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How combined pairwise and higher-order interactions shape transient dynamics

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Chaos
Authors:
Sourin Chatterjee at Aix-Marseille University
Sourin Chatterjee
Sayantan Nag Chowdhury at Constructor University Bremen gGmbH
Sayantan Nag Chowdhury
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Abstract and Figures

Understanding how species interactions shape biodiversity is a core challenge in ecology. While much focus has been on long-term stability, there is rising interest in transient dynamics—the short-lived periods when ecosystems respond to disturbances and adjust toward stability. These transitions are crucial for predicting ecosystem reactions and guiding effective conservation. Our study introduces a model that uses convex combinations to blend pairwise and higher-order interactions (HOIs), offering a more realistic view of natural ecosystems. We find that pairwise interactions slow the journey to stability, while HOIs speed it up. Employing global stability analysis and numerical simulations, we establish that as the proportion of HOIs increases, mean transient times exhibit a significant reduction, thereby underscoring the essential role of HOIs in enhancing biodiversity stabilization. Our results reveal a robust correlation between the most negative real part of the eigenvalues of the Jacobian matrix associated with the linearized system at the coexistence equilibrium and the mean transient times. This indicates that a more negative leading eigenvalue correlates with accelerated convergence to stable coexistence abundances. This insight is vital for comprehending ecosystem resilience and recovery, emphasizing the key role of HOIs in promoting stabilization. Amid growing interest in transient dynamics and its implications for biodiversity and ecological stability, our study enhances the understanding of how species interactions affect both transient and long-term ecosystem behavior. By addressing a critical gap in ecological theory and offering a practical framework for ecosystem management, our work advances knowledge of transient dynamics, ultimately informing effective conservation strategies.
The impact of pairwise interactions ( γ 1) and higher-order interactions ( γ 2) on species coexistence. Dynamics of model systems with N = 3 for (a) pairwise ( γ 1 = 1.0 ), (b) non-pairwise ( γ 2 = 1.0 ), and (c) mixed system ( γ 1 ≠ 0, γ 2 ≠ 0, γ 1 + γ 2 = 1.0 ). The initial condition ( 0.45 , 0.3 , 0.25 ) is chosen randomly within the ( 0 , 1 ) × ( 0 , 1 ) × ( 0 , 1 ) maintaining the constraint x 1 ( 0 ) + x 2 ( 0 ) + x 3 ( 0 ) = 1.0. As shown in subfigure (a), the pairwise interaction system does not converge to a steady state. However, introducing higher-order interactions significantly alters this behavior. Interestingly, the system with both pairwise and higher-order interactions converges more slowly compared to the system with only higher-order interactions, although both ultimately reach the same equilibrium point ( 1 3 , 1 3 , 1 3 ). γ 1 = γ 2 = 0.5 is set for subfigure (c).
The impact of pairwise interactions ( γ 1) and higher-order interactions ( γ 2) on species coexistence. Dynamics of model systems with N = 3 for (a) pairwise ( γ 1 = 1.0 ), (b) non-pairwise ( γ 2 = 1.0 ), and (c) mixed system ( γ 1 ≠ 0, γ 2 ≠ 0, γ 1 + γ 2 = 1.0 ). The initial condition ( 0.45 , 0.3 , 0.25 ) is chosen randomly within the ( 0 , 1 ) × ( 0 , 1 ) × ( 0 , 1 ) maintaining the constraint x 1 ( 0 ) + x 2 ( 0 ) + x 3 ( 0 ) = 1.0. As shown in subfigure (a), the pairwise interaction system does not converge to a steady state. However, introducing higher-order interactions significantly alters this behavior. Interestingly, the system with both pairwise and higher-order interactions converges more slowly compared to the system with only higher-order interactions, although both ultimately reach the same equilibrium point ( 1 3 , 1 3 , 1 3 ). γ 1 = γ 2 = 0.5 is set for subfigure (c).
… 
The influence of varying proportions of pairwise interactions ( γ 1) and higher-order interactions ( γ 2) on species dynamics. Dynamics of mixed systems with N = 3 for different values of γ 1 and γ 2: (a) γ 1 = 0.7 , γ 2 = 0.3, (b) γ 1 = 0.5 , γ 2 = 0.5, (c) γ 1 = 0.3 , γ 2 = 0.7. Starting from the initial condition of ( 0.45 , 0.3 , 0.25 ), as in Fig. 1, the system consistently converges to the coexistence equilibrium point, regardless of the relative contributions of pairwise and higher-order interactions. However, as the proportion of pairwise interactions decreases (i.e., as γ 1 diminishes), the transient time significantly shortens. This demonstrates that higher-order interactions play a crucial role in accelerating convergence toward equilibrium.
The influence of varying proportions of pairwise interactions ( γ 1) and higher-order interactions ( γ 2) on species dynamics. Dynamics of mixed systems with N = 3 for different values of γ 1 and γ 2: (a) γ 1 = 0.7 , γ 2 = 0.3, (b) γ 1 = 0.5 , γ 2 = 0.5, (c) γ 1 = 0.3 , γ 2 = 0.7. Starting from the initial condition of ( 0.45 , 0.3 , 0.25 ), as in Fig. 1, the system consistently converges to the coexistence equilibrium point, regardless of the relative contributions of pairwise and higher-order interactions. However, as the proportion of pairwise interactions decreases (i.e., as γ 1 diminishes), the transient time significantly shortens. This demonstrates that higher-order interactions play a crucial role in accelerating convergence toward equilibrium.
… 
Reduction of transient time with increasing higher-order interactions ( γ 2). The boxplot illustrates how transient times vary with different values of γ 2 across various initial conditions. For the considered interaction matrix A, we use ecologically relevant initial conditions and plot the transient times across varying levels of higher-order interactions. For a fixed value of γ 2, we select 20 distinct initial conditions and determine the transient times required to reach the coexistence equilibrium. We then calculate the arithmetic mean of these transient times, represented by the red dots, and the median, indicated by the yellow line. As γ 2 increases from 0.05 (indicating a minimal presence of higher-order interactions) to 1.0 (representing a system solely dominated by higher-order interactions), the transient times steadily decrease. While the higher values of γ 2 may appear to have minimal variation in transient times due to the large range of the y axis, the inset provides a clearer view, showing that transient times with different initial conditions are actually well-distributed across this range.
Reduction of transient time with increasing higher-order interactions ( γ 2). The boxplot illustrates how transient times vary with different values of γ 2 across various initial conditions. For the considered interaction matrix A, we use ecologically relevant initial conditions and plot the transient times across varying levels of higher-order interactions. For a fixed value of γ 2, we select 20 distinct initial conditions and determine the transient times required to reach the coexistence equilibrium. We then calculate the arithmetic mean of these transient times, represented by the red dots, and the median, indicated by the yellow line. As γ 2 increases from 0.05 (indicating a minimal presence of higher-order interactions) to 1.0 (representing a system solely dominated by higher-order interactions), the transient times steadily decrease. While the higher values of γ 2 may appear to have minimal variation in transient times due to the large range of the y axis, the inset provides a clearer view, showing that transient times with different initial conditions are actually well-distributed across this range.
… 
Variation of (a) mean transient time and (b) the real part of the leading eigenvalue with respect to γ 2. (a) Mean transient time plotted against the proportion of higher-order interactions γ 2, with a rectangular hyperbola fit (red dashed line) illustrating how transient time decreases as γ 2 increases. (b) The most negative real part of the eigenvalues of the Jacobian matrix associated with the linearized system at the coexistence equilibrium plotted against γ 2 (blue dots), with a linear fit (red dashed line). The negative real part of the leading eigenvalue in (b) provides a direct measure of convergence speed. The correlation between the most negative real part of the leading eigenvalue and the mean transient time illustrates that greater negativity in the leading eigenvalue is associated with faster convergence to the same coexistence equilibrium point. This highlights the role of higher-order interactions in accelerating system stabilization and reducing transient oscillations.
Variation of (a) mean transient time and (b) the real part of the leading eigenvalue with respect to γ 2. (a) Mean transient time plotted against the proportion of higher-order interactions γ 2, with a rectangular hyperbola fit (red dashed line) illustrating how transient time decreases as γ 2 increases. (b) The most negative real part of the eigenvalues of the Jacobian matrix associated with the linearized system at the coexistence equilibrium plotted against γ 2 (blue dots), with a linear fit (red dashed line). The negative real part of the leading eigenvalue in (b) provides a direct measure of convergence speed. The correlation between the most negative real part of the leading eigenvalue and the mean transient time illustrates that greater negativity in the leading eigenvalue is associated with faster convergence to the same coexistence equilibrium point. This highlights the role of higher-order interactions in accelerating system stabilization and reducing transient oscillations.
… 
The influence of higher-order interactions ( γ 2) on transient times across different interaction matrices ( A). In this analysis, we explore how different interaction structures affect the transient times at varying levels of higher-order interactions. The boxplot shows the variation in transient times for different values of γ 2 across 20 randomly generated interaction matrices A, with a fixed initial condition ( 0.15 , 0.35 , 0.5 ). The red dots represent mean transient times, while the yellow line marks the median. As γ 2 increases from 0.05 (minimal higher-order interactions) to 1.0 (solely higher-order interactions), transient times decrease, demonstrating the structural robustness of the system in reaching equilibrium. Notably, the coexistence equilibrium remains unchanged across varying A, emphasizing that the choice of the particular circulant matrix preserves the coexistence equilibrium and higher-order interactions accelerate convergence to the ecological stability.
+1
The influence of higher-order interactions ( γ 2) on transient times across different interaction matrices ( A). In this analysis, we explore how different interaction structures affect the transient times at varying levels of higher-order interactions. The boxplot shows the variation in transient times for different values of γ 2 across 20 randomly generated interaction matrices A, with a fixed initial condition ( 0.15 , 0.35 , 0.5 ). The red dots represent mean transient times, while the yellow line marks the median. As γ 2 increases from 0.05 (minimal higher-order interactions) to 1.0 (solely higher-order interactions), transient times decrease, demonstrating the structural robustness of the system in reaching equilibrium. Notably, the coexistence equilibrium remains unchanged across varying A, emphasizing that the choice of the particular circulant matrix preserves the coexistence equilibrium and higher-order interactions accelerate convergence to the ecological stability.
… 
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Chaos ARTICLE pubs.aip.org/aip/cha
How combined pairwise and higher-order
interactions shape transient dynamics
Cite as: Chaos 34, 101102 (2024); doi: 10.1063/5.0238827
Submitted: 14 September 2024 ·Accepted: 29 September 2024 ·
Published Online: 16 October 2024
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Sourin Chatterjee1,2,a)and Sayantan Nag Chowdhury3,4,5,b)
AFFILIATIONS
1Department of Mathematics and Statistics, Indian Institute of Science Education and Research, Kolkata,
West Bengal 741246, India
2Institut de Neurosciences des Systèmes (INS), UMR1106, Aix-Marseille Université, Marseilles, France
3School of Science, Constructor University, 28759 Bremen, Germany
4Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
5Department of Environmental Science and Policy, University of California, Davis, Davis, California 95616, USA
a)Electronic mail: sourin.CHATTERJEE@univ-amu.fr
b)Author to whom correspondence should be addressed: jcjeetchowdhury1@gmail.com
ABSTRACT
Understanding how species interactions shape biodiversity is a core challenge in ecology. While much focus has been on long-term stability,
there is rising interest in transient dynamics—the short-lived periods when ecosystems respond to disturbances and adjust toward stability.
These transitions are crucial for predicting ecosystem reactions and guiding effective conservation. Our study introduces a model that uses
convex combinations to blend pairwise and higher-order interactions (HOIs), offering a more realistic view of natural ecosystems. We find
that pairwise interactions slow the journey to stability, while HOIs speed it up. Employing global stability analysis and numerical simulations,
we establish that as the proportion of HOIs increases, mean transient times exhibit a significant reduction, thereby underscoring the essen-
tial role of HOIs in enhancing biodiversity stabilization. Our results reveal a robust correlation between the most negative real part of the
eigenvalues of the Jacobian matrix associated with the linearized system at the coexistence equilibrium and the mean transient times. This
indicates that a more negative leading eigenvalue correlates with accelerated convergence to stable coexistence abundances. This insight is
vital for comprehending ecosystem resilience and recovery, emphasizing the key role of HOIs in promoting stabilization. Amid growing inter-
est in transient dynamics and its implications for biodiversity and ecological stability, our study enhances the understanding of how species
interactions affect both transient and long-term ecosystem behavior. By addressing a critical gap in ecological theory and offering a practical
framework for ecosystem management, our work advances knowledge of transient dynamics, ultimately informing effective conservation
strategies.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0238827
Our study offers a fresh perspective on how species interactions
influence transient dynamics in ecological systems by integrating
both pairwise and higher-order interactions (HOIs) into a uni-
fied framework. While much of the existing research has focused
on these interactions in isolation, our model reveals that the
inclusion of higher-order interactions accelerates convergence
toward stable coexistence, in contrast to the prolonged transient
phases caused by pairwise interactions alone. This insight is cru-
cial for understanding ecosystem resilience and recovery, as it
highlights the importance of higher-order interactions in stabi-
lizing biodiversity. By applying global stability analysis and linear
algebraic theories along with numerical simulations, we show that
increasing the proportion of HOIs leads to quicker convergence
to coexistence equilibrium, underscoring their critical role in
ecosystem resilience. Given the rising interest in transient dynam-
ics and its implications for biodiversity and ecological stability,
our findings deepen the understanding of how species interac-
tions influence both transient and long-term ecosystem behavior.
Our work not only bridges a significant gap in ecological theory
by offering a more realistic representation of natural ecosystems
but also provides a practical framework for managing ecosystems
by advancing our understanding of transient dynamics.
Chaos 34, 101102 (2024); doi: 10.1063/5.0238827 34, 101102-1
Published under an exclusive license by AIP Publishing
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