Generalized Lotka Volterra Research Articles

Embedding recurrent neural networks into predator–prey models

We study changes of coordinates that allow the embedding of ordinary differential equations describing continuous-time recurrent neural networks into differential equations describing predator–prey models—also called Lotka–Volterra systems. We transform the equations for the neural network first into quasi-monomial form (Brenig, L. (1988). Complete factorization and analytic solutions of generalized Lotka–Volterra equations. Physics Letters A, 133(7–8), 378–382), where we express the vector field of the dynamical system as a linear combination of products of powers of the variables. In practice, this transformation is possible only if the activation function is the hyperbolic tangent or the logistic sigmoid. From this quasi-monomial form, we can directly transform the system further into Lotka–Volterra equations. The resulting Lotka–Volterra system is of higher dimension than the original system, but the behavior of its first variables is equivalent to the behavior of the original neural network. We expect that this transformation will permit the application of existing techniques for the analysis of Lotka–Volterra systems to recurrent neural networks. Furthermore, our results show that Lotka–Volterra systems are universal approximators of dynamical systems, just as are continuous-time neural networks.

Hamiltonian structure and Darboux theorem for families of generalized Lotka–Volterra systems

This work is devoted to the establishment of a Poisson structure for a format of equations known as generalized Lotka–Volterra systems. These equations, which include the classical Lotka–Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux canonical representation, rise as a result of rather simple matrix manipulations.

Lotka-Volterra representation of general nonlinear systems.

In this article we elaborate on the structure of the generalized Lotka-Volterra (GLV) form for nonlinear differential equations. We discuss here the algebraic properties of the GLV family, such as the invariance under quasimonomial transformations and the underlying structure of classes of equivalence. Each class possesses a unique representative under the classical quadratic Lotka-Volterra form. We show how other standard modeling forms of biological interest, such as S-systems or mass-action systems, are naturally embedded into the GLV form, which thus provides a formal framework for their comparison and for the establishment of transformation rules. We also focus on the issue of recasting of general nonlinear systems into the GLV format. We present a procedure for doing so and point at possible sources of ambiguity that could make the resulting Lotka-Volterra system dependent on the path followed. We then provide some general theorems that define the operational and algorithmic framework in which this is not the case.

Chaotic interactions of self-replicating RNA

A general system of high-order differential equations describing complex dynamics of replicating biomolecules is given. Symmetry relations and coordinate transformations of general replication systems leading to topologically equivalent systems are derived. Three chaotic attractors observed in Lotka—Volterra equations of dimension n = 3 are shown to represent three cross-sections of one and the same chaotic regime. Also a fractal torus in a generalized three-dimensional Lotka—Volterra Model has been linked to one of the chaotic attractors. The strange attractors are studied in the equivalent four-dimensional catalytic replicator network. The fractal torus has been examined in adapted Lotka—Volterra equations. Analytic expressions are derived for the Lyapunov exponents of the flow in the replicator system. Lyapunov spectra for different pathways into chaos has been calculated. In the generalized Lotka—Volterra system a second inner rest point—coexisting with (quasi)-periodic orbits—can be observed; with an abundance of different bifurcations. Pathways from chaotic tori, via quasi-periodic tori, via limit cycles, via multi-periodic orbits—emerging out of periodic doubling bifurcations—to “simple” chaotic attractors can be found.

Consistent scaling and parameter choice for linear and Generalized Lotka-Volterra models used in community ecology

Many variants of linear and Generalized Lotka-Volterra (GLV) models have been employed in theoretical community ecology. In order to assess what has been learned from these, this paper proposes standard representations and presents results that facilitate comparison of alternative formulations and evaluation of choice of parameter values. Wigner's “semi-circle law” for large random matrices as adapted by May for ecological models is revised and extended using interactance as a measure of system complexity. Diagonal dominance and May's proposed complexity limit for stability are shown to be too conservative for moderate size systems. Biological arguments for choice of parameter values in GLV models are also found to be difficult to sustain. Exploration of linear and GLV models as mathematical systems does, however, suggest questions about the construction and turnover of ecological communities.

The construction and turnover of complex community models having Generalized Lotka-Volterra dynamics

Results based on Generalized Lotka-Volterra (GLV) models are presented that confirm a developmental insight: although stable complex systems may be rare in the universe of permitted systems, they can be readily constructed over time by addition and elimination of potential components (populations). Trial-and-elimination for GLV models favors positive intrinsic increase terms and positive interaction terms, but otherwise special parameter values, such as interactions arranged into cycles or loosely inter-linked blocks, are not necessary for stability. Ecological complexity may persist through the turnover of one ecological system into another.

Generalized Lotka–Volterra schemes and the construction of two-dimensional explodator cores and their liapunov functions via‘critical’ Hopf bifurcations

A new class of generalized Lotka–Volterra schemes has been investigated. It is shown that such systems are conservative. Their first integrals can be used as Liapunov functions to prove the globally stable or explosive behaviour of a wide class of modifications. On modifying the value of an exponent in one rate law, a ‘critical’ Hopf bifurcation occurs and the stable and the explosive regions are separated by a critical value at which conservative oscillations take place. The explosive schemes can be regarded as two-dimensional explodator cores. Limit-cycle oscillators can be constructed using these cores and one or more limiting reactions.

Dyspareunia.

<h3>Abstract</h3> Growth in individual size or biomass is a key demographic component in population models, with wide-ranging applications from quantifying species performance across abiotic or biotic conditions to assessing landscape-level dynamics under global change. In forest ecology, the responses of tree growth to biotic interactions are widely held to be crucial for understanding forest diversity, function, and structure. To date, most studies on plant–plant interactions only examine the additive competitive or facilitative interactions between species pairs; however, there is increasing evidence of non-additive, higher-order interactions (HOIs) impacting species demographic rates. When HOIs are present, the dynamics of a multi-species community cannot be fully understood or accurately predicted solely from pairwise outcomes because of how additional species ‘interfere’ with the direct, pairwise interactions. Such HOIs should be particularly prevalent where species show nonlinear functional responses to resource availability and resource-acquisition traits themselves are density dependent. With this in mind, we used data from a tropical secondary forest—a system that fulfills both of these conditions—to build a ontogenetic diameter-growth model for individuals across ten woody-plant species. We allowed both direct and indirect interactions within communities to influence the species-specific growth parameters in a generalized Lotka–Volterra model. Specifically, indirect interactions entered the model as higher-order quadratic terms, i.e. non-additive effects of conspecific and heterospecific neighbour size on the focal individual’s growth. For the whole community and for four out of ten focal species, the model that included HOIs had more statistical support than the model that included only direct interactions, despite the former containing a far greater number of parameters. HOIs had comparable effect sizes to direct interactions, and tended to further reduce the diameter growth rates of most species beyond what direct interactions had already reduced. In a simulation of successional stand dynamics, the inclusion of HOIs lead to rank swaps in species’ diameter hierarchies, even when community-level size distributions remained qualitatively similar. Our study highlights the implications, and discusses possible mechanisms, of non-additive density dependence in highly diverse and light-competitive tropical forests.