Niche models

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

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

where ${\displaystyle \mathbf {R} \equiv (R_{1},\dots ,R_{M})}$ 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.

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., ${\displaystyle r_{\alpha }\gg 1}$) 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 ${\displaystyle R_{\alpha }=r_{\alpha }-\sum _{j=1}^{S}N_{j}c_{j\alpha }\geq 0}$. The above expression can be written as ${\displaystyle \mathrm {d} N_{i}/\mathrm {d} t=-\tau _{i}^{-1}N_{i}\,\partial Q/\partial N_{i}}$, 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 ${\displaystyle \partial Q/\partial N_{i}=0}$ for all surviving species ${\displaystyle i}$ and ${\displaystyle \partial Q/\partial N_{i}>0}$ for all extinct species ${\displaystyle i}$.^[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,$${\displaystyle q_{i\alpha }(\mathbf {R} )=-a_{i}(\mathbf {R} )b_{\alpha }(\mathbf {R} ){\frac {\partial g_{i}}{\partial R_{\alpha }}},}$$ species-resource interactions are said to be symmetric. In the above equation ${\displaystyle a_{i}}$ 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]$${\displaystyle {\frac {\partial d}{\partial R_{\alpha }}}=-{\frac {h_{\alpha }(\mathbf {R} )}{b_{\alpha }(\mathbf {R} )}}.}$$After determining this function ${\displaystyle d}$, the steady-state uninvadable resource abundances and species populations are the solution to the constrained optimization problem:$${\displaystyle {\begin{aligned}\min _{\mathbf {R} }&\;d(\mathbf {R} )&&\\{\text{s.t.,}}&\;g_{i}(\mathbf {R} )\leq 0,&&\qquad i=1,\dots ,S,\\&\;R_{\alpha }\geq 0,&&\qquad \alpha =1,\dots ,M.\end{aligned}}}$$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 ${\displaystyle N_{i}}$ to be the Lagrange multipliers:$${\displaystyle {\begin{aligned}0&=N_{i}g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\0&={\frac {\partial d}{\partial R_{\alpha }}}-\sum _{i=1}^{S}N_{i}{\frac {\partial g_{i}}{\partial R_{\alpha }}},&&\qquad \alpha =1,\dots ,M,\\0&\geq g_{i}(\mathbf {R} ),&&\qquad i=1,\dots ,S,\\0&\leq N_{i},&&\qquad i=1,\dots ,S.\end{aligned}}}$$Lines 1, 3, and 4 are the statements of feasibility and uninvadability: if ${\displaystyle {\overline {N}}_{i}>0}$, 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. The Lagrangian for the dual 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.

Zero net-growth isoclines (ZNGIs)

For a community to satisfy the uninvisibility and steady-state conditions, the steady-state resource abundances (denoted ${\displaystyle \mathbf {R} ^{\star }}$) must satisfy, $${\displaystyle g_{i}(\mathbf {R} ^{\star })\leq 0,}$$ 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-growth isoclines (ZNGIs). If species ${\displaystyle i=1,\dots ,S^{\star }}$survive, then the steady-state resource abundances must satisfy, ${\displaystyle g_{1}(\mathbf {R} ^{\star }),\ldots ,g_{S^{\star }}(\mathbf {R} ^{\star })=0}$. 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.

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 ${\displaystyle i=1,\ldots ,S^{\star }}$ is defined to be the set of possible resource supply vectors which will lead to a community containing precisely the species ${\displaystyle i=1,\ldots ,S^{\star }}$.

To see the cone structure, consider that in the MacArthur or Tilman models, the steady-state non-depleted resource abundances must satisfy,$${\displaystyle \mathbf {K} =\mathbf {R} ^{\star }+\sum _{i=1}^{S}N_{i}\mathbf {C} _{i},}$$ where ${\displaystyle \mathbf {K} }$ is a vector containing the carrying capacities/supply rates, and ${\displaystyle \mathbf {C} _{i}=(c_{i1},\ldots ,c_{iM^{\star }})}$ 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 a convex cone with apex ${\displaystyle \mathbf {R} ^{\star }}$ and whose generating vectors are the ${\displaystyle \mathbf {C} _{i}}$ for the surviving species ${\displaystyle i}$.

MacArthur consumer resource model cavity solution

In the MCRM, the model parameters can be taken to be random variables with means and variances:${\displaystyle \langle c_{i\alpha }\rangle =\mu /M,\quad \operatorname {var} (c_{i\alpha })=\sigma ^{2}/M,\quad \langle m_{i}\rangle =m,\quad \operatorname {var} (m_{i})=\sigma _{m}^{2},\quad \langle K_{\alpha }\rangle =K,\quad \operatorname {var} (K_{\alpha })=\sigma _{K}^{2}.}$

With this parameterization, in the thermodynamic limit (i.e., ${\displaystyle M,S\to \infty }$ with ${\displaystyle S/M=\Theta (1)}$), 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]$${\displaystyle {\begin{aligned}0&=R(K-\mu {\tfrac {S}{M}}\langle N\rangle -R+{\sqrt {\sigma _{K}^{2}+{\tfrac {S}{M}}\sigma ^{2}\langle N^{2}\rangle }}Z_{R}+\sigma ^{2}{\tfrac {S}{M}}\nu R),\\0&=N(\mu \langle R\rangle -m-\sigma ^{2}\chi N+{\sqrt {\sigma ^{2}\langle R^{2}\rangle +\sigma _{m}^{2}}}Z_{N}),\end{aligned}}}$$ where ${\displaystyle \langle N\rangle ,\langle N^{2}\rangle ,\langle R\rangle ,\rangle R^{2}\rangle }$ are all moments which are determined self-consistently, ${\displaystyle Z_{R},Z_{N}}$ are independent standard normal random variables, and ${\displaystyle \nu =\langle \partial N/\partial m\rangle }$ and ${\displaystyle \chi =\langle \partial R/\partial K\rangle }$ 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,^[11] the Tilman model,^[12] and the microbial consumer-resource model.^[19] These techniques were first developed to analyze the random generalized Lotka–Volterra model.