Energy flow controls the stability of multitrophic ecosystems with stratified nonreciprocity

Abstract

Complex systems with nonreciprocal interactions are often stratified into layers. Ecosystems are a prime example, where species at one trophic level grow by consuming those at another. Yet the dynamical consequences of such stratified nonreciprocity -- where the correlation between growth and consumption differs across trophic levels -- remain unexplored. Here, using an ecological model with three trophic levels, we reveal an emergent asymmetry: nonreciprocal interactions between consumers and predators (top and middle level) destabilize ecosystems far more readily than nonreciprocity between consumers and resources (middle and bottom level). We analytically derive the phase diagram for the model and show that its stability boundary is controlled by energy flow across trophic levels. Because energy flows upward -- from resources to predators -- diversity is progressively lower at higher trophic levels, which we show explains the asymmetry. Lowering energy flow efficiency flips the asymmetry toward resources and remarkably expands the stable region of the phase diagram, suggesting that the famous "10% energy transfer" seen in natural ecosystems might promote stability. More broadly, our findings show that the location of nonreciprocity within a complex network, not merely its magnitude, determines stability.

Figures (3)

• Figure 1: Stratified nonreciprocity in a multitrophic ecosystem. (a) Schematic of an ecosystem with three trophic levels: resources (squares), consumers (ellipses), and predators (hexagons). The ecosystem has two distinct reciprocity parameters: predator reciprocity $\rho_X$ and resource reciprocity $\rho_R$. (b) At the consumer-predator interface, growth benefits $d_{\alpha j}$ (blue) and predation impacts $e_{\alpha j}$ (red) are correlated with strength $\rho_X = \text{corr}(d, e)$. (c) At the resource interface, growth benefits $c_{ip}$ (blue) and resource depletion $f_{ip}$ (red) are correlated with strength $\rho_R = \text{corr}(c, f)$.
• Figure 2: Ecosystem stability relies more on reciprocal interactions with predators than resources. (a) Phase diagram in the $(\rho_X, \rho_R)$ plane showing stable (white), unstable (purple), and infeasible (light blue) regions. The analytically-derived stability boundary (black curve) is asymmetric: reducing $\rho_X$ leads to instability earlier than reducing $\rho_R$. Green square and yellow star mark points equidistant from the reciprocal corner. Heatmap shows probability of reaching steady state from numerical simulations. Gray indicates infeasible region where the cavity solutions admit no nontrivial solution; the infeasibility boundary, obtained analytically, is next to it in black (Appendix D). (b) Dynamics at the reciprocal point ($\rho_X = 1$, $\rho_R = 1$; circle): all three trophic layers (predators, consumers, producers) reach stable steady states. (c) Dynamics with reduced resource reciprocity ($\rho_X = 1$, $\rho_R = 0.5$, star): the ecosystem remains stable despite nonreciprocal consumer-resource interactions. (d) Dynamics with reduced predator reciprocity ($\rho_X = 0.5$, $\rho_R = 1$, square): the ecosystem is unstable, exhibiting chaotic fluctuations across all trophic levels. The asymmetry between (c) and (d) demonstrates inherent top-down control of stability.
• Figure 3: Energy flow efficiency controls the stability boundary. (a) Schematic of energy flow across an ecosystem: a factor $\eta$ of the energy at one trophic level is transferred to the next higher level. (b) Ratio of critical reciprocities $\rho_X^\dagger/\rho_R^\dagger$ as a function of $\eta$. When the ratio exceeds 1 (pink), stability is top-down controlled; when below 1 (green), bottom-up controlled. (c) The ecosystem stability boundary rotates as energy flow efficiency $\eta$ decreases. At high efficiency ($\eta = 1$, red), the boundary is steep; at low efficiency ($\eta = 0.6$, green), the boundary is shallow. Shown are analytic solutions. (d) Total area of the stable region increases as $\eta$ decreases, showing that energy dissipation is stabilizing.