Stability of ecosystems enhanced by species-interaction constraints

Open Access

Stability of ecosystems enhanced by species-interaction constraints

Susanne Pettersson, Van M. Savage, and Martin Nilsson Jacobi

Phys. Rev. E 102, 062405 – Published 3 December 2020

Abstract

Ecosystem stability is a central question both in theoretical and applied biology. Dynamical systems theory can be used to analyze how growth rates, carrying capacities, and patterns of species interactions affect the stability of an ecosystem. The response to increasing complexity has been extensively studied and the general conclusion is that there is a limit. While there is a complexity limit to stability at which global destabilisation occurs, the collapse rarely happens suddenly if a system is fully viable (no species is extinct). In fact, when complexity is successively increased, we find that the generic response is to go through multiple single-species extinctions before a global collapse. In this paper we demonstrate this finding via both numerical simulations and elaborations of theoretical predictions. We explore more biological interaction patterns, and, perhaps most importantly, we show that constrained interaction structures—a constant row sum in the interaction matrix—prevent extinctions from occurring. This makes an ecosystem more robust in terms of allowed complexity, but it also means singles-species extinctions do not precede or signal collapse—a drastically different behavior compared to the generic and commonly assumed case. We further argue that this constrained interaction structure—limiting the total interactions for each species—is biologically plausible.

Received 4 April 2020
Accepted 22 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.062405

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bifurcations Complex systems Dynamics of networks Ecological population dynamics Emergent biological functions from complex interactions Network stability Network structure

Physics of Living SystemsNetworksStatistical Physics & ThermodynamicsInterdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Susanne Pettersson¹, Van M. Savage², and Martin Nilsson Jacobi¹

¹Department of Space, Earth and Environment, Chalmers University of Technology, Maskingränd 2, 412 58 Gothenburg, Sweden
²Department of Ecology and Evolutionary Biology, Department of Biomathematics, UCLA, Los Angeles, California 90095, USA

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 102, Iss. 6 — December 2020

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Species abundances vs standard deviation of interaction strengths. Example simulation of a system with initial biodiversity, $N = 100$ , connectance $c = 0.5$ , $r_{i} = K_{i} = 1$ , and $μ = 0$ in the interspecific interaction strength distribution. The plot shows species abundances at locally stable fixed-points for varying standard deviation of interaction strengths, $σ$ . The first extinction event and collapse are indicated by blue lines, and the dashed blue line indicates $σ_{M}$ . Prior to the first extinction, the system has feasible fixed points where all $N = 100$ species are nonextinct. The collapse, where no similar stable fixed point exists, occurs for larger $σ$ and the stability boundary introduced by May is in between. The insets show the spectrum of May's Jacobian, $J_{M}$ , at three points of interest: (1) first extinction, (2) $σ_{M}$ , and (3) loss of stability, with the circle indicating the radius of stability. Note how $σ_{M}$ both overestimates the first extinction event and underestimates collapse.
Reuse & Permissions
Figure 2
Extinctions or destabilization first? The top plot shows simulation averages of the mean value (with one-standard-deviation error bars) of the variation in interaction strength, $σ$ , at which first extinction occurs. The simulation averages are from “Random” systems with interaction strengths from a normal distribution $A_{i j} \sim N (0, 1)$ , predator/prey systems where $sgn (A_{i j}) = - sgn (A_{j i})$ ( $ρ = - 2 / π$ ), mutualistic/competitive systems with $sgn (A_{i j}) = sgn (A_{j i})$ ( $ρ = 2 / π$ ), and random systems with different means of the normal distribution $A_{i j} \sim N (μ = \pm 0.1, 1)$ . All simulations were implemented with $r_{i} = k_{i} = 1$ . The dashed lines are theoretical first extinction boundaries, $σ_{f}$ . The dash-dotted lines are May's stability boundaries, $σ_{M}$ , color-coded according to the legend (some colors are “missing” since they coincide; for example all first extinction boundaries coincide and collapse boundaries for systems with varying interaction strength mean, for these only “Random” is visible). May's stability boundary is given by $σ_{M; μ, ρ} = 1 / {\sqrt{c N [1 + (1 - c) μ^{2}]} (1 + ρ) - c μ}$ [11]. Note the spread in $σ_{M}$ compared to $σ_{f}$ , which is equal for all systems. The bottom panels show an example simulation of each type, with the same color coding. The vertical lines indicate $σ_{f}$ , $σ_{M}$ , and the actual loss of stability $σ_{c}$ (operational collapse definition), in order of increasing $σ$ . Three phases are marked by the different shades: strict stability (SS) before $σ_{f}$ , extinction continuum (EC), a phase of single-species extinctions, and collapse (C) where no stable nearby fixed-point exists. Note that $σ_{M}$ always falls in the extinction continuum for every type of system.
Reuse & Permissions
Figure 3
Variability of the extinction continuum. The left plot shows averages from simulations for first extinction event, with one standard deviation error bars, when increasing row-sum constraint $(1 - ξ)$ . Theoretical predictions of first extinction, $σ_{f}$ , are shown as solid lines and $σ_{M}$ as dashed lines for systems of size $N = 100$ and $N = 160$ . The decreasing width for small $ξ$ and convergence to $σ_{M}$ is clearly seen for both system sizes. The right panels show example simulations of the two collapse types for systems with $(ξ = 0.01)$ . The behavioral phases have different shades of grey: strict stability (SS), extinction continuum (EC), and collapse (C) where no stable nearby fixed-point exists. The solid vertical line indicates the first extinction prediction $σ_{f}$ and $σ_{M}$ , which coincide, and the dashed vertical line indicates collapse $σ_{c}$ . Note that collapse type 1 does not have an extinction continuum. Collapse type 2 abruptly enters the extinction continuum at $σ_{f} / σ_{M}$ .
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics