Dynamic changes in microbial communities play an important role in human health and disease. Specifically, deciphering how microbial species in a community interact with each other and their environment can elucidate mechanisms of disease, a problem typically investigated using tools from community ecology. Yet, such methods require measurements of absolute densities, whereas typical datasets only provide estimates of relative abundances. Here, we systematically investigate models of microbial dynamics in the simplex of relative abundances. We derive a new nonlinear dynamical system for microbial dynamics, termed "compositional" Lotka-Volterra (cLV), unifying approaches using generalized Lotka-Volterra (gLV) equations from community ecology and compositional data analysis. On three real datasets, we demonstrate that cLV recapitulates interactions between relative abundances implied by gLV. Moreover, we show that cLV is as accurate as gLV in forecasting microbial trajectories in terms of relative abundances. We further compare cLV to two other models of relative abundance dynamics motivated by common assumptions in the literature-a linear model in a log-ratio transformed space, and a linear model in the space of relative abundances-and provide evidence that cLV more accurately describes community trajectories over time. Finally, we investigate when information about direct effects can be recovered from relative data that naively provide information about only indirect effects. Our results suggest that strong effects may be recoverable from relative data, but more subtle effects are challenging to identify.

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Compositional Lotka-Volterra describes microbial dynamics in the simplex.

This letter studies the stability of voltage dynamics for a power network in which nodal voltages are controlled by means of quadratic droop controllers with nonlinear AC reactive power as inputs. We show that the voltage dynamics is a Lotka-Volterra system, which is a class of nonlinear positive systems. We study the stability of the closed-loop system by proving a uniform ultimate boundedness result and investigating conditions under which the network is cooperative. We then restrict to study the stability of voltage dynamics under a decoupling assumption (i.e., zero relative angles). We analyze the existence and uniqueness of the equilibrium in the interior of the positive orthant for the system and prove an asymptotic stability result.

The Interconnection of Quadratic Droop Voltage Controllers Is a Lotka-Volterra System: Implications for Stability Analysis

Rank of stably dissipative graphs

In this paper we address a class of replicator dynamics, referred as polymatrix replicators,
that contains well known classes of evolutionary game dynamics, such as
the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for $n$-person games. Polymatrix replicators form a simple class of
algebraic o.d.e.'s on prisms (products of simplexes), which describe the evolution of strategical behaviours within a population stratified in $p\geq 1$ social groups.
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In the 80's Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems.
This theory is built around a reduction algorithm that ``infers'' the localization of the system' s attractor in some affine subspace.
It was later proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system.
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In this paper we extend these results to polymatrix replicators.

Conservative and Dissipative Polymatrix Replicators

Tese de doutoramento, Matematica (Analise Matematica), Universidade de Lisboa, Faculdade de Ciencias, 2015

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Lotka-volterra systems and polymatrix replicators

This paper deals with classification and dynamical behaviors of stably dissipative Lotka–Volterra (LV) systems. The sufficient and necessary conditions for a matrix being stably dissipative are discussed firstly. Then, a graph-theoretic classification method for stably dissipative matrices is proposed, based on which, for all five-order stably dissipative matrices, the associated graphs are classified completely as 27 topologically different graphs and for each graph the dynamics of the corresponding LV system is discussed. Finally, the effects of removing some links from the above graphs on the dynamics of the corresponding LV systems are discussed.

Jigui Luo

Papers

Classification and dynamics of stably dissipative Lotka–Volterra systems