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.     

Lotka–Volterra approximations for evolutionary trait-substitution processes

Journal of Mathematical Biology - A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion...     

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.     

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 \(n\le 4\).     

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     

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.     

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.     

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.     

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.     

On a stochastic reaction–diffusion system modeling pattern formation on seashells

Starting from the Gierer–Meinhardt setting, we propose a stochastic model to characterize pattern formation on seashells under the influence of random space–time fluctuations. We prove the existence of a positive solution for the resulting system and perform numerical simulations in order to assess the behavior of the solution in comparison with the deterministic approach.     

Studies on the oxidation–reduction systems of the erythrocyte

1. Starvation for 3 days produces a decrease in methaemoglobin-reductase and glutathione-reductase activities, but it does not alter the glucose 6-phosphate-dehydrogenase activity of the rat erythrocyte. 2. The feeding of a protein-free diet for 11 days causes greater changes in the first two enzymes and also a diminution of the third. Under this experimental condition slight decreases in protein and haemoglobin contents were noted. 3. The experimental animals did not show methaemoglobinaemia, probably because the activity of methaemoglobin diaphorase is preserved. 4. The GSH content was not affected but the stability of the tripeptide in the presence of an oxidizing agent was diminished.     

A Lagrangian particle method for reaction–diffusion systems on deforming surfaces

Reaction–diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction–diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries.     

Maternal effects on offspring consumption can stabilize fluctuating predator–prey systems

Maternal effects, where the conditions experienced by mothers affect the phenotype of their offspring, are widespread in nature and have the potential to influence population dynamics. However, they are very rarely included in models of population dynamics. Here, we investigate a recently discovered maternal effect, where maternal food availability affects the feeding rate of offspring so that well-fed mothers produce fast-feeding offspring. To understand how this maternal effect influences population dynamics, we explore novel predator–prey models where the consumption rate of predators is modified by changes in maternal prey availability. We address the ‘paradox of enrichment'', a theoretical prediction that nutrient enrichment destabilizes populations, leading to cycling behaviour and an increased risk of extinction, which has proved difficult to confirm in the wild. Our models show that enriched populations can be stabilized by maternal effects on feeding rate, thus presenting an intriguing potential explanation for the general absence of ‘paradox of enrichment'' behaviour in natural populations. This stabilizing influence should also reduce a population''s risk of extinction and vulnerability to harvesting.     

Molecular design of an acid–base cooperative catalyst for RNA cleavage based on a dizinc complex

The effects of donor groups of dizinc complexes, formed from a 2:1 mixture of Zn(II) and a dinucleating ligand, on adenylyl(3'-5')adenosine (ApA) cleavage have been studied. Two dinucleating ligands were used: one had two 2-pyridylmethyl and two 2-hydroxyethyl moieties on the 1,3-diamino-2-propanol linker moiety (2), and the other had two 2-pyridylmethyl and two carboxymethyl moieties on the 1,3-diamino-2-propanol linker moiety (3(2-)). The dizinc complex with2 [(Zn(2+))(2)-2] showed higher activities toward ApA cleavage than the dizinc complex using an analogous dinucleating ligand having four 2-pyridylmethyl donor moieties [(Zn(2+))(2)-1] at pH 5-8. The former showed a bell-shaped pH-rate constant profile, whereas the latter showed a sigmoidal pattern. The differences in the pH-rate constant profile are attributable to the various distributions of the monohydroxo-dizinc species, i.e. dideprotonated species, which are responsible for ApA cleavage. The monohydroxo species of (Zn(2+))(2)-2 has two acidic protons, which are not present in the corresponding monohydroxo species of (Zn(2+))(2)-1. The existence of both intracomplex acid (ROH or H(2)O) and base catalysts (RO(-) or OH(-)) in (Zn(2+))(2)-2 can explain its higher activity toward ApA cleavage than that of (Zn(2+))(2)-1. In contrast, (Zn(2+))(2)-3(2-) showed lower activity toward ApA cleavage at pH 7.0, which can be ascribed to the absence of the monohydroxo-dizinc species under these conditions.     

A stochastic model for the spatial distribution of species based on an aggregation–repulsion rule

The heterogeneity associated with the spatial distribution of organisms is an awkward problem in ecology because this heterogeneity directly depends on the sampling scale. To specify the scope of the influence of sampling scale on the level of species aggregation, we need data sets that entail excessive sampling costs in situ. To find a solution for this problem, we can use models to simulate patterns of organisms. These models are often very complex models that take into account heterogeneity of habitats and displacement or longevity of studied species. In this article, we introduce a new stochastic model to simulate patterns for one taxon and we want this model to be parsimonious, i.e., with few parameters and able to simulate observed patterns. This model is based on an aggregation–repulsion rule. This aggregation–repulsion rule is defined by two parameters. On a large scale, the number of aggregates present on the pattern is the first parameter. On a smaller scale, the level of aggregation–repulsion among individuals is determined by a probability distribution. These two parameters are estimated from field data set in a robust way so that the simulated patterns reflect the observed heterogeneity. We apply this model to entomological data: four Diptera families, namely the Sciaridae, Phoridae, Cecidomyiidae, and Empididae. The field data for the Phoridae family are used to simulate sampling using different trap sizes. We record changes in the coefficient of variation (C) as a function of the sampling scale, and we can suggest to ecologists emergence traps of 0.6 m², in other words a square 77 × 77 cm trap, to obtain a C value under 20%. Received: February 28, 2000 / Accepted: October 14, 2000