Therandom generalized Lotka–Volterra model (rGLV) is anecological model andrandom set of coupled ordinary differential equations where the parameters of thegeneralized Lotka–Volterra equation are sampled from aprobability distribution, analogously toquenched disorder. The rGLV models dynamics of a community of species in which each species' abundance grows towards acarrying capacity but is depleted due tocompetition from the presence of other species. It is often analyzed in themany-species limit using tools fromstatistical physics, in particular fromspin glass theory.

The rGLV has been used as a tool to analyzeemergent macroscopic behavior inmicrobial communities with dense, strong interspecies interactions. The model has served as a context for theoretical investigations studyingdiversity-stability relations in community ecology^[1] and properties of static and dynamiccoexistence.^[2][3] Dynamical behavior in the rGLV has been mapped experimentally in community microcosms.^[4] The rGLV model has also served as an object of interest for the spin glass and disordered systems physics community to develop new techniques and numerical methods.^{[5][6][7][8][9]}

The random generalized Lotka–Volterra model is written as the system ofcoupled ordinary differential equations,^{[1][2][4][10]}$${\displaystyle \frac{\mathrm{d}{N}_i}{\mathrm{d}t}=\frac{r_i}{K_i}{N}_i\left(K_i-N_i-\sum _{j(\ne i)}{\alpha }_{ij}{N}_j\right),i=1,\dots ,S,}$$where${\displaystyle N_i}$ is the abundance of species${\displaystyle i}$,${\displaystyle S}$ is the number of species,${\displaystyle K_i}$ is the carrying capacity of species${\displaystyle i}$ in the absence of interactions,${\displaystyle r_i}$ sets a timescale, and${\displaystyle \alpha }$ is arandom matrix whose entries arerandom variables with mean${\displaystyle ⟨{\alpha }_{ij}⟩={\mu }_{\alpha }/S}$, variance${\displaystyle \mathrm{v}\mathrm{a}\mathrm{r}({\alpha }_{ij})={\sigma }_{\alpha }^2/S}$, andcorrelations${\displaystyle \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}({\alpha }_{ij},{\alpha }_{ji})=\gamma }$ for${\displaystyle i\ne j}$ where${\displaystyle -1\le \gamma \le 1}$. Theinteraction matrix,${\displaystyle \alpha }$, may be parameterized as,$${\displaystyle {\alpha }_{ij}=\frac{{\mu }_{\alpha }}{S}+\frac{{\sigma }_{\alpha }}{\sqrt{S}}{a}_{ij},}$$where${\displaystyle a_{ij}}$ are standard random variables (i.e., zero mean and unit variance) with${\displaystyle ⟨a_{ij}{a}_{ji}⟩=\gamma }$ for${\displaystyle i\ne j}$. The matrix entries may have any distribution with common finite first and second moments and will yield identical results in the large${\displaystyle S}$ limit due to thecentral limit theorem. The carrying capacities may also be treated as random variables with${\displaystyle ⟨K_i⟩=K,\mathrm{var}(K_i)={\sigma }_K^2.}$ Analyses by statistical physics-inspired methods have revealedphase transitions between different qualitative behaviors of the model in themany-species limit. In some cases, this may include transitions between the existence of aunique globally-attractive fixed point andchaotic, persistent fluctuations.

In thethermodynamic limit (i.e., the community has a very large number of species) where a unique globally-attractive fixed point exists, the distribution of species abundances can be computed using thecavity method while assuming the system isself-averaging. The self-averaging assumption means that the distribution of any one species' abundance between samplings of model parameters matches the distribution of species abundances within a single sampling of model parameters. In the cavity method, an additionalmean-field species${\displaystyle i=0}$ is introduced and the response of the system is approximated linearly. The cavity calculation yields aself-consistent equation describing the distribution of species abundances as a mean-field random variable,${\displaystyle N_0}$. When${\displaystyle {\sigma }_K=0}$, the mean-field equation is,^[1]$${\displaystyle 0=N_0\left(K-{\mu }_{\alpha }m-N_0+\sqrt{q\left({\mu }_{\alpha }^2+\gamma {\sigma }_{\alpha }^2\right)}Z+{\sigma }_{\alpha }^2\gamma \chi N_0\right),}$$where${\displaystyle m=⟨N_0⟩,q=⟨N_0^2⟩,\chi =⟨\mathrm{\partial }{N}_0/\mathrm{\partial }{K}_0⟩}$, and${\displaystyle Z\sim \mathcal{N}(0,1)}$ is astandard normal random variable. Onlyecologically uninvadable solutions are taken (i.e., the largest solution for${\displaystyle N_0}$ in the quadratic equation is selected). The relevant susceptibility and moments of${\displaystyle N_0}$, which has atruncated normal distribution, are determined self-consistently.

In the thermodynamic limit where there is an asymptotically large number of species (i.e.,${\displaystyle S\to \mathrm{\infty }}$), there are three distinctphases: one in which there is a unique fixed point (UFP), another with a multipleattractors (MA), and a third with unbounded growth. In the MA phase, depending on whether species abundances are replenished at a small rate, may approach arbitrarily small population sizes, or are removed from the community when the population falls below some cutoff, the resulting dynamics may be chaotic with persistent fluctuations or approach an initial conditions-dependent steady state.^[1]

The transition from the UFP to MA phase is signaled by the cavity solution becoming unstable to disordered perturbations. When${\displaystyle {\sigma }_K=0}$, thephase transition boundary occurs when the parameters satisfy,$${\displaystyle {\sigma }_{\alpha }=\frac{\sqrt{2}}{1+\gamma }.}$$In the${\displaystyle {\sigma }_K>0}$ case, the phase boundary can still be calculated analytically, but no closed-form solution has been found; numerical methods are necessary to solve the self-consistent equations determining the phase boundary.

The transition to the unbounded growth phase is signaled by the divergence of${\displaystyle ⟨N_0⟩}$ as computed in the cavity calculation.

The cavity method can also be used to derive adynamical mean-field theory model for the dynamics. The cavity calculation yields a self-consistent equation describing the dynamics as aGaussian process defined by the self-consistent equation (for${\displaystyle {\sigma }_K=0}$),^[8]$${\displaystyle \frac{\mathrm{d}{N}_0}{\mathrm{d}t}=N_0(t)\left[K_0-N_0(t)-{\mu }_{\alpha }m(t)-{\sigma }_{\alpha }\eta (t)+\gamma {\sigma }_{\alpha }^2{\int }_0^t\mathrm{d}{t}^{\prime }\chi (t,t^{\prime })N_0(t^{\prime })\right],}$$where${\displaystyle m(t)=⟨N_0(t)⟩}$,${\displaystyle \eta }$ is a zero-mean Gaussian process withautocorrelation${\displaystyle ⟨\eta (t)\eta (t^{\prime })⟩=⟨N_0(t)N_0(t^{\prime })⟩}$, and${\displaystyle \chi (t,t^{\prime })=⟨{\delta N_0(t)/\delta K_0(t^{\prime })|}_{K_0(t^{\prime })=K_0}⟩}$ is the dynamical susceptibility defined in terms of afunctional derivative of the dynamics with respect to a time-dependent perturbation of the carrying capacity.

Using dynamical mean-field theory, it has been shown that at long times, the dynamics exhibit aging in which the characteristic time scale defining the decay of correlations increases linearly in the duration of the dynamics. That is,${\displaystyle C_N(t,t+t\tau )\to f(\tau )}$ when${\displaystyle t}$ is large, where${\displaystyle C_N(t,t^{\prime })=⟨N(t)N(t^{\prime })⟩}$ is the autocorrelation function of the dynamics and${\displaystyle f(\tau )}$ is a common scaling collapse function.^[8][11]

When a small immigration rate${\displaystyle \lambda \ll 1}$ is added (i.e., a small constant is added to the right-hand side of the equations of motion) the dynamics reach atime transitionally invariant state. In this case, the dynamics exhibit jumps between${\displaystyle O(1)}$ and${\displaystyle O(\lambda )}$ abundances.^[12]

• Generalized Lotka–Volterra equation
• Competitive Lotka–Volterra equations
• Lotka–Volterra equations
• Consumer-resource model
• Theoretical ecology
• Random dynamical system
• Spin glass
• Cavity method
• Dynamical mean-field theory
• Quenched disorder
• Community (ecology)
• Ecological stability

• Stefano Allesina's Community Ecology course lecture notes:https://stefanoallesina.github.io/Theoretical_Community_Ecology/
• Bunin, Guy (2017-04-28). "Ecological communities with Lotka-Volterra dynamics". Physical Review E. 95 (4): 042414. Bibcode:2017PhRvE..95d2414B.doi:10.1103/PhysRevE.95.042414. PMID 28505745.

• Community ecology
• Complex systems theory
• Theoretical ecology
• Random dynamical systems
• Dynamical systems
• Mathematical modeling
• Biophysics
• Ordinary differential equations
• Population ecology
• Ecology

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• Short description matches Wikidata