eCRM 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:$${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=r_{\alpha }(K_{\alpha }-R_{\alpha })-\sum _{i=1}^{S}N_{i}c_{i\alpha },&&\qquad \alpha =1,\dots ,M,\end{aligned}}$$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:$${\begin{aligned}{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=\tau _{i}^{-1}N_{i}\left(\sum _{\alpha =1}^{M}(1-l_{\alpha })w_{\alpha }c_{i\alpha }R_{\alpha }-m_{i}\right),&&\qquad i=1,\dots ,S,\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=\kappa _{\alpha }-rR_{\alpha }-\sum _{i=1}^{S}N_{i}c_{i\alpha }R_{\alpha }+\sum _{i=1}^{S}\sum _{\beta =1}^{M}N_{i}D_{\alpha \beta }l_{\beta }{\frac {w_{\beta }}{w_{\alpha }}}c_{i\beta }R_{\beta },&&\qquad \alpha =1,\dots ,M,\end{aligned}}$$where all parameters shared with the MCRM have similar interpretations; $D_{\alpha \beta }$ is the fraction of the byproducts due to consumption of resource $\beta$ which are converted to resource $\alpha$ and $l_{\alpha }$ is the "leakage fraction" of resource $\alpha$ governing how much of the resource is released into the environment as metabolic byproducts.

Multi-trophic Model

The multi-trophic consumer-resource model extends the typical paradigm of consumer-resource models which describe single trophic levels to describe multiple trophic levels. A description of an ecosystem with two-level trophic structure consists of $M_{R}$ primary producers with abundances $\{R_{\alpha }\}_{\alpha =1}^{M_{R}}$, $M_{N}$ primary consumers with populations $\{N_{i}\}_{i=1}^{M_{N}}$, and $M_{X}$ secondary consumers with populations $\{X_{k}\}_{k=1}^{M_{X}}$. In an example ecosystem, the primary producers could be plants, the primary consumers could be herbivores, and the secondary consumers could be carnivores; in this example, plants self-renew or are supplied externally, herbivores only consume plants, and carnivores only consume herbivores. Such a two-level trophic structure can be described by the following set of coupled ODEs:$${\begin{aligned}{\frac {\mathrm {d} X_{k}}{\mathrm {d} t}}&=X_{k}\left(\sum _{i=1}^{M_{N}}d_{ki}N_{i}-u_{k}\right),&&\qquad k=1,\dots ,M_{X},\\{\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}&=N_{i}\left(\sum _{\alpha =1}^{M_{R}}c_{i\alpha }R_{\alpha }-m_{i}-\sum _{k=1}^{M_{X}}d_{ki}X_{k}\right),&&\qquad i=1,\dots ,M_{N},\\{\frac {\mathrm {d} R_{\alpha }}{\mathrm {d} t}}&=R_{\alpha }\left(K_{\alpha }-R_{\alpha }-\sum _{i=1}^{M_{N}}c_{i\alpha }N_{i}\right),&&\qquad \alpha =1,\dots ,M_{R}.\end{aligned}}$$The form of the multi-trophic CRM has similar structure to the MCRM, specifically in that the primary producers are self-renewing. This model can be generalized further by allowing for any number of intermediate consumer/predator species where species in the $n$th trophic level consume only those in the $(n-1)$th trophic level and are consumed only by those in the $(n+1)$th trophic level. Such intermediate-level trophic interactions could be described by the following set of coupled ODEs:$${\begin{aligned}{\frac {\mathrm {d} N_{i}^{(n)}}{\mathrm {d} t}}&=N_{i}^{(n)}\left(\sum _{j=1}^{M_{(n-1)}}c_{ij}^{(n)}N_{j}^{(n-1)}-m_{i}^{(n)}-\sum _{k=1}^{M_{(n+1)}}c_{ki}^{(n+1)}N_{k}^{(n+1)}\right),\qquad i=1,\dots ,M_{(n)}.\end{aligned}}$$

Connection to Lotka–Volterra models

The (asymmetric) MacArthur consumer resource model can be written in the form of a generalized Lotka–Volterra model:$${\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=x_{i}\left(b_{i}-\sum _{j=1}^{n}A_{ij}x_{j}\right),\qquad i=1,\dots ,n.$$For the aMCRM, there are $n=M+S$ species where $x_{i}$ is identified with $N_{i}$ for $i=1,\dots ,S$ is identified with $R_{\alpha }$ for $i=S+1,\dots ,M+S$. In this form, the relevant Lotka–Volterra parameters are,$${\begin{aligned}\left[b_{i}\right]&={\begin{bmatrix}-\left[m_{i}\right]\\\left[K_{\alpha }\right]\\\end{bmatrix}}={\begin{bmatrix}-m_{1}\\\vdots \\-m_{S}\\K_{1}\\\vdots \\K_{M}\end{bmatrix}},\\\left[A_{ij}\right]&={\begin{bmatrix}0&\left[c_{i\alpha }\right]\\\left[e_{i\alpha }\right]^{\mathrm {T} }&-I\end{bmatrix}}={\begin{bmatrix}0&\cdots &0&c_{11}&\cdots &c_{1M}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&\cdots &0&c_{S1}&\cdots &c_{SM}\\-e_{11}&\cdots &-e_{S1}&-1&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\-e_{1M}&\cdots &-e_{SM}&0&\cdots &-1\end{bmatrix}}.\end{aligned}}$$Some but not all consumer resource models can be written as generalized Lotka–Volterra models.

Symmetric interactions and constrained 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., $r_{\alpha }\gg 1$) that they become entrained to species dynamics and are constantly at steady state (i.e., ${\mathrm {d} }R_{\alpha }/{\mathrm {d} }t=0$) so that $R_{\alpha }$ is expressed as a function of $N_{i}$. With this assumption, one can express species dynamics as,$${\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}=\tau _{i}^{-1}N_{i}\left[\sum _{\alpha \in M^{\ast }}r_{\alpha }^{-1}K_{\alpha }w_{\alpha }c_{i\alpha }\left(r_{\alpha }-\sum _{j=1}^{S}N_{j}c_{j\alpha }\right)-m_{i}\right],$$where $\sum _{\alpha \in M^{\ast }}$ denotes a sum over resource abundances which satisfy $R_{\alpha }=r_{\alpha }-\sum _{j=1}^{S}N_{j}c_{j\alpha }\geq 0$. The above expression can be written as $\mathrm {d} N_{i}/\mathrm {d} t=-\tau _{i}^{-1}N_{i}\,\partial Q/\partial N_{i}$, where,$$Q(\{N_{i}\})={\frac {1}{2}}\sum _{\alpha \in M^{\ast }}r_{\alpha }^{-1}K_{\alpha }w_{\alpha }\left(r_{\alpha }-\sum _{j=1}^{S}c_{j\alpha }N_{j}\right)^{2}+\sum _{i=1}^{S}m_{i}N_{i}.$$

At un-invadable steady state $\partial Q/\partial N_{i}=0$ for all surviving species $i$ and $\partial Q/\partial N_{i}>0$ for all extinct species $i$.

Minimum Environmental Perturbation Principle (MEPP)

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,$$q_{i\alpha }(\mathbf {R} )=-a_{i}(\mathbf {R} )b_{\alpha }(\mathbf {R} ){\frac {\partial g_{i}}{\partial R_{\alpha }}},$$species-resource interactions are said to be symmetric. In the above equation $a_{i}$ and $b_{\alpha }$ are arbitrary functions of resource abundances. When this symmetry condition is satisfied, it can be shown that there exists a function $d(\mathbf {R} )$ such that:$${\frac {\partial d}{\partial R_{\alpha }}}=-{\frac {h_{\alpha }(\mathbf {R} )}{b_{\alpha }(\mathbf {R} )}}.$$After determining this function $d$, the steady-state uninvadable resource abundances and species populations are the solution to the constrained optimization problem:$${\begin{aligned}\min _{\mathbf {R} }&\;d(\mathbf {R} )&&\\{\text{s.t.,}}&\;g_{i}(\mathbf {R} )\leq 0,&&\qquad i=1,\dots ,S,\\&\;R_{\alpha }\geq 0,&&\qquad \alpha =1,\dots ,M.\end{aligned}}$$The species populations are the Lagrange multipliers for the constraints on the second line. This can be seen by looking at the KKT conditions, taking $N_{i}$ to be the Lagrange multipliers:$${\begin{aligned}0&=N_{i}g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\0&={\frac {\partial d}{\partial R_{\alpha }}}-\sum _{i=1}^{S}N_{i}{\frac {\partial g_{i}}{\partial R_{\alpha }}},&&\qquad \alpha =1,\dots ,M,\\0&\geq g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\0&\leq N_{i},&&\qquad i=1,\dots ,S.\end{aligned}}$$Lines 1, 3, and 4 are the statements of feasibility and uninvadability: if ${\overline {N}}_{i}>0$, then $g_{i}(\mathbf {R} )$ must be zero otherwise the system would not be at steady state, and if ${\overline {N}}_{i}=0$, then $g_{i}(\mathbf {R} )$ must be non-positive otherwise species $i$ would be able to invade. Line 2 is the stationarity condition and the steady-state condition for the resources in nice CRMs. The function $d(\mathbf {R} )$ can be interpreted as a distance by defining the point in the state space of resource abundances at which it is zero, $\mathbf {R} _{0}$, to be its minimum. The Lagrangian for the dual problem which leads to the above KKT conditions is,$$L(\mathbf {R} ,\{N_{i}\})=d(\mathbf {R} )-\sum _{i=1}^{S}N_{i}g_{i}(\mathbf {R} ).$$The above Lagrangian does not constitute a Lyapunov function.

Geometric perspectives

Talk about ZNGIs