Gardner volumes and self-organization in a minimal model of complex ecosystems

TL;DR

The paper addresses how large random ecosystems self-organize under a minimally nonlinear, piecewise-linear dynamic with extinction and revival. By decomposing the dynamics into angular (species composition) and radial (total population) parts, it shows that the angular motion is confined to time-varying Gardner volumes, with a critical diversity $\gamma_c=1/2$ independent of reciprocity $\xi$, and that the radial growth follows May-type spectral constraints as diversity changes. Finite event sequences lead to equilibria with leading eigenvectors localized to compact Gardner volumes, while infinite sequences can yield periodic orbits near the critical diversity; a nonlinear extension introduces a saturated attractor when the origin becomes unstable. The results bridge ecological self-organization with neural-network theory, providing analytic insight into stability transitions and the emergence of targeted, localized interaction structures that sustain ecological communities.

Abstract

We study self-organization in a minimally nonlinear model of large random ecosystems. Populations evolve over time according to a piecewise linear system of ordinary differential equations subject to a non-negativity constraint resulting in discrete time extinction and revival events. The dynamics are generated by a random elliptic community matrix with tunable correlation strength. We show that, independent of the correlation strength, solutions of the system are confined to subsets of the phase space that can be cast as time-varying Gardner volumes from the theory of learning in neural networks. These volumes decrease with the diversity (i.e. the fraction of extant species) and become exponentially small in the long-time limit. Using standard results from random matrix theory, the changing diversity is then linked to a sequence of contractions and expansions in the spectrum of the community matrix over time, resulting in a sequence of May-type stability problems determining whether the total population evolves toward complete extinction or unbounded growth. In the case of unbounded growth, we show the model allows for a particularly simple nonlinear extension in which the solutions instead evolve towards a new attractor.

Figures (10)

• Figure 1: Numerical solutions $\bm{x}(t)$ over time for two realizations (a. and b.) of the system \ref{['eq:MNL']} with $d=10$, each with two components $x_1(t),x_2(t)$ highlighted in black. The other components $x_i(t),i>2$ in grey. In a. the first component is initially extinct at time $t_0 = 0$ and revived immediately (red dot), while the second, initially extant component, goes extinct at time $t>5$ (blue dot). In b. the first initially extant component is driven extinct and subsequently revived at a later time. The second component simply remains active for all time. The index set \ref{['eq:Ix']} changes after each event, as extinct species are removed and revived species added. Numerical integration of \ref{['eq:MNL']} is performed using the explicit Euler method. Extinction events are detected by a positive component becoming non-positive in the next time-step. They are then set to zero. Revival events are detected by whether a component at zero becomes positive in the next time-step.
• Figure 2: Sketch of the polar decomposition in $\mathbb{R}^2$. The angular variable $\bm{y}$ is the radial projection of $\bm{x}$ through a line through the origin. The projected dynamics depend only on the ordering of the real parts of the eigenvalue. They are independent of the decay rate $\beta$ and the scaling $\alpha$ can be removed by a rescaling of time. The trajectory of the full system \ref{['eq:MNL']} is unbounded. The corresponding trajectory for the (compactified) angular dynamics tends to an equilibrium.
• Figure 3: Sketch of the decreasing Gardner volumes \ref{['eq:V']} for two values of the diversity $\gamma_n$ and $\gamma_N$ with $\gamma_n > \gamma_N$. a. A solution $\bm{y}(t)$ moves on the set $\tilde{S}_n$ containing the active species $\{i_1,i_2,i_3\} \subset I_n$. The red-dot depicts a revival event where the solution hits any hyperplane $\{ \bm{x} :\bm{k}^\top_i \bm{x} = 0\}$ for $i\in I^c_n$, whose intersections with the sphere are depicted as black lines. b. A solution moves on the set $\tilde{S}_N$ at lower diversity resulting in a smaller Gardner volume. The subset satisfies the feasibility conditions \ref{['eq:feasibility']} and the solution converges to the stable equilibrium $\bm{p}_N$ without further extinction or revival events.
• Figure 4: Minimal fraction of active species $\gamma_n$ over the course all extinction and revival events for $d = 5000$ averaged over 70 realizations for each value of the correlation strength $\xi = -1,-0.5,0,0.5,1$. The black curve corresponds to the event sequence of the angular dynamics \ref{['eq:angular']}. Solutions which penetrate beyond the critical threshold $\gamma_c = 1/2$ are atypical. The dashed curve corresponds to the model without revival events, see equation \ref{['eq:MNL_absorbing']}. In this case solutions freely move below the critical threshold. Inset shows the empirical probability that an event sequence is infinite for $d=200$. For each value of the correlation strength $\xi$, we average over 200 realizations of the system and count the number of event sequences detected as infinite. For $\xi=+1$, all sequences are finite (see Section \ref{['sec:angular_finite']}), and for $\xi=-1$ all sequences are infinite (see Section \ref{['sec:angular_infinite']}). An event sequence is detected as infinite, if the time between events $t_{n+1} -t_n$ does not exceed the threshold $500$ and the distance between solution and a leading eigenvector does not drop below $10^{-10}$.
• Figure 5: The quenched average $\Phi(\gamma_n)$, \ref{['eq:limlogV']} determined by numerically solving equation \ref{['appeq:gardner_entropy']}. As the diversity decreases, the fractional volume decreases monotonically. In the limit $\gamma_n \to \gamma_c = 1/2$ of the critical threshold, $\Phi(\gamma_n)$ diverges to $-\infty$. Below the critical threshold, the fractional volume is typically vanishing. Inset shows the typical overlap \ref{['eq:overlap']} as function of the fraction of active species, numerically solving equation \ref{['appeq:qimplicit']}. As the number of active species decreases, the overlap monotonically increases towards the limit $q=1$. The corresponding typical distance between solutions \ref{['eq:qdist']} plotted in blue. Validity of equation \ref{['appeq:qimplicit']} for the overlap is limit to the regime of large overlaps, but remains in good agreement with also in the case $\gamma_n = 1$ of minimal overlap, see equation \ref{['appeq:q_noconstraints']} in Appendix \ref{['sec:gardner']}.