Consumer-resource model

Niche models are a notable class of CRMs which are described by the system of coupled ordinary differential equations,{{Cite journal |last1=Marsland |first1=Robert |last2=Cui |first2=Wenping |last3=Mehta |first3=Pankaj |date=2020-09-01 |title=The Minimum Environmental Perturbation Principle: A New Perspective on Niche Theory |url=http://dx.doi.org/10.1086/710093 |journal=The American Naturalist |volume=196 |issue=3 |pages=291–305 |doi=10.1086/710093 |pmid=32813998 |s2cid=59316948 |issn=0003-0147|arxiv=1901.09673 }}{{cite arXiv |eprint=2403.05497 |class=q-bio.PE |first1=Wenping |last1=Cui |first2=Robert |last2=Marsland III |title=Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics |date=2024-03-08 |last3=Mehta |first3=Pankaj}}

: $$

\begin{align}

\frac{\mathrm dN_i}{\mathrm dt}

&=

N_i g_i(\mathbf R), &&\qquad i =1,\dots,S,\\

\frac{\mathrm dR_\alpha}{\mathrm dt}

&=

h_\alpha(\mathbf R) + \sum_{i=1}^S N_i q_{i\alpha}(\mathbf R),

&&\qquad \alpha = 1,\dots,M,

\end{align}

where $\backslash mathbf\; R\; \backslash equiv\; (R\_1,\backslash dots,R\_M)$ is a vector abbreviation for resource abundances, $g\_i$ is the per-capita growth rate of species $i$, $h\_\backslash alpha$ is the growth rate of species $\backslash alpha$ in the absence of consumption, and $-q\_\{i\backslash alpha\}$ is the rate per unit species population that species $i$ depletes the abundance of resource $\backslash alpha$ 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 after Robert H. MacArthur, is a foundational CRM for the development of niche and coexistence theories.{{Cite web |date=1982-08-21 |title=Resource Competition and Community Structure. (MPB-17), Volume 17 {{!}} Princeton University Press |url=https://press.princeton.edu/books/paperback/9780691083025/resource-competition-and-community-structure-mpb-17-volume-17 |access-date=2024-03-18 |website=press.princeton.edu |language=en}}{{Cite book |last1=Chase |first1=Jonathan M. |url=https://press.uchicago.edu/ucp/books/book/chicago/E/bo3638660.html |title=Ecological Niches: Linking Classical and Contemporary Approaches |last2=Leibold |first2=Mathew A. |publisher=University of Chicago Press |series=Interspecific Interactions |location=Chicago, IL |language=en}} The MCRM is given by the following set of coupled ordinary differential equations:{{Cite journal |last1=Cui |first1=Wenping |last2=Marsland |first2=Robert |last3=Mehta |first3=Pankaj |date=2020-07-21 |title=Effect of Resource Dynamics on Species Packing in Diverse Ecosystems |journal=Physical Review Letters |volume=125 |issue=4 |pages=048101 |doi=10.1103/PhysRevLett.125.048101 |pmc=8999492 |pmid=32794828|arxiv=1911.02595 |bibcode=2020PhRvL.125d8101C }}{{Cite journal |last1=Cui |first1=Wenping |last2=Marsland |first2=Robert |last3=Mehta |first3=Pankaj |date=2021-09-27 |title=Diverse communities behave like typical random ecosystems |journal=Physical Review E |volume=104 |issue=3 |pages=034416 |doi=10.1103/PhysRevE.104.034416 |pmc=9005152 |pmid=34654170|bibcode=2021PhRvE.104c4416C }}$$\backslash begin\{align\}$$

\frac{\mathrm dN_i}{\mathrm dt} &=

\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 dR_\alpha}{\mathrm dt} &=

\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{align}where $c\_\{i\backslash alpha\}$ is the relative preference of species $i$ for resource $\backslash alpha$ and also the relative amount by which resource $\backslash alpha$ is depleted by the consumption of consumer species $i$; $K\_\backslash alpha$ is the steady-state carrying capacity of resource $\backslash alpha$ in absence of consumption (i.e., when $c\_\{i\backslash alpha\}$ is zero); $\backslash tau\_i$ and $r\_\backslash alpha^\{-1\}$ are time-scales for species and resource dynamics, respectively; $w\_\backslash alpha$ is the quality of resource $\backslash alpha$; and $m\_i$ is the natural mortality rate of species $i$. This model is said to have self-replenishing resource dynamics because when $c\_\{i\backslash alpha\}\; =\; 0$, 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.

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:$$\backslash begin\{align\}$$

\frac{\mathrm dN_i}{\mathrm dt} &=

\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 dR_\alpha}{\mathrm dt} &=

r_\alpha (\kappa_\alpha - R_\alpha)

-

\sum_{i=1}^S N_i c_{i\alpha}R_\alpha,

&&

\qquad \alpha = 1,\dots,M,

\end{align}where all the parameters shared with the MCRM are the same, and $\backslash kappa\_\backslash alpha$ is the rate at which resource $\backslash alpha$ is supplied to the ecosystem. In the eCRM, in the absence of consumption, $R\_\backslash alpha$ decays to $\backslash kappa\_\backslash alpha$ exponentially with timescale $r\_\backslash alpha^\{-1\}$. This model is also known as a chemostat model.

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:$$\backslash begin\{align\}$$

\frac{\mathrm dN_i}{\mathrm dt} &=

\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 dR_\alpha}{\mathrm dt} &=

r_\alpha (K_\alpha - R_\alpha)

-

\sum_{i=1}^S N_i c_{i\alpha},

&&

\qquad \alpha = 1,\dots,M,

\end{align}where all parameters are shared with the MCRM. In the TCRM, resource abundances can become nonphysically negative.

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:$$\backslash begin\{align\}$$

\frac{\mathrm dN_i}{\mathrm dt} &=

\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 dR_\alpha}{\mathrm dt} &=

\kappa_\alpha - r R_\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{align}where all parameters shared with the MCRM have similar interpretations; $D\_\{\backslash alpha\backslash beta\}$ is the fraction of the byproducts due to consumption of resource $\backslash beta$ which are converted to resource $\backslash alpha$ and $l\_\backslash alpha$ is the "leakage fraction" of resource $\backslash alpha$ governing how much of the resource is released into the environment as metabolic byproducts.{{Cite journal |last1=Marsland |first1=Robert |last2=Cui |first2=Wenping |last3=Mehta |first3=Pankaj |date=2020-02-24 |title=A minimal model for microbial biodiversity can reproduce experimentally observed ecological patterns |journal=Scientific Reports |language=en |volume=10 |issue=1 |pages=3308 |doi=10.1038/s41598-020-60130-2 |pmid=32094388 |pmc=7039880 |arxiv=1904.12914 |bibcode=2020NatSR..10.3308M |issn=2045-2322}}

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., $r\_\backslash alpha\; \backslash gg\; 1$) that they become entrained to species dynamics and are constantly at steady state (i.e., $\{\backslash mathrm\; d\}R\_\backslash alpha/\{\backslash mathrm\; d\}t\; =\; 0$) so that $R\_\backslash alpha$ is expressed as a function of $N\_i$. With this assumption, one can express species dynamics as,

\frac{\mathrm dN_i}{\mathrm dt}

=

\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 $\backslash sum\_\{\backslash alpha\; \backslash in\; M^\backslash ast\}$ denotes a sum over resource abundances which satisfy $R\_\backslash alpha\; =\; r\_\backslash alpha\; -\; \backslash sum\_\{j=1\}^S\; N\_j\; c\_\{j\backslash alpha\}\; \backslash geq\; 0$. The above expression can be written as $\backslash mathrm\{d\}N\_i/\backslash mathrm\{d\}t=-\backslash tau\_i^\{-1\}N\_i\; \backslash ,\backslash partial\; Q/\backslash partial\; N\_i$, 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 $\backslash partial\; Q/\backslash partial\; N\_i\; =\; 0$ for all surviving species $i$ and $\backslash partial\; Q/\backslash partial\; N\_i\; >\; 0$ for all extinct species $i$.{{Cite journal |last=Arthur |first=Robert Mac |title=Species Packing, and What Competition Minimizes |date=December 1969 |journal=Proceedings of the National Academy of Sciences |language=en |volume=64 |issue=4 |pages=1369–1371 |doi=10.1073/pnas.64.4.1369 |issn=0027-8424 |pmc=223294 |pmid=16591810 |doi-access=free }}{{Cite journal |last=MacArthur |first=Robert |date=1970-05-01 |title=Species packing and competitive equilibrium for many species |url=https://dx.doi.org/10.1016/0040-5809%2870%2990039-0 |journal=Theoretical Population Biology |volume=1 |issue=1 |pages=1–11 |doi=10.1016/0040-5809(70)90039-0 |pmid=5527624 |issn=0040-5809|url-access=subscription }}

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,$$q\_\{i\backslash alpha\}(\backslash mathbf\; R)\; =\; -\; a\_i(\backslash mathbf\; R)b\_\backslash alpha(\backslash mathbf\; R)\; \backslash frac\{\backslash partial\; g\_i\}\{\backslash partial\; R\_\backslash alpha\}$$

, species-resource interactions are said to be symmetric. In the above equation $a\_i$ and $b\_\backslash alpha$ are arbitrary functions of resource abundances. When this symmetry condition is satisfied, it can be shown that there exists a function $d(\backslash mathbf\; R)$ such that:$$\backslash frac\{\backslash partial\; d\}\{\backslash partial\; R\_\backslash alpha\}$$

=

-\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:$$\backslash begin\{align\}$$

\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{align}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:$$\backslash begin\{align\}$$

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{align}Lines 1, 3, and 4 are the statements of feasibility and uninvadability: if $\backslash overline\; N\_i\; >\; 0$, then $g\_i(\backslash mathbf\; R)$ must be zero otherwise the system would not be at steady state, and if $\backslash overline\; N\_i\; =\; 0$, then $g\_i(\backslash mathbf\; R)$ 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 $d(\backslash mathbf\; R)$ can be interpreted as a distance by defining the point in the state space of resource abundances at which it is zero, $\backslash mathbf\; R\_0$, to be its minimum. The Lagrangian for the dual problem which leads to the above KKT conditions is,$$L(\backslash mathbf\; R,\backslash \{N\_i\backslash \})\; =$$

d(\mathbf R) - \sum_{i = 1}^S N_i g_i(\mathbf R). In this picture, the unconstrained value of $\backslash mathbf\; R$ that minimizes $d(\backslash mathbf\; R)$ (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.{{Cite journal |last1=Mancuso |first1=Christopher P |last2=Lee |first2=Hyunseok |last3=Abreu |first3=Clare I |last4=Gore |first4=Jeff |last5=Khalil |first5=Ahmad S |date=2021-09-03 |editor-last=Shou |editor-first=Wenying |editor2-last=Walczak |editor2-first=Aleksandra M |editor3-last=Shou |editor3-first=Wenying |title=Environmental fluctuations reshape an unexpected diversity-disturbance relationship in a microbial community |journal=eLife |volume=10 |pages=e67175 |doi=10.7554/eLife.67175 |issn=2050-084X |pmc=8460265 |pmid=34477107 |doi-access=free}}{{Cite journal |last1=Tikhonov |first1=Mikhail |last2=Monasson |first2=Remi |date=2017-01-27 |title=Collective Phase in Resource Competition in a Highly Diverse Ecosystem |url=https://link.aps.org/doi/10.1103/PhysRevLett.118.048103 |journal=Physical Review Letters |volume=118 |issue=4 |pages=048103 |doi=10.1103/PhysRevLett.118.048103|pmid=28186794 |arxiv=1609.01270 |bibcode=2017PhRvL.118d8103T }}

For a community to satisfy the uninvisibility and steady-state conditions, the steady-state resource abundances (denoted $$

\mathbf R^\star

) must satisfy,

g_i(\mathbf R^\star) \leq 0,

for all species $$

i

. The inequality is saturated if and only if species $$

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 $$

g_i(\mathbf R^\star) = 0

, are known as the zero net-growth isoclines (ZNGIs). If species $$

i = 1,\dots,S^\star

survive, then the steady-state resource abundances must satisfy, $$

g_1(\mathbf R^\star),\ldots, 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 $i\; =\; 1,\backslash ldots,\; S^\backslash star$ is defined to be the set of possible resource supply vectors which will lead to a community containing precisely the species $i\; =1,\backslash ldots,S^\backslash star$.

To see the cone structure, consider that in the MacArthur or Tilman models, the steady-state non-depleted resource abundances must satisfy,$$\backslash mathbf\; K\; =\; \backslash mathbf\; R^\backslash star\; +\; \backslash sum\_\{i=1\}^S\; N\_i\; \backslash mathbf\; C\_i,$$ where $\backslash mathbf\; K$ is a vector containing the carrying capacities/supply rates, and $\backslash mathbf\; C\_i\; =\; (c\_\{i1\},\backslash ldots,c\_\{iM^\backslash star\})$ is the $i$th row of the consumption matrix $c\_\{i\backslash alpha$

}, 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 a convex cone with apex $\backslash mathbf\; R^\backslash star$ and whose generating vectors are the $\backslash mathbf\; C\_i$ for the surviving species $i$.

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.{{Cite journal |last1=Advani |first1=Madhu |last2=Bunin |first2=Guy |last3=Mehta |first3=Pankaj |date=2018-03-20 |title=Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2018 |issue=3 |pages=033406 |doi=10.1088/1742-5468/aab04e |issn=1742-5468 |pmc=6329381 |pmid=30636966|bibcode=2018JSMTE..03.3406A }}{{Cite journal |last1=Marsland |first1=Robert |last2=Cui |first2=Wenping |last3=Mehta |first3=Pankaj |date=2020-02-24 |title=A minimal model for microbial biodiversity can reproduce experimentally observed ecological patterns |journal=Scientific Reports |language=en |volume=10 |issue=1 |pages=3308 |doi=10.1038/s41598-020-60130-2 |pmid=32094388 |pmc=7039880 |arxiv=1904.12914 |bibcode=2020NatSR..10.3308M |issn=2045-2322}}{{Citation |last1=Mehta |first1=Pankaj |title=Cross-feeding shapes both competition and cooperation in microbial ecosystems |date=2021-10-10 |arxiv=2110.04965 |last2=Marsland III |first2=Robert}} In the large ecosystem limit, there is an explosion of the number of parameters. For example, in the MacArthur model, $O(SM)$ 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:$\backslash langle\; c\_\{i\backslash alpha\}\backslash rangle\; =\; \backslash mu/M,\backslash quad\; \backslash operatorname\{var\}(c\_\{i\backslash alpha\})\; =\; \backslash sigma^2/M,$

\quad \langle m_i \rangle = m, \quad \operatorname{var}(m_i) = \sigma_m^2,

\quad \langle K_\alpha\rangle = K,\quad\operatorname{var}(K_\alpha) = \sigma_K^2.

With this parameterization, in the thermodynamic limit (i.e., $M,S\; \backslash to\; \backslash infty$ with $S/M\; =\; \backslash Theta(1)$), the steady-state resource and species abundances are modeled as a random variable, $N,\; R$, which satisfy the self-consistent mean-field equations,$$\backslash begin\{aligned\}$$

0 &= R(K - \mu \tfrac{S}{M} \langle N\rangle - R + \sqrt{\sigma_K^2 + \tfrac{S}{M} \sigma^2 \langle N^2\rangle} Z_R + \sigma^2 \tfrac{S}{M} \nu R ), \\

0 &= N(\mu \langle R\rangle - m - \sigma^2 \chi N + \sqrt{\sigma^2 \langle R^2\rangle + \sigma_m^2} Z_N ),

\end{aligned} where $\backslash langle\; N\backslash rangle,\; \backslash langle\; N^2\backslash rangle,\; \backslash langle\; R\backslash rangle,\; \backslash rangle\; R^2\backslash rangle$ are all moments which are determined self-consistently, $Z\_R,Z\_N$ are independent standard normal random variables, and $\backslash nu\; =\; \backslash langle\; \backslash partial\; N/\backslash partial\; m\; \backslash rangle$ and $\backslash chi\; =\; \backslash langle\; \backslash partial\; R/\backslash partial\; K\; \backslash rangle$ 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, the Tilman model, and the microbial consumer-resource model. These techniques were first developed to analyze the random generalized Lotka–Volterra model.

• Theoretical ecology
• Community (ecology)
• Competition (biology)
• Lotka–Volterra equations
• Competitive Lotka–Volterra equations
• Generalized Lotka–Volterra equation
• Random generalized Lotka–Volterra model

{{reflist}}

• {{cite arXiv |eprint=2403.05497 |last1=Cui |first1=Wenping |author2=Robert Marsland III |last3=Mehta |first3=Pankaj |title=Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics |date=2024 |class=q-bio.PE }}
• Stefano Allesina's Community Ecology course lecture notes: https://stefanoallesina.github.io/Theoretical_Community_Ecology/

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