Lotka-Volterra Competition Research Articles

Existence of forced waves and gap formations for a Lotka-Volterra competition system with nonlocal dispersal in a time periodic shift habitat

This paper is devoted to the study of a class of time periodic Lotka-Volterra competition system with nonlocal dispersal and shifting habitats. By using some known results of the periodic KPP model and employing the iterative techniques, we prove that there exist two positive numbers $c_{0}(d_{1})$ and $c_{0}(d_{2})$ such that the system admits a forced wave provided that the forcing speed $c\in(-c_{0}(d_{2}),c_{0}(d_{1}))$. In addition, based on the theoretical results, we show that the gap formations exist for $c>c_{0}(d_{1})$ and $c<-c_{0}(d_{2})$.

Stable coexistence in indefinitely large systems of competing species

The Lotka–Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. One of the debated questions is understanding how the number of species in the system influences the stability of the model. Robert May studied how large systems may become unstable when species–species interactions do not vanish. This outcome has frequently been interpreted as a universal phenomenon and summarized as ‘large systems are unstable’. By exploring general interaction networks, we show that the competitive Lotka–Volterra system may maintain stability even for large networks, despite non-vanishing interaction strength. We establish sufficient conditions for stability as a threshold on the interspecific interaction strength, formulated in terms of the maximum and minimum degrees (or weights) rather than the network’s size. For values below this threshold, coexistence of all species is attained, regardless of the system size. Our finding generalizes May’s result by showing that it is the outlier nodes with large degree that cause instability rather than the large number of species in the system.

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.

Global dynamics and spatiotemporal patterns of a two‐species chemotaxis system with chemical signaling loop and Lotka–Volterra competition

AbstractThis paper considers a two‐species chemotaxis system with chemical signaling loop and Lotka–Volterra competition kinetics under the homogeneous Newman boundary condition in smooth bounded domains. The global existence and boundedness of solutions for the parabolic–elliptic/parabolic–parabolic system are established. In the strong competition case, the global stability of the semitrivial constant steady state is obtained under certain parameter conditions. Linear analyzes and numerical simulations demonstrate that chemical signaling loop can significantly impact population dynamics, and admit the coexistence in the exclusion competitive case, including nonconstant steady states, chaos, and spatially inhomogeneous time‐periodic types.

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).

Multispecies Lotka–Volterra competition–diffusion system with forcing terms depending on the variables

In this paper, we study the diffusive Lotka–Volterra system containing the variable coefficients. In the Sobolev‐space settings, we achieve the well‐posedness properties (existence and regularity) and large‐time asymptotics (continuation and global existence or finite‐time blowup) of the solutions in the Sobolev spaces and time‐weighted Sobolev spaces.

The effects of diffusion and advection on the dynamics of a Lotka–Volterra competition system

This study focuses on a two-species competition-diffusion-advection system, in which two species may have different population dynamics. Using the diffusion rate and advection rate as two varying parameters, we establish a clear classification of the dynamics of this system. It turns out that there is a tradeoff between growth capacity and dispersal strategy driven invasion and coexistence mechanisms. Stronger growth and slower advection are more conducive to the successful invasion of species. Balanced growth and advection are favorable for species coexistence. In addition, species with weak growth and slow advection can coexist with competitive opponents with strong growth and moderate advection. These results show that dispersal strategies and growth capacities are important for determining dynamics.

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.

Global dynamics of a two-species clustering model with Lotka–Volterra competition

This paper is concerned with the global dynamics of a two-species Grindrod clustering model with Lotka–Volterra competition. The model takes the advective flux to depend directly upon local population densities without requiring intermediate signals like attractants or repellents to form the aggregation so as to increase the chances of survival of individuals like human populations forming small nucleated settlements. By imposing appropriate boundary conditions, we establish the global boundedness of solutions in two-dimensional bounded domains. Moreover, we prove the global stability of spatially homogeneous steady states under appropriate conditions on system parameters, and show that the rate of convergence to the coexistence steady state is exponential while the rate of convergence to the competitive exclusion steady state is algebraic.

Asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra type competition and two signals

This paper deals with the study of asymptotic behavior of the following two-species chemotaxis system with Lotka-Volterra type competition and two signals{∂tu1=Δu1−∇⋅(χ11(υ1)u1∇υ1)−∇⋅(χ12(υ2)u1∇υ2)+μ1u1(1−u1−a1u2),x∈Ω,t>0,∂tu2=Δu2−∇⋅(χ21(υ1)u2∇υ1)−∇⋅(χ22(υ2)u2∇υ2)+μ2u2(1−u2−a2u1),x∈Ω,t>0,∂tυ1=Δυ1−λ1υ1+α11u1+α12u2,x∈Ω,t>0,∂tυ2=Δυ2−λ2υ2+α21u1+α22u2,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, where parameters μi≥0,ai≥0,λi>0, αij>0(i,j=1,2) and χij(i,j=1,2) are the chemotactic sensitivity functions. The boundedness and global existence of solutions to this system have been studied [22]. By constructing appropriate energy function, the purpose of this paper is to prove the asymptotic behavior for the cases•If a1,a2∈(0,1) and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (1−a11−a1a2,1−a21−a1a2,α11(1−a1)+α12(1−a2)λ1(1−a1a2),α21(1−a1)+α22(1−a2)λ2(1−a1a2)) as t→∞.•If a1>1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (0,1,α12λ1,α22λ2) as t→∞.•If a1=1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is algebraically converges to (0,1,α12λ1,α22λ2) as t→∞.

Minimal-Speed Selection to a Lotka–Volterra Competition System with Local Versus Nonlocal Diffusions and Cubic Nonlinearity

Minimal-Speed Selection to a Lotka–Volterra Competition System with Local Versus Nonlocal Diffusions and Cubic Nonlinearity

A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion

This paper presents the computational solutions of a time-dependent nonlinear system of partial differential equations (PDEs) known as the Lotka-Volterra competition system with diffusion. We propose a combined semi-discretized spectral matrix collocation algorithm to solve this system of PDEs. The first part of the algorithm deals with the time-marching procedure, which is performed using the well-known Taylor series formula. The resulting linear systems of ordinary differential equations (ODEs) are then solved using the spectral matrix collocation technique based on the novel Touchard family of polynomials. We discuss and establish the error analysis and convergence of the proposed method. Additionally, we examine the stability analysis and the equilibrium points of the model to determine the stability condition for the system. We perform numerical simulations using diverse model parameters and with different Dirichlet and Neumann boundary conditions to demonstrate the utility and applicability of our combined Taylor-Touchard spectral collocation algorithm.

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

Coexistence states in a cross-diffusion system of a competition model

The main goal of this paper is to study the stationary problem for a Lotka–Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by Kuto and Tsujikawa (2015) and Wang and Yan (2015), where the corresponding Neumann problem is analyzed.

Dynamics of Lotka-Volterra Competition Patch Models in Streams with Two Branches.

Streams may have many branches and form complex river networks. We investigate two competition patch models associated with two different river network modules, where one is a distributary stream with two branches at the downstream end, and the other is a tributary stream with two branches at the upstream end. Treating one species as resident species and the other one as mutant species, it is shown that, for each model, there exists a invasion curve such that the mutant species can invade when rare if and only if its dispersal strategy is below this curve, but the shapes of the invasion curves are different. Moreover, we show that the global dynamics of the two models can be similar or different depending on river networks. Especially, if the drift rates of the two species are equal, then the global dynamics are similar for small drift rate and different for large drift rate. Our results also confirm a conjecture in Jiang et al. (Bull Math Biol 82:131, 2020).

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

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