random generalized Lotka–Volterra model

The random generalized Lotka–Volterra model is written as the system of coupled ordinary differential equations,{{Cite journal |last=Bunin |first=Guy |date=2017-04-28 |title=Ecological communities with Lotka-Volterra dynamics |url=https://link.aps.org/doi/10.1103/PhysRevE.95.042414 |journal=Physical Review E |volume=95 |issue=4 |pages=042414 |doi=10.1103/PhysRevE.95.042414|pmid=28505745 |bibcode=2017PhRvE..95d2414B |url-access=subscription }}{{Cite journal |last1=Serván |first1=Carlos A. |last2=Capitán |first2=José A. |last3=Grilli |first3=Jacopo |last4=Morrison |first4=Kent E. |last5=Allesina |first5=Stefano |date=August 2018 |title=Coexistence of many species in random ecosystems |url=https://www.nature.com/articles/s41559-018-0603-6 |journal=Nature Ecology & Evolution |language=en |volume=2 |issue=8 |pages=1237–1242 |doi=10.1038/s41559-018-0603-6 |pmid=29988167 |bibcode=2018NatEE...2.1237S |s2cid=49668570 |issn=2397-334X|url-access=subscription }}{{Cite journal |last1=Hu |first1=Jiliang |last2=Amor |first2=Daniel R. |last3=Barbier |first3=Matthieu |last4=Bunin |first4=Guy |last5=Gore |first5=Jeff |date=2022-10-07 |title=Emergent phases of ecological diversity and dynamics mapped in microcosms |url=https://www.science.org/doi/10.1126/science.abm7841 |journal=Science |language=en |volume=378 |issue=6615 |pages=85–89 |doi=10.1126/science.abm7841 |pmid=36201585 |bibcode=2022Sci...378...85H |s2cid=240251815 |issn=0036-8075}}{{Cite journal |last1=Altieri |first1=Ada |last2=Roy |first2=Felix |last3=Cammarota |first3=Chiara |last4=Biroli |first4=Giulio |date=2021-06-23 |title=Properties of Equilibria and Glassy Phases of the Random Lotka-Volterra Model with Demographic Noise |url=https://link.aps.org/doi/10.1103/PhysRevLett.126.258301 |journal=Physical Review Letters |volume=126 |issue=25 |pages=258301 |doi=10.1103/PhysRevLett.126.258301|pmid=34241496 |arxiv=2009.10565 |bibcode=2021PhRvL.126y8301A |hdl=11573/1623024 |s2cid=221836142 }}$$\backslash frac\{\backslash mathrm\; dN\_i\}\{\backslash mathrm\; dt\}\; =\; \backslash frac\{r\_i\}\{K\_i\}N\_i\; \backslash left(K\_i\; -\; N\_i\; -\; \backslash sum\_\{j\; (\backslash neq\; i)\}\; \backslash alpha\_\{ij\}\; N\_j\backslash right),$$

\qquad i = 1,\dots,S,where $N\_i$ is the abundance of species $i$, $S$ is the number of species, $K\_i$ is the carrying capacity of species $i$ in the absence of interactions, $r\_i$ sets a timescale, and $\backslash alpha$ is a random matrix whose entries are random variables with mean $\backslash langle\; \backslash alpha\_\{ij\}\backslash rangle\; =\; \backslash mu\_\backslash alpha/S$, variance $\backslash mathrm\{var\}(\backslash alpha\_\{ij\})\; =\; \backslash sigma\_\backslash alpha^2/S$, and correlations $\backslash mathrm\{corr\}(\backslash alpha\_\{ij\},\; \backslash alpha\_\{ji\})\; =\backslash gamma$ for $i\; \backslash neq\; j$ where $-1\backslash leq\; \backslash gamma\; \backslash leq\; 1$. The interaction matrix, $\backslash alpha$, may be parameterized as,$$\backslash alpha\_\{ij\}\; =\; \backslash frac\{\backslash mu\_\backslash alpha\}\{S\}\; +\; \backslash frac\{\backslash sigma\_\backslash alpha\}\{\backslash sqrt\{S\}\}\; a\_\{ij\},$$where $a\_\{ij\}$ are standard random variables (i.e., zero mean and unit variance) with $\backslash langle\; a\_\{ij\}\; a\_\{ji\}\backslash rangle\; =\; \backslash gamma$ for $i\; \backslash neq\; j$. The matrix entries may have any distribution with common finite first and second moments and will yield identical results in the large $S$ limit due to the central limit theorem. The carrying capacities may also be treated as random variables with $\backslash langle\; K\_i\; \backslash rangle\; =\; K,\backslash ,\backslash operatorname\{var\}(K\_i)\; =\backslash sigma\_K^2.$ Analyses by statistical physics-inspired methods have revealed phase transitions between different qualitative behaviors of the model in the many-species limit. In some cases, this may include transitions between the existence of a unique globally-attractive fixed point and chaotic, persistent fluctuations.

In the thermodynamic 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 the cavity method while assuming the system is self-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 additional mean-field species $i\; =\; 0$ is introduced and the response of the system is approximated linearly.

The cavity calculation yields a self-consistent equation describing the distribution of species abundances as a mean-field random variable, $N\_0$. When $\backslash sigma\_K\; =0$, the mean-field equation is,$$$$

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 $$

m = \langle N_0\rangle ,\,q=\langle N_0^2\rangle, \,\chi = \langle \partial N_0/\partial K_0\rangle

, and $$

Z \sim \mathcal{N}(0,1)

is a standard normal random variable. Only ecologically uninvadable solutions are taken (i.e., the largest solution for $$

N_0

in the quadratic equation is selected). The relevant susceptibility and moments of $$

N_0

, which has a truncated normal distribution, are determined self-consistently.

In the thermodynamic limit where there is an asymptotically large number of species (i.e., $S\; \backslash to\; \backslash infty$), there are three distinct phases: one in which there is a unique fixed point (UFP), another with a multiple attractors (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.

The transition from the UFP to MA phase is signaled by the cavity solution becoming unstable to disordered perturbations. When $$

\sigma_K = 0

, the phase transition boundary occurs when the parameters satisfy,$$$$

\sigma_\alpha = \frac{\sqrt{2}}{1+\gamma}.

In the $\backslash sigma\_K\; >\; 0$ 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 $$

\langle N_0 \rangle

as computed in the cavity calculation.

The cavity method can also be used to derive a dynamical mean-field theory model for the dynamics. The cavity calculation yields a self-consistent equation describing the dynamics as a Gaussian process defined by the self-consistent equation (for $\backslash sigma\_K\; =\; 0$),$$\backslash frac\{\backslash mathrm\; dN\_0\}\{\backslash mathrm\; d\; t\}$$

=

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 dt'\, \chi(t,t') N_0(t')

\right],where $m(t)\; =\; \backslash langle\; N\_0(t)\backslash rangle$, $\backslash eta$ is a zero-mean Gaussian process with autocorrelation $\backslash langle\; \backslash eta(t)\backslash eta(t\text{'})\backslash rangle\; =\; \backslash langle\; N\_0(t)N\_0(t\text{'})\backslash rangle$, and $\backslash chi(t,t\text{'})\; =\; \backslash langle\; \backslash left.\backslash delta\; N\_0(t)/\backslash delta\; K\_0(t\text{'})\backslash right|\_\{K\_0(t\text{'})\; =\; K\_0\}\; \backslash rangle$ is the dynamical susceptibility defined in terms of a functional 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, $C\_N(t,t+t\backslash tau)\; \backslash to\; f(\backslash tau)$ when $t$ is large, where $C\_N(t,t\text{'})\; =\; \backslash langle\; N(t)N(t\text{'})\backslash rangle$ is the autocorrelation function of the dynamics and $f(\backslash tau)$ is a common scaling collapse function.{{Cite journal |last=Arnoulx de Pirey |first=Thibaut |last2=Bunin |first2=Guy |date=2024-03-05 |title=Many-Species Ecological Fluctuations as a Jump Process from the Brink of Extinction |url=https://link.aps.org/doi/10.1103/PhysRevX.14.011037 |journal=Physical Review X |volume=14 |issue=1 |pages=011037 |doi=10.1103/PhysRevX.14.011037|arxiv=2306.13634 }}

When a small immigration rate $\backslash lambda\; \backslash ll\; 1$ is added (i.e., a small constant is added to the right-hand side of the equations of motion) the dynamics reach a time transitionally invariant state. In this case, the dynamics exhibit jumps between $O(1)$ and $O(\backslash lambda)$ abundances.{{Cite journal |last=Arnoulx de Pirey |first=Thibaut |last2=Bunin |first2=Guy |date=2024-03-05 |title=Many-Species Ecological Fluctuations as a Jump Process from the Brink of Extinction |url=https://link.aps.org/doi/10.1103/PhysRevX.14.011037 |journal=Physical Review X |volume=14 |issue=1 |pages=011037 |doi=10.1103/PhysRevX.14.011037|arxiv=2306.13634 }}

• 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

{{reflist}}

• 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.

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