Consumer-resource model explained

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.[1] [2]

A general consumer-resource model consists of resources whose abundances are

R1,...,RM

and consumer species whose populations are

N1,...,NS

. A general consumer-resource model is described by the system of coupled ordinary differential equations,\begin\frac&=N_i g_i(R_1,\dots,R_M),&&\qquad i =1,\dots,S,\\\frac&=f_\alpha(R_1,\dots,R_M,N_1,\dots,N_S), &&\qquad \alpha = 1,\dots,M\endwhere

gi

, 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. Through resource interactions, there are emergent inter-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]

Models

Niche models

Niche models are a notable class of CRMs which are described by the system of coupled ordinary differential equations,[7] [8]

\begin{align} dNi
dt

&= Nigi(R),&&    i=1,...,S,\\

dR\alpha
dt

&= h\alpha(R)+

S
\sum
i=1

Niqi\alpha(R), &&    \alpha=1,...,M, \end{align}

where

R\equiv(R1,...,RM)

is a vector abbreviation for resource abundances,

gi

is the per-capita growth rate of species

i

,

h\alpha

is the growth rate of species

\alpha

in the absence of consumption, and

-qi\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.[9] [10] The MCRM is given by the following set of coupled ordinary differential equations:[11] [12] \begin\frac &= \tau_i^ N_i \left(\sum_^M w_\alpha c_ R_\alpha - m_i\right),&&\qquad i = 1,\dots,S,\\\frac &=\frac \left(K_\alpha - R_\alpha \right)R_\alpha-\sum_^S N_i c_R_\alpha,&&\qquad \alpha = 1,\dots,M,\endwhere

ci\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

ci\alpha

is zero);

\taui

and
-1
r
\alpha
are time-scales for species and resource dynamics, respectively;

w\alpha

is the quality of resource

\alpha

; and

mi

is the natural mortality rate of species

i

. This model is said to have self-replenishing resource dynamics because when

ci\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:\begin\frac &= \tau_i^ N_i \left(\sum_^M w_\alpha c_ R_\alpha - m_i\right),&&\qquad i = 1,\dots,S,\\\frac &=r_\alpha (\kappa_\alpha - R_\alpha)-\sum_^S N_i c_R_\alpha,&&\qquad \alpha = 1,\dots,M,\endwhere 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
-1
r
\alpha
. 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:\begin\frac &= \tau_i^ N_i \left(\sum_^M w_\alpha c_ R_\alpha - m_i\right),&&\qquad i = 1,\dots,S,\\\frac &=r_\alpha (K_\alpha - R_\alpha)-\sum_^S N_i c_,&&\qquad \alpha = 1,\dots,M,\endwhere 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\frac &= \tau_i^ N_i \left(\sum_^M (1-l_\alpha) w_\alpha c_ R_\alpha - m_i\right),&&\qquad i = 1,\dots,S,\\\frac &=\kappa_\alpha - r R_\alpha-\sum_^S N_i c_R_\alpha+\sum_^S\sum_^MN_i D_ l_\beta \frac c_ R_\beta,&&\qquad \alpha = 1,\dots,M, \endwhere 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.[13]

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\alpha\gg1

) that they become entrained to species dynamics and are constantly at steady state (i.e.,

{d}R\alpha/{d}t=0

) so that

R\alpha

is expressed as a function of

Ni

. With this assumption, one can express species dynamics as,\frac=\tau_i^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\inM\ast
denotes a sum over resource abundances which satisfy

R\alpha=r\alpha-

S
\sum
j=1

Njcj\alpha\geq0

. The above expression can be written as

dNi/dt=-\tau

-1
i

Ni\partialQ/\partialNi

, where,Q(\)=\frac\sum_r_\alpha^K_\alpha w_\alpha\left(r_\alpha - \sum_^S c_ N_j\right)^2+\sum_^S m_i N_i.

At un-invadable steady state

\partialQ/\partialNi=0

for all surviving species

i

and

\partialQ/\partialNi>0

for all extinct species

i

.[14] [15]

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_(\mathbf R) = - a_i(\mathbf R)b_\alpha(\mathbf R) \frac, species-resource interactions are said to be symmetric. In the above equation

ai

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(R)

such that:\frac=-\frac.After determining this function

d

, the steady-state uninvadable resource abundances and species populations are the solution to the constrained optimization problem:\begin\min_& \; d(\mathbf R)&&\\\text&\; g_i(\mathbf R) \leq 0,&&\qquad i=1,\dots,S,\\&\; R_\alpha \geq 0,&&\qquad \alpha =1,\dots,M.\endThe species populations are the Lagrange multipliers for the constraints on the second line. This can be seen by looking at the KKT conditions, taking

Ni

to be the Lagrange multipliers:\begin0 &= N_i g_i(\mathbf R), && \qquad i =1,\dots,S,\\0 &= \frac - \sum_^S N_i \frac,&&\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.\endLines 1, 3, and 4 are the statements of feasibility and uninvadability: if

\overlineNi>0

, then

gi(R)

must be zero otherwise the system would not be at steady state, and if

\overlineNi=0

, then

gi(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(R)

can be interpreted as a distance by defining the point in the state space of resource abundances at which it is zero,

R0

, to be its minimum. The Lagrangian for the dual problem which leads to the above KKT conditions is,L(\mathbf R,\) = d(\mathbf R) - \sum_^S N_i g_i(\mathbf R). In this picture, the unconstrained value of

R

that minimizes

d(R)

(i.e., the steady-state resource abundances in the absence of any consumers) is known as the resource supply vector.

Geometric perspectives

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

Zero net-growth isoclines (ZNGIs)

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

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
\star)
g
i(R

=0

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

i=1,...,S\star

survive, then the steady-state resource abundances must satisfy,
\star),\ldots,
g
1(R
g
S\star

(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.

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,\ldots,S\star

is defined to be the set of possible resource supply vectors which will lead to a community containing precisely the species

i=1,\ldots,S\star

.

To see the cone structure, consider that in the MacArthur or Tilman models, the steady-state non-depleted resource abundances must satisfy, \mathbf K = \mathbf R^\star + \sum_^S N_i \mathbf C_i, where

K

is a vector containing the carrying capacities/supply rates, and

Ci=(ci1

,\ldots,c
iM\star

)

is the

i

th row of the consumption matrix

ci\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

R\star

and whose generating vectors are the

Ci

for the surviving species

i

.

Complex ecosystems

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.[18] [19] 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.

MacArthur consumer resource model cavity solution

In the MCRM, the model parameters can be taken to be random variables with means and variances:

\langleci\alpha\rangle=\mu/M,\operatorname{var}(ci\alpha)=\sigma2/M, \langlemi\rangle=m,\operatorname{var}(mi)=

2,
\sigma
m

\langleK\alpha\rangle=K,\operatorname{var}(K\alpha)=

2.
\sigma
K

With this parameterization, in the thermodynamic limit (i.e.,

M,S\toinfty

with

S/M=\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,\begin0 &= R(K - \mu \tfrac \langle N\rangle - R + \sqrt Z_R + \sigma^2 \tfrac \nu R), \\0 &= N(\mu \langle R\rangle - m - \sigma^2 \chi N + \sqrt Z_N),\end where

\langleN\rangle,\langleN2\rangle,\langleR\rangle,\rangleR2\rangle

are all moments which are determined self-consistently,

ZR,ZN

are independent standard normal random variables, and

\nu=\langle\partialN/\partialm\rangle

and

\chi=\langle\partialR/\partialK\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.

See also

Further reading

Notes and References

  1. Book: Chase . Jonathan M. . Ecological Niches . Leibold . Mathew A. . 2003 . University of Chicago Press . 978-0-226-10180-4 . en . 10.7208/chicago/9780226101811.001.0001.
  2. Pimm . Stuart L. . September 1983 . TILMAN, D. 1982. Resource competition and community structure. Monogr. Pop. Biol. 17. Princeton University Press, Princeton, N.J. 296 p. $27.50. . Limnology and Oceanography . en . 28 . 5 . 1043–1045 . 10.4319/lo.1983.28.5.1043. 1983LimOc..28.1043P .
  3. MacArthur . Robert . 1970-05-01 . Species packing and competitive equilibrium for many species . Theoretical Population Biology . 1 . 1 . 1–11 . 10.1016/0040-5809(70)90039-0 . 0040-5809 . 5527624.
  4. Book: Levins, Richard . Evolution in Changing Environments: Some Theoretical Explorations. (MPB-2) . 1968 . Princeton University Press . 978-0-691-07959-2 . 10.2307/j.ctvx5wbbh . j.ctvx5wbbh.
  5. Goldford . Joshua E. . Lu . Nanxi . Bajić . Djordje . Estrela . Sylvie . Tikhonov . Mikhail . Sanchez-Gorostiaga . Alicia . Segrè . Daniel . Mehta . Pankaj . Sanchez . Alvaro . 2018-08-03 . Emergent simplicity in microbial community assembly . Science . en . 361 . 6401 . 469–474 . 10.1126/science.aat1168 . 0036-8075 . 6405290 . 30072533. 2018Sci...361..469G .
  6. Dal Bello . Martina . Lee . Hyunseok . Goyal . Akshit . Gore . Jeff . October 2021 . Resource–diversity relationships in bacterial communities reflect the network structure of microbial metabolism . Nature Ecology & Evolution . en . 5 . 10 . 1424–1434 . 2021NatEE...5.1424D . 10.1038/s41559-021-01535-8 . 2397-334X . 34413507 . 256708107 . free . 1721.1/141887.
  7. Marsland . Robert . Cui . Wenping . Mehta . Pankaj . 2020-09-01 . The Minimum Environmental Perturbation Principle: A New Perspective on Niche Theory . The American Naturalist . 196 . 3 . 291–305 . 10.1086/710093 . 32813998 . 59316948 . 0003-0147. 1901.09673 .
  8. 2403.05497 . q-bio.PE . Wenping . Cui . Robert . Marsland III . Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics . 2024-03-08 . Mehta . Pankaj.
  9. Web site: 1982-08-21 . Resource Competition and Community Structure. (MPB-17), Volume 17 Princeton University Press . 2024-03-18 . press.princeton.edu . en.
  10. Book: Chase . Jonathan M. . Ecological Niches: Linking Classical and Contemporary Approaches . Leibold . Mathew A. . University of Chicago Press . Interspecific Interactions . Chicago, IL . en.
  11. Cui . Wenping . Marsland . Robert . Mehta . Pankaj . 2020-07-21 . Effect of Resource Dynamics on Species Packing in Diverse Ecosystems . Physical Review Letters . 125 . 4 . 048101 . 10.1103/PhysRevLett.125.048101 . 8999492 . 32794828. 1911.02595 . 2020PhRvL.125d8101C .
  12. Cui . Wenping . Marsland . Robert . Mehta . Pankaj . 2021-09-27 . Diverse communities behave like typical random ecosystems . Physical Review E . 104 . 3 . 034416 . 10.1103/PhysRevE.104.034416 . 9005152 . 34654170. 2021PhRvE.104c4416C .
  13. Marsland . Robert . Cui . Wenping . Mehta . Pankaj . 2020-02-24 . A minimal model for microbial biodiversity can reproduce experimentally observed ecological patterns . Scientific Reports . en . 10 . 1 . 3308 . 10.1038/s41598-020-60130-2 . 32094388 . 7039880 . 1904.12914 . 2020NatSR..10.3308M . 2045-2322.
  14. Arthur . Robert Mac . Species Packing, and What Competition Minimizes . December 1969 . Proceedings of the National Academy of Sciences . en . 64 . 4 . 1369–1371 . 10.1073/pnas.64.4.1369 . 0027-8424 . 223294 . 16591810 . free .
  15. MacArthur . Robert . 1970-05-01 . Species packing and competitive equilibrium for many species . Theoretical Population Biology . 1 . 1 . 1–11 . 10.1016/0040-5809(70)90039-0 . 5527624 . 0040-5809.
  16. Mancuso . Christopher P . Lee . Hyunseok . Abreu . Clare I . Gore . Jeff . Khalil . Ahmad S . 2021-09-03 . Shou . Wenying . Walczak . Aleksandra M . Shou . Wenying . Environmental fluctuations reshape an unexpected diversity-disturbance relationship in a microbial community . eLife . 10 . e67175 . 10.7554/eLife.67175 . 2050-084X . 8460265 . 34477107 . free.
  17. Tikhonov . Mikhail . Monasson . Remi . 2017-01-27 . Collective Phase in Resource Competition in a Highly Diverse Ecosystem . Physical Review Letters . 118 . 4 . 048103 . 10.1103/PhysRevLett.118.048103. 28186794 . 1609.01270 . 2017PhRvL.118d8103T .
  18. Advani . Madhu . Bunin . Guy . Mehta . Pankaj . 2018-03-20 . Statistical physics of community ecology: a cavity solution to MacArthur's consumer resource model . Journal of Statistical Mechanics: Theory and Experiment . 2018 . 3 . 033406 . 10.1088/1742-5468/aab04e . 1742-5468 . 6329381 . 30636966. 2018JSMTE..03.3406A .
  19. Marsland . Robert . Cui . Wenping . Mehta . Pankaj . 2020-02-24 . A minimal model for microbial biodiversity can reproduce experimentally observed ecological patterns . Scientific Reports . en . 10 . 1 . 3308 . 10.1038/s41598-020-60130-2 . 32094388 . 7039880 . 1904.12914 . 2020NatSR..10.3308M . 2045-2322.