Generalized Lotka-Volterra model with hierarchical interactions

Lyle Poley, Joseph W. Baron, and Tobias Galla

Phys. Rev. E 107, 024313 – Published 24 February 2023

Abstract

In the analysis of complex ecosystems it is common to use random interaction coefficients, which are often assumed to be such that all species are statistically equivalent. In this work we relax this assumption by imposing hierarchical interspecies interactions. These are incorporated into a generalized Lotka-Volterra dynamical system. In a hierarchical community species benefit more, on average, from interactions with species further below them in the hierarchy than from interactions with those above. Using dynamic mean-field theory, we demonstrate that a strong hierarchical structure is stabilizing, but that it reduces the number of species in the surviving community, as well as their abundances. Additionally, we show that increased heterogeneity in the variances of the interaction coefficients across positions in the hierarchy is destabilizing. We also comment on the structure of the surviving community and demonstrate that the abundance and probability of survival of a species are dependent on its position in the hierarchy.

Received 2 August 2022
Accepted 21 December 2022

DOI:https://doi.org/10.1103/PhysRevE.107.024313

Physics Subject Headings (PhySH)

Complex systems Phase diagrams Population dynamics

Disordered systems

Dynamical mean field theory Path-integral methods

Statistical Physics & ThermodynamicsInterdisciplinary PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Lyle Poley ^1,*, Joseph W. Baron ^2,†, and Tobias Galla^2,1,‡

¹Department of Physics and Astronomy, School of Natural Science, The University of Manchester, Manchester M13 9PL, United Kingdom
²Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC, UIB, 07122 Palma de Mallorca, Spain

^*lyle.poley@manchester.ac.uk
^†joseph-william.baron@phys.ens.fr
^‡tobias.galla@ifisc.uib-csic.es

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Issue

Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
Illustration of the block-structured interaction matrix $A_{i j}^{a b}$ . The figure shows the mean value (indicated by color) of interaction matrix elements $A_{i j}^{a b}$ in three cases. (a) General block-structured interaction matrix with $B = 4$ subcommunities. (b) Example of an interaction matrix with hierarchical statistics, again with $B = 4$ . All blocks above the diagonal blocks have the same mean and variance, and similarly for blocks below the diagonal. (c) Model with an infinite number of subcommunities ( $B \to \infty$ ) and hierarchical statistics [see Eq. (4)].
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Figure 2
Variation of average abundance $[[M]]$ and the fraction of surviving species $[[ϕ]]$ with $ν$ (the strength of the hierarchy). (a) Dependence of average abundance on $ν$ . The vertical dashed line indicates a divergence in average abundance when $γ = 1$ and $ν \approx 0.4$ . The system does not converge for $ν ≲ 0.4$ . (b) Dependence of the fraction of surviving species on $ν$ . When abundances diverge (for $γ = 1$ and $ν \leq 0.4$ ), the fraction of surviving species cannot be meaningfully calculated and so the lower curve does not extend to the left of the vertical line. Markers are data from the numerical integration of Eq. (1), with $N = B = 250$ , averaged over 40 runs, and lines are predictions from theory [Eqs. (27b) and (27a), with $u, Δ_{0}$ , and $Δ_{1}$ determined from Eq. (26)]. The model parameters used in both plots are $σ = 0.3, ρ = 1.5$ , and $μ = 1.0$ .
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Figure 3
Abundance distributions with and without hierarchical interactions. All plots are for $μ = 0.5, σ = 0.15$ , and $γ = - 0.3$ . (a) Two SADs are plotted; the taller one is for the model without hierarchy ( $ν = 0$ and $ρ = 1$ ) and is a clipped Gaussian. The flatter distribution in (a) is for $ν = 1.5$ and $ρ = \frac{2}{3}$ ; bars are from simulation. Also shown are the corresponding (b) RADs and (c) HADs. Square markers are from simulations for $ν = 0$ and $ρ = 1$ and diamond markers are from simulations for $ν = 1.5$ and $ρ = \frac{2}{3}$ . Simulations are for $N = B = 1000$ species, averaged over 200 runs. Lines are from the theory.
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Figure 4
Survival distributions with and without hierarchical interactions. The parameters used are $σ = 0.75, γ = - 0.6$ , and, from top to bottom along the left edge of the plot, $(ν, ρ) = (0, 1), (0, 3), (3, 3), (3, 1), (3, \frac{1}{3})$ . Markers are the average fraction of survivors for every 40th species in simulations with $N = B = 500$ , averaged over 400 runs.
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Figure 5
Hierarchical interactions and stability. (a) Similar to the phase diagrams in Ref. [23], we show the three possible dynamical behaviors of the system without hierarchy in the $μ − σ$ plane. We have $γ = 0.8$ ( $20 %$ predator-prey interactions), $ν = 0$ , and $ρ = 1$ . In (b) and (c) we choose the points (b) $μ = - 0.5$ and $σ = 0.7$ and (c) $μ = 0.5$ and $σ = 0.8$ and show stability in the $ν − ρ$ plane. The left circle (yellow) in (a) and the circle in (b) are the same point in parameter space ( $μ = - 0.5, ν = 0, σ = 0.7, ρ = 1$ , and $γ = 0.8$ ) and similarly for the right (red) circle in (a) and the circle at the center of panel (c) with $μ = 0.5, ν = 0, σ = 0.8, ρ = 1$ , and $γ = 0.8$ .
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Figure 6
Phase diagrams (horizontal axis shows the proportion of predator-prey pairs). The onset of instability [(a) linear instability and (b) diverging abundances] are shown as black lines for $(ν, ρ) = (0, 1)$ , as well as for $(ν, ρ) = (3, 1)$ and $(ν, ρ) = (0, 3)$ as indicated. Stability is colored as labeled for the case $(ν, ρ) = (0, 1)$ . The lengths of arrows indicate the effect of increasing the values of $ν$ or $ρ$ by one ( $ν \to ν + 1$ or $ρ \to ρ + 1$ , respectively).
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