For classic Lotka-Volterra systems governing many interacting species, we establish an exclusion principle that rules out the existence of linearly asymptotically stable steady states in subcommunitites of communities that admit a stable state which is internally D-stable. This type of stability is known to be ensured, e.g., by diagonal dominance or Volterra-Lyapunov stability conditions. By consequence, the number of stable steady states of this type is bounded by Sperner's lemma on anti-chains in a poset. The number of stable steady states can nevertheless be very large if there are many groups of species that strongly inhibit outsiders but have weak interactions among themselves. By examples we also show that in general it is possible for a stable community to contain a stable subcommunity consisting of a single species. Thus a recent empirical finding to the contrary, in a study of random competitive systems by Lischke and Löffler (Theo. Pop. Biol. 115 (2017) 24--34), does not hold without qualification.
We consider systems where a cycle born via the Hopf bifurcation blows up to infinity as a parameter ranges over a finite interval. Two examples demonstrating this effect are presented: planar Lotka-Volterra type systems with a competition-cooperation term and quasi-linear higher order equations.
Centre for Mathematical Sciences, Lund University（隆德大学数学科学中心）；Centre for Mathematical Modeling - Health and Disease, Institute for Mathematics and Physics, Dept. of Environment and Science, Roskilde University（罗斯基勒大学环境科学与系数学与物理研究所健康与疾病数学建模中心）；Department of Infectious Diseases, Uppsala University Hospital（乌普萨拉大学医院传染病科）；Faculty of Medicine, Department of Clinical Sciences, Division of Infection Medicine, Lund University（隆德大学医学院临床科学学院感染医学部）
Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while th…
In the last years it has been shown that Lotka-Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as Physics, Population Dynamics, Chemistry and Economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remar…
Although social network analysis is a promising tool to study the structure and dynamics of wildlife communities, the current methods require costly and detailed network data, which often are not available over long time periods (e.g. decades). The present work aims to resolve this issue by developing a new methodology that requires much less detailed data and it relies on well-established Lotka-Volterra models. Using the long-term abundance data (1973-2003) of northeastern Kansas rodents, the changes in the magnitude and direction of interactions (e.g., changes from cooperative behavior to co…
For a population with any given number of types, we construct a new multivariate Moran process with frequency-dependent selection and establish, analytically, a correspondence to equilibrium Lotka-Volterra phenomenology. This correspondence, on the one hand, allows us to infer the phenomenology of our Moran process based on much simpler Lokta-Volterra phenomenology, and on the other, allows us to study Lotka-Volterra dynamics within the finite populations of a Moran process. Applications to community ecology, population genetics, and evolutionary game theory are discussed.
We perform an analysis of a recent spatial version of the classical Lotka-Volterra model, where a finite scale controls individuals' interaction. We study the behavior of the predator-prey dynamics in physical spaces higher than one, showing how spatial patterns can emerge for some values of the interaction range and of the diffusion parameter.
Population dynamics are often affected by sudden environmental perturbations. Parameters of stochastic models are often imprecise due to various uncertainties. In this paper, we formulate a stochastic Lotka Volterra food chain chemostat model that includes Levy jumps and interval parameters. Firstly, we prove the existence and uniqueness of the positive solution. Moreover, the threshold between extinction and persistence in the mean of the microorganism is obtained. Finally, some simulations are carried out to demonstrate our theoretical results.
We study the properties of Lotka-Volterra competitive models in which the intensity of the interaction among species depends on their position along an abstract niche space through a competition kernel. We show analytically and numerically that the properties of these models change dramatically when the Fourier transform of this kernel is not positive definite, due to a pattern forming instability. We estimate properties of the species distributions, such as the steady number of species and their spacings, for different types of kernels.
Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.
We study a model of a multi-species ecosystem described by Lotka-Volterra-like equations. Interactions among species form a network whose evolution is determined by the dynamics of the model. Numerical simulations show power-law distribution of intervals between extinctions, but only for ecosystems with sufficient variability of species and with networks of connectivity above certain threshold that is very close to the percolation threshold of the network. Effect of slow environmental changes on extinction dynamics, degree distribution of the network of interspecies interactions, and some emer…
The relationship between the Moran model and stochastic Lotka-Volterra competition (SLVC) model is explored via timescale separation arguments. For neutral systems the two are found to be equivalent at long times. For systems with selective pressure, their behavior differs. It is argued that the SLVC is preferable to the Moran model since in the SLVC population size is regulated by competition, rather than arbitrarily fixed as in the Moran model. As a consequence, ambiguities found in the Moran model associated with the introduction of more complex processes, such as selection, are avoided.
Competitive birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey competition. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of…
We study the time evolution of two ecosystems in the presence of external noise and climatic periodical forcing by a generalized Lotka-Volterra (LV) model. In the first ecosystem, composed by two competing species, we find noise induced phenomena such as: (i) quasi deterministic oscillations, (ii) stochastic resonance, (iii) noise delayed extinction and (iv) spatial patterns. In the second ecosystem, composed by three interacting species (one predator and two preys), using a discrete model of the LV equations we find that the time evolution of the spatial patterns is strongly dependent on the …
We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.
The generalized Lotka-Volterra (gLV) equations, a classic model from theoretical ecology, describe the population dynamics of a set of interacting species. As the number of species in these systems grow in number, their dynamics become increasingly complex and intractable. We introduce Steady State Reduction (SSR), a method that reduces a gLV system of many ecological species into two-dimensional (2D) subsystems that each obey gLV dynamics and whose basis vectors are steady states of the high-dimensional model. We apply this method to an experimentally-derived model of the gut microbiome in or…
Dynamical mean field theory (DMFT) is a tool that allows to analyze the stochastic dynamics of $N$ interacting degrees of freedom in terms of a self-consistent $1$-body problem. In this work, focusing on models of ecosystems, we present the derivation of DMFT through the dynamical cavity method, and we develop a method for solving it numerically. Our numerical procedure can be applied to a large variety of systems for which DMFT holds. We implement and test it for the generalized random Lotka-Volterra model, and show that complex dynamical regimes characterized by chaos and aging can be captur…
The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection and the Moran model with frequency-dependent selection (equivalently, a game-theoretic formulation of the Moran model). We demonstrate how these mappings …
Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving s…
We analyse the asymptotic behaviour of integro-differential equations modelling $N$ populations in interaction, all structured by different traits. Interactions are modelled by non-local terms involving linear combinations of the total number of individuals in each population. These models have already been shown to be suitable for the modelling of drug resistance in cancer, and they generalise the usual Lotka-Volterra ordinary differential equations. Our aim is to give conditions under which there is persistence of all species. Through the analysis of a Lyapunov function, our first main resul…
Stochastic simulations of cyclic three-species spatial predator-prey models are usually performed in square lattices with nearest neighbor interactions starting from random initial conditions. In this Letter we describe the results of off-lattice Lotka-Volterra stochastic simulations, showing that the emergence of spiral patterns does occur for sufficiently high values of the (conserved) total density of individuals. We also investigate the dynamics in our simulations, finding an empirical relation characterizing the dependence of the characteristic peak frequency and amplitude on the total de…
We use dynamical generating functionals to study the stability and size of communities evolving in Lotka-Volterra systems with random interaction coefficients. The size of the eco-system is not set from the beginning. Instead, we start from a set of possible species, which may undergo extinction. How many species survive depends on the properties of the interaction matrix; the size of the resulting food web at stationarity is a property of the system itself in our model, and not a control parameter as in most studies based on random matrix theory. We find that prey-predator relations enhance s…
Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator-prey systems. In the mean-field rate equation approximation, the classic Lotka-Volterra model is characterized by neutral cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator-prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi-Peliti path integral repr…
We investigate the outcome of generalised Lotka-Volterra dynamics of ecological communities with random interaction coefficients and non-linear feedback. We show in simulations that the saturation of non-linear feedback stabilises the dynamics. This is confirmed in an analytical generating-functional approach to generalised Lotka-Volterra equations with piecewise linear saturating response. For such systems we are able to derive self-consistent relations governing the stable fixed-point phase, and to carry out a linear stability analysis to predict the onset of unstable behaviour. We investiga…
