Lotka-Volterra Interaction Research Articles

Spatiotemporal ecological chaos enables gradual evolutionary diversification without niches or tradeoffs.

Ecological and evolutionary dynamics are intrinsically entwined. On short timescales, ecological interactions determine the fate and impact of new mutants, while on longer timescales evolution shapes the entire community. Here, we study the evolution of large numbers of closely related strains with generalized Lotka Volterra interactions but no niche structure. Host-pathogen-like interactions drive the community into a spatiotemporally chaotic state characterized by continual, spatially-local, blooms and busts. Upon the slow serial introduction of new strains, the community diversifies indefinitely, accommodating an arbitrarily large number of strains in spite of the absence of stabilizing niche interactions. The diversifying phase persists - albeit with gradually slowing diversification - in the presence of general, nonspecific, fitness differences between strains, which break the assumption of tradeoffs inherent in much previous work. Building on a dynamical-mean field-theory analysis of the ecological dynamics, an approximate effective model captures the evolution of the diversity and distributions of key properties. This work establishes a potential scenario for understanding how the interplay between evolution and ecology - in particular coevolution of a bacterial and a generalist phage species - could give rise to the extensive fine-scale diversity that is ubiquitous in the microbial world.

Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes

We study the uniform convergence to quasi-stationarity of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka–Volterra birth and death processes and Feller diffusions with competitive Lotka–Volterra interaction. To this aim, we develop an original non-linear Lyapunov criterion involving two functions, which applies to general Markov processes.

Making quantitative predictions on the yield of a species immersed in a multispecies community: The focal species method

• Predicting the species yields of a community is central for its management. • Estimating all the required parameters to attain such goal may be unfeasible. • The focal species approximation requires only a fraction of these parameters. • Predictions are compared against experiments covering a wide variety of taxa. • Predictions using only a fraction of the set of model parameters are quite accurate. The abundance or yield of a species is an ecological quantity of paramount importance when making management and conservation decisions. The linear generalized Lotka–Volterra equations (LGLVE) constitute the simplest theoretical framework for predicting such yields. However, in many ecological communities ( e.g. tropical forests), the number of coexisting species S can be very large and estimating the entire interaction matrix A from experiments is practically unfeasible. Here, I introduce a method for predicting the yield of a particular species of interest, or focal species. The method proposed here only requires data on the interaction coefficients of the focal species k , referring to the elements of row and column k of the approximate focal interaction matrix A ( k ) . These interaction coefficients can be obtained from measuring the yields of species in monoculture ( S experimental treatments) and in biculture experiments involving the focal species ( S − 1 treatments). The remaining elements of A ( k ) are considered equal to − 1. I compare the species yields predicted by the focal species method against a data set of experimental studies I compiled from the literature across several taxa −plants, algae, protozoa and crustacean. The accuracy of this method is slightly lower than that attained when using the entire Lotka-Volterra interaction matrix. The advantage of this method is that it requires a much smaller number of experimental treatments to estimate parameters (of the order of S rather than of S 2 ). Therefore, it can be a useful quantitative tool for applied sciences like agriculture or fisheries.

Boundary effect in competition processes

AbstractThis paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

ПРИМЕНЕНИЕ «УНИВЕРСАЛЬНЫХ» МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ В ИССЛЕДОВАНИИ ПРОЦЕССОВ МУНИЦИПАЛЬНЫХ ОБРАЗОВАНИЙ

В статье рассматривается возможность применения известной математической модели взаимодействия Лотки-Вольтерра для анализа процессов в муниципальном образовании как административно-территориальной единице государства, реализующей нижний уровень управления – местное самоуправление.Цель: обосновать применение математических моделей в социологии управления, социологии местного самоуправления. Реализация данного подхода возможна благодаря свойству универсальности математических моделей, их применимости в различных областях, независимо от конкретной специфики.Метод или методология проведения работы: качественная теория дифференциальных уравнений изучает свойства решений ОДУ без нахождения самих решений. Основы заложены в конце XIX века в классических работах А. Пуанкаре и А.М. Ляпунова. В настоящее время ее методы широко применяются для исследования эволюционных систем, описывающих динамические процессы не только в механике и физике, но и в экономике, экологии, медицине, социологии.Результаты: подтверждена приемлемость применения разработанных «универсальных» математических моделей в исследованиях процессов муниципальных образований.Область применения результатов: интерпретация результатов анализа модели позволяет дополнить и расширить научные знания социологии управления, социологии местного самоуправления.

Global dynamics of three species omnivory models with Lotka-Volterra interaction

In this work, we consider the community of three species food web model with Lotka-Volterra type predator-prey interaction. In the absence of other species, each species follows the traditional logistical growth model and the top predator is an omnivore which is defined as feeding on the other two species. It can be seen as a model with one basal resource and two generalist predators, and pairwise interactions of all species are predator-prey type. It is well known that the omnivory module blends the attributes of several well-studied community modules, such as food chains (food chain models), exploitative competition (two predators-one prey models), and apparent competition (one predator-two preys models). With a mild biological restriction, we completely classify all parameters. All local dynamics and most parts of global dynamics are established corresponding to the classification. Moreover, the whole system is uniformly persistent when the unique coexistence appears. Finally, we conclude by discussing the strategy of inferior species to survive and the mechanism of uniform persistence for the three species ecosystem.

Predator–prey dynamics with square root functional responses

A predator–prey model is considered in which a modified Lotka–Volterra interaction term is used as the functional response of the predator to the prey. The interaction term is proportional to the square root of the prey population, which appropriately models systems in which the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd. Because of the square root term, the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.

Towards unification of the socio‐ecological sciences: the value of coupled models

.The major environmental problems of the twenty‐first century, including climate change, water scarcity, pollution and resource exhaustion, represent a new category of crisis and highlight the desperate need for an integrated science of socio‐ecological phenomena. To help establish the foundations of such a science, we explore three traditions of mathematical theory: the Lotka–Volterra interactions of ecological theory, niche construction models of population genetics, and theory from the gene–culture coevolution tradition. We review the theoretical tools of each of these traditions in explaining cultural articulation with the environment. Although the theoretical core of the science we propose does not exist, cultural evolutionary theory supplies useful tools to analyse endogenous culture, cultural dynamics, and deeply rooted behavioural links to the environment. We also present a coupled model for demonstration, and suggest that coupled socio‐ecological models can provide a formal theory to help address the emergent socio‐ecological problems of the future.

Vertical cultural transmission effects on demic front propagation: Theory and application to the Neolithic transition in Europe

It is shown that Lotka-Volterra interaction terms are not appropriate to describe vertical cultural transmission. Appropriate interaction terms are derived and used to compute the effect of vertical cultural transmission on demic front propagation. They are also applied to a specific example, the Neolithic transition in Europe. In this example, it is found that the effect of vertical cultural transmission can be important (about 30%). On the other hand, simple models based on differential equations can lead to large errors (above 50%). Further physical, biophysical, and cross-disciplinary applications are outlined.

Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis

In this paper, we consider a system of nonlinear partial differentialequations modeling the Lotka Volterra interactions of preys and actively movingpredators with prey-taxis and spatial diffusion.The interaction between predators are modelizedby the statement of a food pyramid condition. We establish the existence of weaksolutions by using Schauder fixed-point theorem and uniqueness viaduality technique. This paper is a generalization of the resultsobtained in [2].

A reaction–diffusion system modeling predator–prey with prey-taxis

We are concerned with a system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator's velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation.

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We study the uniform convergence to quasi-stationarity of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka–Volterra birth and death processes and Feller diffusions with competitive Lotka–Volterra interaction. To this aim, we develop an original non-linear Lyapunov criterion involving two functions, which applies to general Markov processes.

Directed Movement of Predators and the Emergence of Density-Dependence in Predator–Prey Models

We consider a bitrophic spatially distributed community consisting of prey and actively moving predators. The model is based on the assumption that the spatial and temporal variations of the predators' velocity are determined by the prey gradient. Locally, the populations follow the simple Lotka–Volterra interaction. We also assume predator reproduction and mortality to be negligible in comparison with the time scale of migration. The model demonstrates heterogeneous oscillating distributions of both species, which occur because of the active movements of predators. One consequence of this heterogeneity is increased viability of the prey population, compared to the equivalent homogeneous model, and increased consumption. Further numerical analysis shows that, on the spatially aggregated scale, the average predator density adversely affects the individual consumption, leading to a nonlinear predator-dependent trophic function, completely different from the Lotka–Volterra rule assumed at the local scale.

An interfacial approach to regional segregation of two competing species mediated by a predator

We consider the problem of coexistence of two competing species mediated by the presence of a predator. We employ a reaction-diffusion model equation with Lotka-Volterra interaction, and speculate that the possibility of coexistence is,enhanced by differences in the diffusion rates of the prey and their predator. In the limit where the diffusion rate of the prey tends to zero, a new equation is derived and the dynamics of spatial segregation is discussed by means of the interfacial dynamics approach. Also, we show that spatial segregation permits periodic and chaotic dynamics for certain parameter ranges.

Evolutionary Instability in Predator-prey Systems

The dynamical stability properties of Lotka-Volterra equations have been used frequently to predict the persistence of predator-prey assemblages and trophic webs, yet they do not take into account evolutionary change. The incorporation of genetic variation into a stable food web will make it less stable in many cases. Since populations containing genetic variation do persist in nature, such theoretical results appear paradoxical. In order to attempt to resolve this paradox, we develop a model based on phenotypic change in investment in predatory or anti-predator traits, in coevolving species. The model is akin to previous models of arms-race coevolution using evolutionarily stable strategy (ESS) theory, but we seek to understand the evolutionary dynamics in phenotype space, as well as to identify any evolutionarily stable states that may occur. The relationship between investment and the Lotka-Volterra interaction terms is defined in a very general form, so as to cover a wide range of cases in nature. From the general case and more specific examples based on hypotheses about the factors affecting predator-prey interactions, we derive conditions for the occurrence of coevolutionarily stable states, where both species are playing evolutionarily stable strategies with respect to their interaction with the other species. Coevolutionarily stable states are found to be unusual outcomes of predator-prey coevolution. However, lack of evolutionary stability is not seen to imply ecological impermanence, as natural selection is found to prevent mutual extinction of predator and prey under certain conditions.

Lie group analysis for a multispecies, spatially inhomogeneous, mutually interacting gas mixture

Abstract We consider systems of general nonlinear PDEs that arise in the extended kinetic theory of gases describing mixtures of spatially inhomogeneous, mutually interacting species. For general, autonomous, space-independent interaction laws, these systems are always invariant under the translation group. Via Lie group analysis methods, we show that for a Lotka-Volterra interaction law and three or more species, the system is invariant under a three-parameter group of transformations representing scaling, and translation in time and space. Moreover, the Lie group analysis shows that this is the only invariance group of the system.

Elementary models for population growth and distribution analysis

Abstract A version of the Lotka-Volterra interaction model is adapted to describe population growth and migration processes in a two-region system. The regions are identified as a metropolis and its non-metropolitan hinterland. Several conditions on growth and migration regimes are imposed. The time behavior of the systems are analyzed, noting especially situations where total depopulation or population explosion eventually occur in one or both populations. Neither growth control nor migration control alone results in a condition of long-run stability in both regions. If at least a momentary condition of zero growth is achieved in both regions, it is possible to maintain finite populations if each population follows a logistic natural growth process and migration flow is proportional to the volume of interaction. It is necessary also that the natural increase limitation is strong relative to migration rates. This result holds even if one population has a net migration advantage over the other.