Lotka-Volterra Competition Model Research Articles

Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps

Optimization problem for a stochastic N-dimensionalcompetitive Lotka-Volterra system is studied in this paper.The considered system is driven by both white noise and jumping noise,and the jumping noise is modeled by astochastic integral with respect to a Poisson counting measure generatedby a Poisson point process.For two types of objective functions, namely,time-averaged yield and sustained yield,the optimal harvesting efforts as well as thecorresponding maximum yields are given respectively.Moreover, almost sure equivalence between thesetwo objective functions is proved by ergodic method.This paper provides us a new idea to study the stochastic optimal harvesting problem with sustained yield,and this idea can be popularized to other stochastic systems.

Limiting structure of steady-states to the Lotka–Volterra competition model with large diffusion and advection

This paper is concerned with the Neumann problem of a stationary Lotka–Volterra competition model with diffusion and advection. First we obtain some sufficient conditions of the existence of nonconstant solutions by the Leray–Schauder degree theory. Next we derive a limiting system as diffusion and advection of one of the species tend to infinity. The limiting system can be reduced to a semilinear elliptic equation with nonlocal constraint. In the simplified 1D case, the global bifurcation structure of nonconstant solutions of the limiting system can be classified depending on the coefficients. For example, this structure involves a global bifurcation curve which connects two different singularly perturbed states (boundary layer solutions and internal layer solutions). Our proof employs a levelset analysis for the associate integral mapping.

The effects of time-dependent delays on global stability of stochastic Lotka–Volterra competitive model

In this paper, a stochastic Lotka–Volterra competitive model with time-dependent delays is investigated. Sufficient conditions for global asymptotic stability of the positive equilibrium are established. The obtained result demonstrates that time-dependent delays have important impacts on the global asymptotic stability of the positive equilibrium of the considered system.

Positive solutions of a Lotka–Volterra competition model with cross-diffusion

This paper concerns a Lotka–Volterra competition reaction–diffusion system with nonlinear diffusion effects. We first briefly discuss the stability of trivial solutions by spectrum analysis. Based on the boundedness of the solutions, the existence of the positive steady state solution is also investigated by the monotone iteration method.

The effect of competition on the neutral intraspecific diversity of invasive species.

This paper deals with the effect of interspecific competition on the dynamics of neutral genetic diversity in a range-expanding population. The spread of an invasive species in an environment already hosting a resident competitor is described by a traveling wave solution with minimal speed, u(t,x) = U(x - c ∗ t), of a diffusive Lotka-Volterra competition model. The description of the dynamics of neutral genetic fractions in this wave is based on a decomposition of the wave into several components, as proposed by Roques et al. (Proc Natl Acad Sci USA 109(23):8828-8833, 2012). Our analytical results reveal that the wave can be either the pulled type, corresponding to strong erosion of the diversity, or the pushed type, corresponding to maintenance of the initial diversity. The pulled/pushed nature of the wave depends on the linear or nonlinear nature of the speed c *. Our results show that, for sufficiently strong competition, the speed is nonlinear, and therefore all of the genetic diversity in the invasive population is maintained. Conversely, in the absence of competition, or when competition is mild, the speed is linear, which means that only the furthest forward fraction in the initial invasive population eventually remains in the colonization front. Our numerical results also show that the sufficient conditions of Lewis et al. (J Math Biol 45(3):219-233, 2002) and Huang (J Dyn Differ Equ 22(2):285-297, 2010) for the linearity of the speed c * can still be improved, and they show that nonlinear speeds occur across a wide region of the parameter space, providing a counterpoint to recent analytical results suggesting that nonlinear speeds only occur in certain limiting cases.

Development of interspecific competition models for the growth of Listeria monocytogenes and Lactobacillus on vacuum-packaged chilled pork by quantitative real-time PCR

The competitive growth of Listeria monocytogenes and Lactobacillus on vacuum-packaged chilled pork (VPCP) was studied by the use of Real-time PCR assay and selective culture counting method, where the Jameson-effect model and modified Lotka-Volterra model were chosen for the fitting of growth data. Real-time PCR assay was successfully applied to model the growth of the L.monocytogenes and Lactobacillus respectively on VPCP. The Jameson effect was occurred between L.monocytogenes and Lactobacillus when they co-existed in the same culture, and Lactobacillus clearly induced an early stationary phase of L. monocytogenes. The Jameson-effect model could be used to fit the growth curves of L. monocytogenes and Lactobacillus on VPCP. Thus the modified Lotka-Volterra competition model could be used to evaluate the interaction of two bacterial populations during the growth process, and found that the influence from Lactobacillus on L. monocytogenes was much higher than L. monocytogenes on Lactobacillus through the interaction coefficients. In conclusion, bacterial interactions need to be considered in microbial predictive models, which will provide a more practical assessment in predictive microbiology, and the model for the prediction based on the Real-time PCR assay can be used in future studies, especially for research of interaction models.

Bifurcation of Lotka–Volterra competition model with nonlinear boundary conditions

In this paper, we discuss the bifurcation of semi-trivial solutions for the Lotka–Volterra competition model with nonlinear boundary conditions over a smooth bounded domain. Applying the Crandall–Rabinowitz local bifurcation theorem Crandall and Rabinowitz (1971) we prove the existence of a smooth curve bifurcating from the appropriate semi-trivial branch.

Multispecies population dynamics of prebiotic compositional assemblies

Present life portrays a two-tier phenomenology: molecules compose supramolecular structures, such as cells or organisms, which in turn portray population behaviors, including selection, evolution and ecological dynamics. Prebiotic models have often focused on evolution in populations of self-replicating molecules, without explicitly invoking the intermediate molecular-to-supramolecular transition. Here, we explore a prebiotic model that allows one to relate parameters of chemical interaction networks within molecular assemblies to emergent population dynamics. We use the graded autocatalysis replication domain (GARD) model, which simulates the network dynamics within amphiphile-containing molecular assemblies, and exhibits quasi-stationary compositional states termed compotype species. These grow by catalyzed accretion, divide and propagate their compositional information to progeny in a replication-like manner. The model allows us to ask how molecular network parameters influence assembly evolution and population dynamics parameters. In 1000 computer simulations, each embodying different parameter set of the global chemical interaction network parameters, we observed a wide range of behaviors. These were analyzed by a multi species logistic model often used for analyzing population ecology (r–K or Lotka–Volterra competition model). We found that compotypes with a larger intrinsic molecular repertoire show a higher intrinsic growth (r) and lower carrying capacity (K), as well as lower replication fidelity. This supports a prebiotic scenario initiated by fast-replicating assemblies with a high molecular diversity, evolving into more faithful replicators with narrower molecular repertoires.

On the complete classification of nullcline stable competitive three-dimensional Gompertz models

We investigate a competitive model of three species, each of which, in isolation, admits Gompertz growth. A well-known theorem by M.W. Hirsch guarantees the existence of carrying simplex. Based on this, we compare three dimensional competitive Gompertz models with three dimensional competitive Lotka–Volterra models, and we find that each Gompertz model has a corresponding Lotka–Volterra model with identical nullclines. We then present the complete classification of nullcline stable models and arrive at a total of 33 stable nullcline classes, and show that in 27 of these classes all the compact limit sets are equilibria. Despite the common results, we go on to show that the behavior on the carrying simplex of Gompertz systems is subtly different from that on Lotka–Volterra systems. The number of limit cycles is finite in 5 of the remaining 6 classes, and that only the classes 26 and 27 admit Hopf bifurcations and the other 4 do not. The class 27, which has a heteroclinic cycle, contains a system having May–Leonard phenomenon: the existence of nonperiodic oscillation, and still admitting at least two limit cycles. The numerical simulation reveals that there are some systems in class 28 with two limit cycles.

Self-localized states in species competition.

We study the conditions under which species interaction, as described by continuous versions of the competitive Lotka-Volterra model (namely the nonlocal Kolmogorov-Fisher model, and its differential approximation), can support the existence of localized states, i.e., patches of species with enhanced population surrounded in niche space by species at smaller densities. These states would arise from species interaction, and not by any preferred niche location or better fitness. In contrast to previous works we include only quadratic nonlinearities, so that the localized patches appear on a background of homogeneously distributed species coexistence, instead of on top of the no-species empty state. For the differential model we find and describe in detail the stable localized states. For the full nonlocal model, however, competitive interactions alone do not allow the conditions for the observation of self-localized states, and we show how the inclusion of additional facilitative interactions leads to the appearance of them.

Free Boundary Problems for a Lotka–Volterra Competition System

In this paper we investigate two free boundary problems for a Lotka-Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of J. Guo and C. Wu.

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.

Existence of Positive Solutions of Lotka-Volterra Competition Model with Nonlinear Boundary Conditions

A Lotka-Volterra competition model with nonlinear boundary conditions is considered. First, by using upper and lower solutions method for nonlinear boundary problems, we investigate the existence of positive solutions in weak competition case. Next, we prove that-d1Δu=u(a1-b1u-c1v),x∈Ω;-d2Δv=v(a2-b2u-c2v),x∈Ω;∂u/∂ν+f(u)=0,x∈∂Ω;∂v/∂ν+g(v)=0,x∈∂Ω, has no positive solution when one of the diffusion coefficients is sufficiently large.

Cooperative and Competitive Dynamics Model for Information Propagation in Online Social Networks

Traditional empirical models of propagation consider individual contagion as an independent process, thus spreading in isolation manner. In this paper, we study how different contagions interact with each other as they spread through the network in order to propose an alternative dynamics model for information propagation. The proposed model is a novel combination of Lotka-Volterra cooperative model and competitive model. It is assumed that the interaction of one message on another is flexible instead of always negative. We prove that the impact of competition depends on the critical speed of the messages. By analyzing the differential equations, one or two stable equilibrium points can be found under certain conditions. Simulation results not only show the correctness of our theoretical analyses but also provide a more attractive conclusion. Different types of messages could coexist in the condition of high critical speed and intense competitive environment, or vice versa. The messages will benefit from the high critical speed when they are both competitive, and adopting a Tit-for-Tat strategy is necessary during the process of information propagation.

String networks in ZN Lotka–Volterra competition models

In this letter we give specific examples of Z_N Lotka-Volterra competition models leading to the formation of string networks. We show that, in order to promote coexistence, the species may arrange themselves around regions with a high number density of empty sites generated by predator-prey interactions between competing species. These configurations extend into the third dimension giving rise to string networks. We investigate the corresponding dynamics using both stochastic and mean field theory simulations, showing that the coarsening of these string networks follows a scaling law which is analogous to that found in other physical systems in condensed matter and cosmology.

Market penetration among competitive innovation products: The case of the Smartphone Operating System

Competition models have seldom considered the future development of products. However, to make decisions about products, it is key to understand their future demand and competition with other products. In the present study, we considered four Smartphone Operating Systems (OSs) (Android, iOS, Symbian, and Blackberry), and proposed an integrated model that combines scenario analysis and the Delphi to predict possible scenarios for the future development of the four OSs. Then we used the Lotka–Volterra competitive model and an innovation diffusion model to forecast the adoption volume of each OS over the next 5 years. We suggest strategies for decision makers.

Adaptive dynamics for a non-autonomous Lotka–Volterra model with size-selective disturbance

This paper intends to develop a theoretical framework for investigating the evolutionary impact of size-selective disturbance on an evolving trait of individuals in a non-autonomous Lotka–Volterra competition model. We first construct an invasion fitness function, which involves the average growth rate and settles in a nonequilibrium attractor. Then using methods of adaptive dynamics and critical function analysis we investigate the evolution of body size related traits in a competitive community, and when having size-selective disturbance we obtain the conditions for continuously stable strategy and evolutionary branching. Our results show that (1) heavy harvesting can lead to rapidly stable evolution towards smaller body size, but planting has an opposite effect; (2) smaller harvest pressure can give rise to high levels of polymorphism during evolution; (3) planting can make for evolutionary branching, while harvesting can go against evolutionary branching and promote evolutionary stability. Thus we can conclude, from an evolutionary point of view, that planting can promote species diversity and spatial differentiation among populations; however, harvesting does the opposite.

The paradox of enrichment in phytoplankton by induced competitive interactions.

The biodiversity loss of phytoplankton with eutrophication has been reported in many aquatic ecosystems, e.g., water pollution and red tides. This phenomenon seems similar, but different from the paradox of enrichment via trophic interactions, e.g., predator-prey systems. We here propose the paradox of enrichment by induced competitive interactions using multiple contact process (a lattice Lotka-Volterra competition model). Simulation results demonstrate how eutrophication invokes more competitions in a competitive ecosystem resulting in the loss of phytoplankton diversity in ecological time. The paradox is enhanced under local interactions, indicating that the limited dispersal of phytoplankton reduces interspecific competition greatly. Thus, the paradox of enrichment appears when eutrophication destroys an ecosystem either by elevated interspecific competition within a trophic level and/or destabilization by trophic interactions. Unless eutrophication due to human activities is ceased, the world's aquatic ecosystems will be at risk.

Loss of Competition in the Outside Host Environment Generates Outbreaks of Environmental Opportunist Pathogens

Environmentally transmitted pathogens face ecological interactions (e.g., competition, predation, parasitism) in the outside-host environment and host immune system during infection. Despite the ubiquitousness of environmental opportunist pathogens, traditional epidemiology focuses on obligatory pathogens incapable of environmental growth. Here we ask how competitive interactions in the outside-host environment affect the dynamics of an opportunist pathogen. We present a model coupling the classical SI and Lotka–Volterra competition models. In this model we compare a linear infectivity response and a sigmoidal infectivity response. An important assumption is that pathogen virulence is traded off with competitive ability in the environment. Removing this trade-off easily results in host extinction. The sigmoidal response is associated with catastrophic appearances of disease outbreaks when outside-host species richness, or overall competition pressure, decreases. This indicates that alleviating outside-host competition with antibacterial substances that also target the competitors can have unexpected outcomes by providing benefits for opportunist pathogens. These findings may help in developing alternative ways of controlling environmental opportunist pathogens.

Principles of parametric estimation in modeling language competition

It is generally difficult to define reasonable parameters and interpret their values in mathematical models of social phenomena. Rather than directly fitting abstract parameters against empirical data, we should define some concrete parameters to denote the sociocultural factors relevant for particular phenomena, and compute the values of these parameters based upon the corresponding empirical data. Taking the example of modeling studies of language competition, we propose a language diffusion principle and two language inheritance principles to compute two critical parameters, namely the impacts and inheritance rates of competing languages, in our language competition model derived from the Lotka-Volterra competition model in evolutionary biology. These principles assign explicit sociolinguistic meanings to those parameters and calculate their values from the relevant data of population censuses and language surveys. Using four examples of language competition, we illustrate that our language competition model with thus-estimated parameter values can reliably replicate and predict the dynamics of language competition, and it is especially useful in cases lacking direct competition data.