Permanence and global attractivity for Lotka–Volterra difference systems

 The permanence and global attractivity for two-species difference systems of Lotka–Volterra type are considered. It is proved that a cooperative system cannot be permanent. For a permanent competitive system, the explicit expression of the permanent set E is obtained and sufficient conditions are given to guarantee the global attractivity of the positive equilibrium of the system. Received: 21 May 1997 / Revised version: 25 November 1998     

Qualitative permanence of Lotka–Volterra equations

In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Lotka—Volterra系统

本文总结了近年来Ｌｏｒｋａ－Ｖｏｌｔｅｒｒａ系统方面的一些进展，讨论了各种Ｌｏｒｋａ－Ｖｏｌｔｅｒｒａ系统动力学行为的异同，同时介绍了一些尚未解决的公开问题。     

A Lotka–Volterra competition model with seasonal succession

A complete classification for the global dynamics of a Lotka–Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.     

Persistence in Stochastic Lotka–Volterra Food Chains with Intraspecific Competition

This paper is devoted to the analysis of a simple Lotka–Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J Math Biol 76:697–754, 2018b) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper, we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and \(n-1\) predators. The jth predator eats the \(j-1\)st species and is eaten by the \(j+1\)st predator; this way each species only interacts with at most two other species—the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann Appl Probab 28:1893–1942, 2018a). Unlike most of the results available in the literature, we provide an in-depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analyzing trophic cascades in a plant–herbivore–predator system and providing persistence/extinction criteria for food chains of length \(n\le 4\).     

Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction

We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka–Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.     

Which evolutionary processes influence natural genetic variation for phenotypic traits?

Although many studies provide examples of evolutionary processes such as adaptive evolution, balancing selection, deleterious variation and genetic drift, the relative importance of these selective and stochastic processes for phenotypic variation within and among populations is unclear. Theoretical and empirical studies from humans as well as natural animal and plant populations have made progress in examining the role of these evolutionary forces within species. Tentative generalizations about evolutionary processes across species are beginning to emerge, as well as contrasting patterns that characterize different groups of organisms. Furthermore, recent technical advances now allow the combination of ecological measurements of selection in natural environments with population genetic analysis of cloned QTLs, promising advances in identifying the evolutionary processes that influence natural genetic variation.     

Effect of parity on productivity and sustainability of Lotka–Volterra food chains

Hairston, Slobodkin, and Smith conjectured that top down forces act on food chains, which opposed the previously accepted theory that bottom up forces exclusively dictate the dynamics of populations. We model food chains using the Lotka–Volterra predation model and derive sustainability constants which determine which species will persist or go extinct. Further, we show that the productivity of a sustainable food chain with even trophic levels is predator regulated, or top down, while a sustainable food chain with odd trophic levels is resource limited, which is bottom up, which is consistent with current ecological theory.     

Asymptotic behaviour of the non-autonomous competing two-species Lotka–Volterra models with impulsive effect

In this paper, the nonautonomous competing two-species Lotka–Volterra models with impulsive effect are considered, where all the parameters are time-dependent and asymptotically approach the corresponding periodic functions. Under some conditions, it is shown that the semi-trivial positive solutions of the models asymptotically approach the semi-trivial positive periodic solutions of the corresponding periodic system. It is also shown that the positive solution of the models asymptotically approach the positive periodic solution of the corresponding periodic system.     

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

The Lotka–Volterra model is a differential system of two coupled equations representing the interaction of two species: a prey one and a predator one. We formulate an optimal control problem adding the effect of hunting both species as the control variable. We analyse the optimal hunting problem paying special attention to the nature of the optimal state and control trajectories in long time intervals. To do that, we apply recent theoretical results on the frame to show that, when the time horizon is large enough, optimal strategies are nearly steady-state. Such path is known as turnpike property. Some experiments are performed to observe such turnpike phenomenon in the hunting problem. Based on the turnpike property, we implement a variant of the single shooting method to solve the previous optimisation problem, taking the middle of the time interval as starting point.     

Do evolutionary processes minimize expected losses?

Evolution by variation and natural selection is often viewed as an optimization process that favors those organisms which are best adapted to their environment. This leaves open the issue of how to measure adaptation and what criterion is implied for optimization. This problem has been framed and analysed mathematically under the assumption that individuals compete to minimize expected losses across a series of decisions (e.g. choice of behavior), where each decision offers a stochastic payoff. But the fact that a particular analysis is tractable for a specified criterion does not imply the fidelity of that criterion. Computer simulations involving a version of the k -armed bandit problem can address the veracity of the hypothesis that individuals are selected to minimize expected losses. The results offered here do not support this hypothesis.     

The dynamics of the lynx–hare system: an application of the Lotka–Volterra model

The Lotka–Volterra model of predator–prey dynamics was used for approximation of the wellknown empirical time series on the lynx–hare system in Canada that was collected by the Hudson Bay Company in 1845–1935. The model was assumed to demonstrate satisfactory data approximation if the sets of deviations of the model and empirical data for both time series satisfied a number of statistical criteria (for the selected significance level). The frequency distributions of deviations between the theoretical (model) trajectories and empirical datasets were tested for symmetry (with respect to the Y-axis; the Kolmogorov–Smirnov and Lehmann–Rosenblatt tests) and the presence or absence of serial correlation (the Swed–Eisenhart and “jumps up–jumps down” tests). The numerical calculations show that the set of points of the space of model parameters, when the deviations satisfy the statistical criteria, is not empty and, consequently, the model is suitable for describing empirical data.     

时变环境Lotka—Volterra系统的渐近性态

本文研究时变环境下的Lotka-Volterra系统,给出一系列关于这类系统持久性以及正T-周期解全局渐近稳定的充分条件．     

A Lotka–Volterra competition model and its global convergence to a definite axial equilibrium

 We consider a four-species model based on competition and show that the whole four-species system collapses to a definite single species equilibrium at its carrying capacity. To do so, we use the results of Hirsch [4], Van Den Driessche and Zeeman [20], Hofbauer and Sigmund [5], and the product theorem of the Conley connection matrix theory by Mischaikow and Reineck [10]. Received: 16 June 2001 / Revised version: 25 August 2001 / Published online: 14 March 2002     

Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka–Volterra model

We propose an optimal control framework to describe intra-seasonal predator–prey interactions, which are characterized by a continuous-time dynamical model comprising predator and prey density, as well as the energy budget of the prey over the length of a season. The model includes a time-dependent decision variable for the prey, representing the portion of the prey population in time that is active, as opposed to diapausing (a state of physiological rest). The predator follows autonomous dynamics and accordingly it remains active during the season. The proposed model is a generalization of the classical Lotka–Volterra predator–prey model towards non-autonomous dynamics that furthermore includes the effect of an energy variable. The model has been inspired by a specific biological system of predatory mites (Acari: Phytoseiidae) and prey mites (so-called fruit-tree red spider mites) (Acari: Tetranychidae) that feed on leaves of apple trees—its parameters have been instantiated based on laboratory and field studies. The goal of the work is to understand the decisions of the prey mites to enter diapause (a state of physiological rest) given the dynamics of the predatory mites: this is achieved by solving an optimization problem hinging on the maximization of the prey population contribution to the next season. The main features of the optimal strategy for the prey are shown to be that (1) once in diapause, the prey does not become active again within the same season and hence diapause is an irreversible process; (2) for the vast majority of parameter space, the portion of prey individuals entering diapause within the season does not decrease in time; (3) with an increased number of predators, the optimal population strategy for the prey is to start diapause earlier and to enter diapause more gradually. This optimal population strategy will be studied for its ESS properties in a sequel to the work presented in this article.     

非自治Lotka—Volterra扩散系统的渐近性质

本文总结了近年来我们对斑块环境的非自治Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型所做的一些研究工作，得到了这类系统的持续生存性、周期解的存在唯一性和全局渐近稳定性的条件，并讨论了扩散的影响。     

中立型时滞Lotka—Volterra系统解的振动性

一、引言所谓生态系统的持久性就是这个生态系统的所有种群都能长久生存下去.所谓生态系统的灭绝性就是这个生态系统的某一种群遭受毁灭而绝种。当今世界每45分钟就有一个种群绝种或濒临绝种.种群绝种或濒临绝种会破坏生态平衡以致酿成灾难.例如,日118杨正清7卷本船队在南极海城大量捕捞磷虾,首先磷虾被大量捕捞使海     

Nasonia–microbiome associations: a model for evolutionary hologenomics research

In recent years, with the development of microbial research technologies, microbiota research has received widespread attention. The parasitoid wasp genus Nasonia is a good model organism for studying insect behavior, development, evolutionary genetics, speciation, and symbiosis. This review describes key advances and progress in the field of the Nasonia–microbiome interactions. We provide an overview of the advantages of Nasonia as a model organism for microbiome studies, list research methods to study the Nasonia microbiome, and discuss recent discoveries in Nasonia microbiome research. This summary of the complexities of Nasonia–microbiome relationships will help to contribute to a better understanding of the interactions between animals and their microbiomes and establish a clear research direction for Nasonia–microbiome interactions in the future.     

Modeling the batch bacteriocin production system by lactic acid bacteria by using modified three-dimensional Lotka–Volterra equations

Different batch cultures of Lactococcus lactis CECT 539, a nisin-producing strain, were carried out in culture media prepared with whey and mussel processing wastes. From these cultures, a reasonable system of differential equations, similar to the three-dimensional Lotka–Volterra two predators-one prey model, was set up to describe, for the first time, the relationship between the absolute rates of growth, pH drop and nisin production.Thus, the nisin production system was described as a three-species (pH, biomass and nisin) ecosystem. In this case, both nisin and biomass production were considered as two pH-dependent species that compete for the nitrogen source. Excellent agreement (R² values ≥0.9885) resulted between model predictions and the experimental data, and significant values for all the model parameters were obtained. The developed model was demonstrated (R² values ≥0.9874) for five batch cultivations of the strains L. lactis CECT 539 in MRS broth and Lactobacillus sakei LB 706 (sakacin A producer), Pediococcus acidilactici LB42-923 (pediocin AcH producer), L. lactis ATCC 11454 (nisin producer) and Leuconostoc carnosum Lm1 (leuconocin Lcm1 producer) in TGE broth. These results suggest that the batch bacteriocin production system in these culture media can be successfully described by using the Lotka–Volterra approach.