Volterra Dynamics Research Articles

Feedback classification and optimal control with applications to the controlled Lotka–Volterra model

Let M be a σ-compact C ∞ manifold of dimension n ≥ 2 and consider a single-input control system: x ˙ ( t ) = X ( x ( t ) ) + u ( t ) Y ( x ( t ) ) , where X, Y are C ∞ vector fields on M. We prove that there exists an open set of pairs ( X , Y ) for the C ∞ -Whitney topology is such that they admit singular abnormal rays so that the spectrum of the projective singular Hamiltonian dynamics is feedback invariant. This result is applied to the controlled Lotka–Volterra dynamics, where such rays are related to shifted equilibria of the free dynamics.

The persistence of bipartite ecological communities with Lotka–Volterra dynamics

The assembly and persistence of ecological communities can be understood as the result of the interaction and migration of species. Here we study a single community subject to migration from a species pool in which inter-specific interactions are organised according to a bipartite network. Considering the dynamics of species abundances to be governed by generalised Lotka–Volterra equations, we extend work on unipartite networks to we derive exact results for the phase diagram of this model. Focusing on antagonistic interactions, we describe factors that influence the persistence of the two guilds, locate transitions to multiple-attractor and unbounded phases, as well as identifying a region of parameter space in which consumers are essentially absent in the local community.

On the limits to invasion prediction using coexistence outcomes

The dynamics of ecological communities in nature are typically characterized by probabilistic processes involving invasion dynamics. Because of technical challenges, however, the majority of theoretical and experimental studies have focused on coexistence dynamics. Therefore, it has become central to understand the extent to which coexistence outcomes can be used to predict analogous invasion outcomes relevant to systems in nature. Here, we study the limits to this predictability under a geometric and probabilistic Lotka–Volterra framework. We show that while individual survival probability in coexistence dynamics can be fairly closely translated into invader colonization probability in invasion dynamics, the translation is less precise between community persistence and community augmentation, and worse between exclusion probability and replacement probability. These results provide a guiding and testable theoretical framework regarding the translatability of outcomes between coexistence and invasion outcomes when communities are represented by Lotka–Volterra dynamics under environmental uncertainty.

Perspectives of the Fritz‐Scheffer Awardee 2021: Profile‐ to ecosystem‐scale perspectives on soil organic matter formation as demonstrated by woody debris in forest dynamics

AbstractSoil organic matter (SOM) forms along a continuum from individual particles, pores, and aggregates to litter–soil profiles and larger ecosystems such as forests. However, forest management of SOM stocks and the carbon therein requires knowledge on which processes and factors at which scales determine SOM formation from forest biomass. As evident from woody debris at the profile scale, SOM forms through additions, transformations, translocations, and removals of litter by soil organisms and environmental components. Yet SOM stocks only increase if litter additions‐to‐removals are out of steady state or enter a new steady state that ignores older litter. Both happen through disturbance and self‐selecting feedback processes in ecosystems consisting of autotrophs, heterotrophs, and their physical environment. One such positive feedback process is litter‐SOM transformation by heterotrophs that releases nutrients that promote plant productivity and thus litter input. Stocks of litter‐SOM, heterotrophs, nutrients, and plants thus exhibit Lotka–Volterra dynamics (i.e., predator–prey interactions) and only increase when attractor states (i.e., steady series or sets of states) change due to disturbance. Evidence of evolving feedback processes and disturbance in SOM would help identify limits, potentials, and precariousness of ecosystems in light of global change, but remains to be found.

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control.

Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of the nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world datasets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900 to 1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.

Is the addition of higher-order interactions in ecological models increasing the understanding of ecological dynamics?

Recent work has shown that higher-order terms in population dynamics models can increase the stability, promote the diversity, and better explain the dynamics of ecological systems. While it is known that these perceived benefits come from an increasing number of alternative solutions given by the nature of multivariate polynomials, this mathematical advantage has not been formally quantified. Here, we develop a general method to quantify the mathematical advantage of adding higher-order interactions in ecological models based on the number of free-equilibrium points that can emerge in a system (i.e., equilibria that can be feasible or unfeasible as a function of model parameters). We apply this method to calculate the number of free-equilibrium points in Lotka–Volterra dynamics. While it is known that Lotka–Volterra models without higher-order interactions only have one free-equilibrium point regardless of the number of parameters, we find that by adding higher-order terms this number increases exponentially with the dimension of the system. Hence, the number of free-equilibrium points can be used to compare more fairly between ecological models. Our results suggest that while adding higher-order interactions in ecological models may be good for prediction purposes, they cannot provide additional explanatory power of ecological dynamics if model parameters are not ecologically restricted.

Global well-posedness of advective Lotka–Volterra competition systems with nonlinear diffusion

AbstractThis paper investigates the global well-posedness of a class of reaction–advection–diffusion models with nonlinear diffusion and Lotka–Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic–elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka–Volterra competition systems.

Disentangling eco-evolutionary effects on trait fixation

In population genetics, fixation of traits in a demographically changing population under frequency-independent selection has been extensively analysed. In evolutionary game theory, models of fixation have typically focused on fixed population sizes and frequency-dependent selection. A combination of demographic fluctuations with frequency-dependent interactions such as Lotka–Volterra dynamics has received comparatively little attention. We consider a stochastic, competitive Lotka–Volterra model with higher order interactions between two traits. The emerging individual-based model allows for stochastic fluctuations in the frequencies of the two traits and the total population size. We calculate the fixation probability of a trait under differing competition coefficients. This fixation probability resembles, qualitatively, the deterministic evolutionary dynamics. Furthermore, we partially disentangle the selection effects into their ecological and evolutionary components. We find that changing the evolutionary selection strength also changes the population dynamics and vice versa. Thus, a clean separation of the ecological and evolutionary effects is not possible. Instead, our results imply a nested interaction of the evolutionary and ecological effects. The entangled eco-evolutionary processes thus cannot be ignored when determining fixation properties in a co-evolutionary system.

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem ut=d1Δu−χ1∇⋅(u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=d2Δv−χ2∇⋅(v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,0=d3Δw−w+u+v,x∈Ω,t>0,in a bounded domain Ω⊂Rn,n≥1, with smooth boundary, where d1,d2,d3,χ1,χ2,μ1,μ2,a1,a2 are positive constants.When a1∈(0,1) and a2∈(0,1), it is shown that under some explicit largeness assumptions on the logistic growth coefficients μ1 and μ2, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state (u∗,v∗,w∗) of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When a1≥1 and a2∈(0,1), if μ2 is suitable large, for all sufficiently regular nonnegative initial data u0 and v0 with u0≢0 and v0≢0, the globally bounded solution of above system will stabilize toward (0,1,1) as t→∞ in algebraic.

The dynamics of the rise and fall of empires

The rise of empires can be elucidated by treating them as living organisms, and the celebrated Verhulst or Lotka–Volterra dynamics can be used to understand the growth mechanisms of empires. The fast growth can be expressed by an exponential function as in the case of Macedonian empire of the Alexander the Great whereas a sigmoidal growth can be expressed by power-law equation as in the case of Roman and Ottoman empires. The superpowers Russia and the USA follow somehow different mechanisms, Russia displays two different exponential growth behaviors whereas the USA follows two different power-law behaviors. They did not disturb and mobilize their social capacity much during the course of their rise. The decline and the collapse of an empire occur through a kind of fragmentation process, and the consequently formed small states become rather free in their behavior. The lands of the new states formed exhibit a hierarchical pattern, and the number of the states having an area smaller than the largest one can be given either by an exponential or power-law function. The exponential distribution pattern occurs when the states are quite free in their pursuits, but the power-law behavior occurs when they are under the pressure of an empire or a strong state in the region. The geological and geographical conditions also affect whether there occurs exponential or power-law behavior. The new unions formed such as the European Union and the Shanghai Cooperation increase the power-law exponent implying that they increase the stress in the international affairs. The viscoelastic behavior of the empires can be found from the scattering diagrams, and the storage [Formula: see text]and loss modulus [Formula: see text], and the associated work-like and heat-like terms can be determined in the sense of thermodynamics. The [Formula: see text] of Ottomans was larger than that of Romans implying that they confronted severe resistance during their expansion. The [Formula: see text] of Russia is also larger than that of the USA; in fact the USA did not face severe resistance as they had an overwhelming superiority over native Americans. The [Formula: see text] indicates solidity in the social structure and Romans, Ottomans, and Russians all have [Formula: see text] larger than [Formula: see text]. The [Formula: see text] is slightly larger than [Formula: see text] for the USA indicating that they have had a very flexible social structure. By the same token the ratio of the work-like term to the internal energy is larger for Ottomans than that of Romans, and larger for the USA than that of Russia. That means the fraction of the total energy allocated to improve the social capacity is larger for Romans than that of Ottomans, and is larger for Russians than that of the USA.

ON THE DYNAMICS OF EVOLUTIONARY COMPETITION MODELS

AbstractAs exemplified by classic Lotka–Volterra theory, there are several canonical outcomes possible to a two species (interference) competitive interaction: coexistence, initial condition‐dependent competitive exclusion of one species, or the global exclusion of one species. Evolutionary versions of Lotka–Volterra dynamics have been investigated in order to see the role that evolutionary adaptation can play in influencing the competitive outcome. For the most part, however, these investigations have been carried out by numerical simulations. In this paper, we provide some rigorous mathematical criteria concerning the outcome of a competition between two species x1 and x2 when evolution is taken into account. Motivated by two classic experiments in which the outcome of two competing beetle species was observed to change due to phenotypic or genetic changes in one species, we consider the case when only the species x2 can evolutionarily adapt. Using methods from persistence theory, we obtain criteria under which x2 will persist and criteria under which x1 will persist.

Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics

We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in ${R}^{N}$ with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we establish the emergence of strange attractors and Sinai--Ruelle--Bowen (SRB) measures with strong statistical properties (central limit theorem, exponential decay of correlations, etc.). We identify and study the mechanism responsible for the nonuniform hyperbolicity: saddle point shear. Our results apply to concrete systems of interest in the biological and physical sciences, such as May--Leonard models of Lotka--Volterra dynamics.

The Jungle Universe: coupled cosmological models in a Lotka–Volterra framework

In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaître universes is a special case of generalized Lotka–Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka–Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann–Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka–Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.

Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota

The intestinal microbiota is a microbial ecosystem of crucial importance to human health. Understanding how the microbiota confers resistance against enteric pathogens and how antibiotics disrupt that resistance is key to the prevention and cure of intestinal infections. We present a novel method to infer microbial community ecology directly from time-resolved metagenomics. This method extends generalized Lotka–Volterra dynamics to account for external perturbations. Data from recent experiments on antibiotic-mediated Clostridium difficile infection is analyzed to quantify microbial interactions, commensal-pathogen interactions, and the effect of the antibiotic on the community. Stability analysis reveals that the microbiota is intrinsically stable, explaining how antibiotic perturbations and C. difficile inoculation can produce catastrophic shifts that persist even after removal of the perturbations. Importantly, the analysis suggests a subnetwork of bacterial groups implicated in protection against C. difficile. Due to its generality, our method can be applied to any high-resolution ecological time-series data to infer community structure and response to external stimuli.

Model uncertainty in economic impacts of climate change: Bernoulli versus Lotka Volterra dynamics

The dynamic economic behavior in most integrated assessment models linking economic growth to climate change involves a differential equation solved by Jacob Bernoulli in 1695. Using the dynamic integrated climate economy (DICE) model and freezing exogenous variables at initial values, this dynamic is shown to produce implausible projections on a 60-year time frame. If world capital started at US$1, after 60 years the world economy would be indistinguishable from one starting with 10 times the current capitalization. Such behavior points to uncertainty at the level of the fundamental dynamics, and suggests that discussions of discounting, utility, damage functions, and ethics should be conducted within a more general modeling vocabulary. Lotka Volterra dynamics is proposed as an alternative with greater prime facie plausibility. With near universality, economists assume that economic growth will go on forever. Lotka Volterra dynamics alert us to the possibility of collapse.

Detecting synchronization in coupled stochastic ecosystem networks

Instantaneous phase difference, synchronization index and mutual information are considered in order to detect phase transitions, collective behaviours and synchronization phenomena that emerge for different levels of diffusive and reactive activity in stochastic networks. The network under investigation is a spatial 2D lattice which serves as a substrate for Lotka–Volterra dynamics with 3rd order nonlinearities. Kinetic Monte Carlo simulations demonstrate that the system spontaneously organizes into a number of asynchronous local oscillators, when only nearest neighbour interactions are considered. In contrast, the oscillators can be correlated, phase synchronized and completely synchronized when introducing different interactivity rules (diffusive or reactive) for nearby and distant species. The quantitative measures of synchronization show that long distance diffusion coupling induces phase synchronization after a well defined transition point, while long distance reaction coupling induces smeared phase synchronization.

Multi-objective optimization of an ecological assembly model

The aim of the present work is to use multi-objective evolutionary algorithms (MOEA) to parameterise an ecological assembly model based on Lotka–Volterra dynamics. In community assembly models, species are introduced from a pool of species according to a sequence of invasion. By manipulating the assembly sequences, we look at the structure of the final communities obtained by a multi-objective process where the goal is to optimize the productivity of the final communities. The MOEA must also meet the constraint that the communities constructed in this fashion have a specified connectance. The Non-dominated Sorting Algorithm (NSGA-II) and the Strength Pareto Evolutionary Algorithm (SPEA2) were employed to optimize sequences according to the multi-objective optimization problem. The results show that the assembly process using optimized sequences generated different community structure than those generated via random sequences. First, the assembled communities are much more productive than those obtained from random sequences. We show that this increase of productivity is due to the degree distribution of the community food web, which was reshaped by the optimization process. In addition, using identical regional species pools the MOEAs were able to generate communities of different expected connectances. These results demonstrate the effectiveness of NSGA-II and SPEA2 for optimizing parameters in ecological models.

How does global change affect the strength of trophic interactions?

Recent research has generally shown that a small change in the number of species in a food web can have consequences both for community structure and ecosystem processes. However ‘change’ is not limited to just the number of species in a community, but might include an alteration to such properties as precipitation, nutrient cycling and temperature, all of which are correlated with productivity. Here we argue that predicted scenarios of global change will result in increased plant productivity. We model three scenarios of change using simple Lotka–Volterra dynamics, which explore how a global change in productivity might affect the strength of local species interactions and detail the consequences for community and ecosystem level stability. Our results indicate that (i) at local scales the average population size of consumers may decline because of poor quality food resources, (ii) that the strength of species interactions at equilibrium may become weaker because of reduced population size, and (iii) that species populations may become more variable and may take longer to recover from environmental or anthropogenic disturbances. At local scales interaction strengths encompass such properties as feeding rates and assimilation efficiencies, and encapsulate functionally important information with regard to ecosystem processes. Interaction strengths represent the pathways and transfer of energy through an ecosystem. We examine how such local patterns might be affected given various scenarios of ‘global change’ and discuss the consequences for community stability and ecosystem functioning.

Predator–prey body size, interaction strength and the stability of a real food web

SummaryWe examined the empirical relationship between predator–prey body size ratio and interaction strength in the Ythan Estuary food web.We have refined a previously published version of the food web and explored how size‐based predatory effects might affect food web dynamics. To do so, we used four predatory speciesCrangon crangon(Linnaeus),Carcinus maenas(Linnaeus),Pomatoschistus microps(Krøyer) andPlatichthys flesus(Linnaeus) and one common prey speciesCorophium volutator(Pallas) from the food web.All predators and prey were sorted into small, medium and large size classes and placed into mesocosms in all possible pairwise combinations of size and species identity to determine per capita effects of predators on prey (aij).Using Lotka–Volterra dynamics the empirical body size relationships obtained from these experiments and other relationships already available for the Ythan Estuary, we parameterized a food web model for this system. The local stability properties of the resulting food web models were then determined.We found that by choosing interaction strengths using an empirically defined scaling law, the resulting food web models are always dynamically stable, despite the residual uncertainties in the modelling approach. This contrasts with the statistical expectation that random webs with random parameters have a vanishingly improbable chance of stability.The patterning of predator and prey body sizes in real ecosystems affects the arrangement of interaction strengths, which in turn determines food web stability.

Disease Induced Oscillations between Two Competing Species

The interaction of disease and competition dynamics is investigated in a system of two competing species in which only one species is susceptible to disease. The model is kept as simple as possible, combining Lotka--Volterra competition between the species with disease dynamics of susceptible and infective individuals within one of the species. It is assumed that pure vertical disease transmission (from parent to offspring) dominates horizontal transmission (by contact between infective and susceptible individuals) and that infective individuals have the same competition strength as susceptibles but a lower intrinsic growth rate. These assumptions yield three-dimensional competitive Lotka--Volterra dynamics modeling the disease-competition interaction. It is proved that if in the absence of disease there is competitive exclusion between the two species, then the presence of disease can lead to stable or oscillatory coexistence of both species. The case of oscillatory coexistence can be viewed either as di...