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On the analysis of a higher-order Lotka-Volterra model: an application of S-tensors and the polynomial complementarity problem

Shaoxuan Cui, Qi Zhao, Guofeng Zhang, Hildeberto Jardón-Kojakhmetov, Ming Cao

TL;DR

The paper addresses the insufficiency of pairwise, additive interactions in classical Lotka-Volterra models by introducing higher-order interactions (HOIs) in a general HO-LV framework, and uses $\\mathcal{S}$-, $\\mathcal{M}$-, and $\\mathcal{H}^{+}$-tensors along with the polynomial complementarity problem (PCP) to analyze equilibria and stability. It proves existence for positive equilibria in non-homogeneous polynomial systems, provides bounds for PCP solutions, and derives existence, uniqueness, and stability results for cooperative, two-faction, and purely competitive regimes. The approach combines tensor algebra with monotone-system theory and PCP to extend classical LV theory to HOIs. Numerical experiments illustrate how HOIs can alter coexistence, promote or hinder persistence, and demonstrate multi-stability in certain regimes. The results offer a rigorous mathematical framework for HOIs in ecological networks and broader higher-order dynamical systems.

Abstract

It is known that the effect of species' density on species' growth is non-additive in real ecological systems. This challenges the conventional Lotka-Volterra model, where the interactions are always pairwise and their effects are additive. To address this challenge, we introduce HOIs (Higher-Order Interactions) which are able to capture, for example, the indirect effect of one species on a second one correlating to a third species. Towards this end, we propose a general higher-order Lotka-Volterra model. We provide an existence result of a positive equilibrium for a non-homogeneous polynomial equation system with the help of S-tensors. Afterward, by utilizing the latter result, as well as the theory of monotone systems and results from the polynomial complementarity problem, we provide comprehensive results regarding the existence, uniqueness, and stability of the corresponding equilibrium. These results can be regarded as natural extensions of many analogous ones for the classical Lotka-Volterra model, especially in the case of full cooperation, competition among two factions, and pure competition. Finally, illustrative numerical examples are provided to highlight our contributions.

On the analysis of a higher-order Lotka-Volterra model: an application of S-tensors and the polynomial complementarity problem

TL;DR

The paper addresses the insufficiency of pairwise, additive interactions in classical Lotka-Volterra models by introducing higher-order interactions (HOIs) in a general HO-LV framework, and uses -, -, and -tensors along with the polynomial complementarity problem (PCP) to analyze equilibria and stability. It proves existence for positive equilibria in non-homogeneous polynomial systems, provides bounds for PCP solutions, and derives existence, uniqueness, and stability results for cooperative, two-faction, and purely competitive regimes. The approach combines tensor algebra with monotone-system theory and PCP to extend classical LV theory to HOIs. Numerical experiments illustrate how HOIs can alter coexistence, promote or hinder persistence, and demonstrate multi-stability in certain regimes. The results offer a rigorous mathematical framework for HOIs in ecological networks and broader higher-order dynamical systems.

Abstract

It is known that the effect of species' density on species' growth is non-additive in real ecological systems. This challenges the conventional Lotka-Volterra model, where the interactions are always pairwise and their effects are additive. To address this challenge, we introduce HOIs (Higher-Order Interactions) which are able to capture, for example, the indirect effect of one species on a second one correlating to a third species. Towards this end, we propose a general higher-order Lotka-Volterra model. We provide an existence result of a positive equilibrium for a non-homogeneous polynomial equation system with the help of S-tensors. Afterward, by utilizing the latter result, as well as the theory of monotone systems and results from the polynomial complementarity problem, we provide comprehensive results regarding the existence, uniqueness, and stability of the corresponding equilibrium. These results can be regarded as natural extensions of many analogous ones for the classical Lotka-Volterra model, especially in the case of full cooperation, competition among two factions, and pure competition. Finally, illustrative numerical examples are provided to highlight our contributions.
Paper Structure (12 sections, 38 theorems, 56 equations, 3 figures)

This paper contains 12 sections, 38 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Tensor-based approach
  3. Tensor
  4. $\mathcal{M}$-tensors and $\mathcal{H}^{+}$-tensors
  5. $\mathcal{S}$-tensors and their properties
  6. Polynomial complementarity problem
  7. General Higher-order Lotka-Volterra model
  8. Cooperative Lotka-Volterra Model
  9. Higher-order Two-faction-competition Lotka-Volterra Model
  10. Purely competitive higher-order Lotka-Volterra
  11. Numerical example
  12. Conclusion

Key Result

Lemma 1

If $A\in\mathbb{R}^{[m, n]}$ is an $\mathcal{H}^{+}$-tensor and $\langle A\rangle$ is its comparison tensor, then there exists a positive vector $x>\mathbf{0}$ such that $A x^{m-1}>\mathbf{0}$ and $\langle A\rangle x^{m-1}>\mathbf{0}$.

Figures (3)

  • Figure 1: (a) Only one faction wins and all members in the faction are the winners. (b) All species coexist. (c) Species from different factions win but some species die out. (d) From a different initial condition but the same parameters with (a), the solution converges to the different one-faction-wins-all boundary equilibrium. That is, bistability is reflected in an interchange of the winner faction.
  • Figure 2: (a) Only one species wins and the winner takes all. (b) Winners share all, but not all species are winners. (c) All species coexist.
  • Figure 3: (a) From a low-level initial condition, species 2 and 5 are winners. (b) From a high-level initial condition, species 4 and 5 are winners. The system has a bi-stability.

Theorems & Definitions (78)

  • Lemma 1: wang2019existence
  • Lemma 2: liu2022further
  • Lemma 3: wang2019existence
  • Lemma 4: Banach fixed point theorem, Lemma 3.1 wang2019existence
  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • Theorem 2
  • proof
  • ...and 68 more