Lotka-Volterra Competition Model Research Articles

The Diffusive Lotka–Volterra Competitive Model with Advection Term: Exclusion, Coexistence and Multiplicity

Earlier, the research was mainly focused on the classic diffusive two-species Lotka–Volterra competitive system (without advection). Averill et al. [2017] regarded a diffusive two-species Lotka–Volterra competitive system with advection term (with one species having a combination of random dispersal and biased movement upward along the resource gradient, while the other species adopts purely random dispersal). In this paper, we consider a more general diffusive two-species Lotka–Volterra competitive system with advection term (involving both species with a combination of random dispersal and biased movement upward along their respective resource gradients). By utilizing the theory of monotone dynamical systems, spectral analysis, methods of super- and sub-solutions and some nontrivial analytic skills, we finally obtain a rather deep understanding on the global dynamics. Moreover, our results reveal that the dynamical behavior of the system is very complex: exclusion, coexistence and bistability all may happen depending closely on the interspecific competition intensities. More interestingly, by bifurcation theory, we find that the system always admits multiple coexistence steady states when the interspecific competition intensities are near the critical point.

Coexistence states for a Lotka–Volterra competitive model with aggressive movement strategy

This paper is concerned with a model of reaction–diffusion–advection equations arising in ecology. The equations model a situation in which the foreign species and the native species are both assumed to be competing for space in the same ecological niche, and the foreign species is just random diffusion while the native species is “braver” — a combination of random diffusion and a directed movement up the gradient of the foreign species. Our aim is to give the necessary and sufficient conditions for the coexistence of the foreign species and native species. For this, we first establish an a priori estimate result. Then we apply eigenvalue theory and homogenization principle to derive the necessary conditions for the coexistence. Finally, the sufficient conditions for the coexistence are given by using a global bifurcation method.

Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle ‘LPA’ model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.

Research on the Sex Ratio Variation of Sea Lampreys and Its Ecological Impact Based on the Lotka-Volterra Competition Model

Research on the Sex Ratio Variation of Sea Lampreys and Its Ecological Impact Based on the Lotka-Volterra Competition Model

Global dynamics of a diffusive competitive Lotka–Volterra model with advection term and more general nonlinear boundary condition

We investigate a competitive diffusion–advection Lotka–Volterra model with more general nonlinear boundary condition. Based on some new ideas, techniques, and the theory of the principal spectral and monotone dynamical systems, we establish the influence of the following parameters on the dynamical behavior of system (1.2): advection rates αu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{u}$\\end{document} and αv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{v}$\\end{document}, interspecific competition intensities cu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{u}$\\end{document} and cv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{v}$\\end{document}, the resources functions ru\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{u}$\\end{document} and rv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{v}$\\end{document} of the two competitive species, and nonlinear boundary functions g1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{1}$\\end{document} and g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{2}$\\end{document}. The models of (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018) are particular cases of our results when gi≡const\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{i}\\equiv const$\\end{document} for i=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$i=1,2$\\end{document}, and hence this paper extends some of the conclusions from (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018).

Extension of the Lotka-Volterra competition model

In this paper, we introduce the (p,q)-Lotka-Volterra competition model which is extension of classical Lotka-Volterra competition model. The main purpose is to give some results on the existence and non-existence of positive solutions. Upper and lower solutions technique and comparison arguments plays a significant in our main proof.

STABILITY ANALYSIS OF CELLULAR OPERATING SYSTEM MARKET SHARE IN INDONESIA WITH THE COMPETITIVE LOTKA-VOLTERRA MODEL

The current increase in smartphone users has caused various mobile device operating system companies to compete with each other to create a mobile device operating system that is suitable and acceptable to the public. Competition among mobile operating system market shares can be analyzed using the Lotka-Volterra model. This study aims to reconstruct the Lotka-Volterra model for the cellular operating system market share in Indonesia. In addition, a stability analysis was carried out, which aims to determine the stability of the competitive model for the market share of the operating system in Indonesia. The results of the study show that a competitive Lotka-Volterra model can be built on the Android and iOS operating system market share in Indonesia. In this model, there are four equilibrium points, one of which is unstable, and the other three equilibrium points are conditionally stable.

In vitro competition between two transmissible cancers and potential implications for their host, the Tasmanian devil.

Since the emergence of a transmissible cancer, devil facial tumour disease (DFT1), in the 1980s, wild Tasmanian devil populations have been in decline. In 2016, a second, independently evolved transmissible cancer (DFT2) was discovered raising concerns for survival of the host species. Here, we applied experimental and modelling frameworks to examine competition dynamics between the two transmissible cancers invitro. Using representative cell lines for DFT1 and DFT2, we have found that in monoculture, DFT2 grows twice as fast as DFT1 but reaches lower maximum cell densities. Using co-cultures, we demonstrate that DFT2 outcompetes DFT1: the number of DFT1 cells decreasing over time, never reaching exponential growth. This phenomenon could not be replicated when cells were grown separated by a semi-permeable membrane, consistent with exertion of mechanical stress on DFT1 cells by DFT2. A logistic model and a Lotka-Volterra competition model were used to interrogate monoculture and co-culture growth curves, respectively, suggesting DFT2 is a better competitor than DFT1, but also showing that competition outcomes might depend on the initial number of cells, at least in the laboratory. We provide theories how the invitro results could be translated to observations in the wild and propose that these results may indicate that although DFT2 is currently in a smaller geographic area than DFT1, it could have the potential to outcompete DFT1. Furthermore, we provide a framework for improving the parameterization of epidemiological models applied to these cancer lineages, which will inform future disease management.

An optimal control problem for a Lotka-Volterra competition model with chemo-repulsion

An optimal control problem for a Lotka-Volterra competition model with chemo-repulsion

Asymptotic properties of the Lotka-Volterra competition and mutualism model under stochastic perturbations.

Stochastically perturbed models, where the white noise type stochastic perturbations are proportional to the current system state, the most realistically describe real-life biosystems. However, such models essentially have no equilibrium states apart from one at the origin. This feature makes analysis of such models extremely difficult. Probably, the best result that can be found for such models is finding of accurate estimations of a region in the model phase space that serves as an attractor for model trajectories. In this paper, we consider a classical stochastically perturbed Lotka-Volterra model of competing or symbiotic populations, where the white noise type perturbations are proportional to the current system state. Using the direct Lyapunov method in a combination with a recently developed technique, we establish global asymptotic properties of this model. In order to do this, we, firstly, construct a Lyapunov function that is applicable to the both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method allows to find estimates for this region. We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization ('stabilization by noise'). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (that for this model occurs via the process that is known as the 'stochastic stabilization', or the 'stabilization by noise') are found applying the Khasminskii-type Lyapunov functions.

Self-organization of ecosystems to exclude half of all potential invaders

Species-rich Lotka-Volterra competition models of ecosystem dynamics transition with increasing species pool size from a phase with well-defined stable equilibrium to a dynamic phase that remains incompletely understood. We analytically describe the statistical mechanics of the steady state deep inside this dynamic phase, characterized by incessant turnover in species composition, and extract the distribution of invasion fitness of random invaders. We find that steady state invasion probability universally equals 1/2. This striking result agrees well with observations in plants and animals. Published by the American Physical Society 2024

Dynamical Analysis of a Delayed Stochastic Lotka–Volterra Competitive Model in Polluted Aquatic Environments

Dynamical Analysis of a Delayed Stochastic Lotka–Volterra Competitive Model in Polluted Aquatic Environments

Predicting Market Share: Application of Alternative Lotka-Volterra Competition Model

Predicting Market Share: Application of Alternative Lotka-Volterra Competition Model

Inverse priority effects: A role for historical contingency during species losses.

Communities worldwide are losing multiple species at an unprecedented rate, but how communities reassemble after these losses is often an open question. It is well established that the order and timing of species arrival during community assembly shapes forthcoming community composition and function. Yet, whether the order and timing of species losses can lead to divergent community trajectories remains largely unexplored. Here, we propose a novel framework that sets testable hypotheses on the effects of the order and timing of species losses-inverse priority effects-and suggests its integration into the study of community assembly. We propose that the order and timing of species losses within a community can generate alternative reassembly trajectories, and suggest mechanisms that may underlie these inverse priority effects. To formalize these concepts quantitatively, we used a three-species Lotka-Volterra competition model, enabling to investigate conditions in which the order of species losses can lead to divergent reassembly trajectories. The inverse priority effects framework proposed here promotes the systematic study of the dynamics of species losses from ecological communities, ultimately aimed to better understand community reassembly and guide management decisions in light of rapid global change.

Turing instability for a space and time discrete delay Lotka-Volterra competitive model with periodic boundary conditions

Turing instability for a space and time discrete delay Lotka-Volterra competitive model with periodic boundary conditions

Coexistence in two-species competition with delayed maturation

Inter- and intraspecific competition is most important during the immature life stage for many species of interest, such as multiple coexisting mosquito species that act as vectors of diseases. Mortality caused by competition that occurs during maturation is explicitly modelled in some alternative formulations of the Lotka–Volterra competition model. We generalise this approach by using a distributed delay for maturation time. The kernel of the distributed delay is represented by a truncated Erlang distribution. The shape and rate of the distribution, as well as the position of the truncation, are found to determine the solution at equilibrium. The resulting system of delay differential equations is transformed into a system of ordinary differential equations using the linear chain approximation. Numerical solutions are provided to demonstrate cases where competitive exclusion and coexistence occur. Stability conditions are determined using the nullclines method and local stability analysis. The introduction of a distributed delay promotes coexistence and survival of the species compared to the limiting case of a discrete delay, potentially affecting management of relevant pests and threatened species.

Joint effects of resource supply and resource types on properties of population dynamic models

Abstract The Lotka–Volterra competition model is the foundation of many ecological theories including competitive exclusion principle, limiting similarity and modern coexistence theory. However, competition between species is often modelled phenomenologically without explicitly considering the underlying mechanisms (e.g. resource competition with various resource supply forms and resource types). Deriving phenomenological models from the first principles and linking parameters in phenomenological models to the mechanisms of competition are critical to advance our understanding of population dynamics and community assembly. Here, we used time‐scale separation to theoretically derive phenomenological competition models from mechanistic consumer‐resource models with different resource supply forms (logistic and chemostatic) and resource types (substitutable and essential). We then compared the performance of the resulting phenomenological models with simulation experiments and observational plant dynamic data during 50 years of succession. We further explored how the resource supply ratio such as nutrient imbalance induced by human activities affected the critical parameters, including species' intrinsic growth rate and interaction strength in phenomenological models and consequently the population dynamics. We found that consumer population dynamics can be described by the Lotka–Volterra competition model when consumers are competing for substitutable or essential resources under logistic supply. In contrast, a novel reciprocal model was derived when consumers are competing for essential resources under chemostatic supply. Results from simulation experiments supported the model derivations. We also found that the reciprocal model outperformed the Lotka–Volterra model in explaining and predicting the population growth of 90 herbaceous plants during 50 years of succession. Moreover, the resource supply ratio affected species' intrinsic growth rates and competitive strengths differently under various resource supply forms and resource types and thus determined competitive outcomes. Synthesis. Our study provides an alternative model (reciprocal model) when organisms compete for essential resources under chemostatic supply and sets up a basis for the selection of phenomenological competition models through the lens of ecological processes and contributes to a more accurate prediction of population dynamics, especially those driven by human‐induced nutrient imbalance.

Climate warming enhances microbial network complexity by increasing bacterial diversity and fungal interaction strength in litter decomposition

Microbial communities are important drivers of plant litter decomposition; however, the mechanisms of microbial co-occurrence networks and their network interaction dynamics in response to climate warming in wetlands remain unclear. Here, we conducted a 1.5-year warming experiment on the bacterial and fungal communities involved in litter decomposition in a typical wetland. The results showed that warming accelerated the decomposition of litter and had a greater effect on the diversity of bacteria than on that of fungi. Dominant bacterial communities, such as Bacteroidia, Alphaproteobacteria, and Actinobacteria, and dominant fungal communities, such as Leotiomycetes and Sordariomycetes, showed significant positive correlations with lignin and cellulose. Co-occurrence networks revealed that the average path length and betweenness centrality under warming conditions increased in the bacterial community but decreased in the fungal community. Both bacterial and fungal networks in the 2.0 °C warming treatment had the highest ratio of positive links (58.53 % and 98.14 %), indicating that moderate warming can promote the positive correlations and symbiotic relationships observed in the microbial community. This also suggests that small-world characteristics and weak-link advantages accelerate diffusion, and scale-free features facilitate propagation in microbial communities in response to climate warming. Logistic growth and Lotka-Volterra competition models revealed that climate warming enhances microbial network complexity mainly by increasing bacterial diversity and fungal interaction strength in litter decomposition. However, the symbiotic relationship decreased slightly under 4.0 °C warming, indicating that climate warming is a random attack rather than a targeted attack, and the microbial network has strong resistance to random attack, as shown by the highly robust dynamic performance of the microbial network in litter decomposition. Overall, the microbial community in litter decomposition responded to climate warming and shifted its network interactions, leading to further changes in emergent network topology and dynamics, thus accelerating litter decomposition in wetlands.

Comparative Analysis of Taylor Series and Runge-Kutta Fehlberg Methods in Solving the Lotka-Volterra Competitive Model

This paper primarily scrutinizes the comparative efficacy of numerical methods, namely the Taylor Series and the Runge-Kutta Fehlberg methods, against the exact solution in solving mathematical models. These methods exhibit the capacity to handle the non-linearity of the Lotka-Volterra competitive model with a high degree of accuracy and reliability. The comparison of results obtained from both methods and the exact solution shows that the Runge-Kutta Fehlberg method provides a more precise approximation for the model than the Taylor Series method. This conclusion is supported by computations carried out using the Mathematica 13.2 software. The research data involves two species, Paramecium Caudatum and Stylonychia Pustulata, derived from Gause's experiment. Both species demonstrate an intraspecific interaction, with their populations rising steadily until reaching a constant level. For Paramecium Caudatum, the population peaks at 202 cells on the 16th day, while for Stylonychia Pustulata, it reaches a maximum of 41 cells on the 8th day. Equilibrium and stability analysis offer vital insights into the long-term behavior of the system and its reaction to perturbations. In mixed populations, when the carrying capacity of both species is less than the carrying capacity of another species divided by the competition coefficient, the species coexist in a stable equilibrium. Conversely, if the carrying capacity of one species is less than the carrying capacity of the other divided by the competition coefficient, one species may outcompete the other, leading to an unstable equilibrium or even extinction of the weaker species. The research thus provides valuable insights into the dynamics of competition and survival, with profound implications for the fields of ecology, conservation, and environmental management.

Dynamical analysis of a Lotka–Volterra competition model with both Allee and fear effects

Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.