Lotka-Volterra Competition Model Research Articles

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.

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

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.

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.

Closing the gap in the Janzen-Connell hypothesis: What determines pathogen diversity?

The high tree diversity in tropical forests has long been a puzzle to ecologists. In the 1970s, Janzen and Connell proposed that tree species (hosts) coexist due to the stabilizing actions of specialized enemies. This Janzen-Connell hypothesis was subsequently supported by theoretical studies. Yet, such studies have taken the presence of specialized pathogens for granted, overlooking that pathogen coexistence also requires an explanation. Moreover, stable ecological coexistence does not necessarily imply evolutionary stability. What are the conditions that allow Janzen-Connell effects to evolve? We link theory from community ecology, evolutionary biology and epidemiology to tackle this question, structuring our approach around five theoretical frameworks. Phenomenological Lotka-Volterra competition models provide the most basic framework, which can be restructured to include (single- or multi-)pathogen dynamics. This ecological foundation can be extended to include pathogen evolution. Hosts, of course, may also evolve, and we introduce a coevolutionary model, showing that host-pathogen coevolution can lead to highly diverse systems. Our work unpacks the assumptions underpinning Janzen-Connell and places theoretical bounds on pathogen and host ecology and evolution. The five theoretical frameworks taken together provide a stronger theoretical basis for Janzen-Connell, delivering a wider lens that can yield important insights into the maintenance of diversity in these increasingly threatened systems.

Resistance and resilience to invasion is stronger in synchronous than compensatory communities.

While community synchrony is a key framework for predicting ecological constancy, the interplay between community synchrony and ecological invasions remains unclear. Yet the degree of synchrony in a resident community may influence its resistance and resilience to the introduction of an invasive species. Here we used a generalizable mathematical framework, constructed with a modified Lotka-Volterra competition model, to first simulate resident communities across a range of competitive strengths and species' responses to environmental fluctuations, which yielded communities that ranged from strongly synchronous to compensatory. We then invaded these communities at different timesteps with invaders of varying demographic traits, after which we quantified the resident community's susceptibility to initial invasion attempts (resistance) and the degree to which community synchrony was altered after invasion (resiliency of synchrony). We found that synchronous communities were not only more resistant but also more resilient to invasion than compensatory communities, likely due to stronger competition between resident species and thus lower cumulative abundances in compensatory communities, providing greater opportunities for invasion. The growth rate of the invader was most influenced by the resident and invader competition coefficients and the growth rate of the invader species. Our findings support prioritizing the conservation of compensatory and weakly synchronous communities which may be at increased risk of invasion.

Global Dynamics of a Diffusive Lotka–Volterra Competition Model with Stage-Structure

Global Dynamics of a Diffusive Lotka–Volterra Competition Model with Stage-Structure

Frequency‐dependent community dynamics driven by sexual interactions

AbstractResearch in community ecology has tended to focus on trophic interactions (e.g., predation, resource competition) as driving forces of community dynamics, and sexual interactions have often been overlooked. Here we discuss how sexual interactions can affect community dynamics, especially focusing on frequency‐dependent dynamics of horizontal communities (i.e., communities of competing species in a single ecological guild). By combining mechanistic and phenomenological models of competition, we place sexual reproduction into the framework of modern coexistence theory. First, we review how population dynamics of two species competing for two resources can be represented by the Lotka–Volterra competition model as well as frequency dynamics, and how niche differentiation and overlap produce negative and positive frequency‐dependence (i.e., stable coexistence and priority effect), respectively. Then, we explore two situations where sexual interactions change the frequency‐dependence in community dynamics: (1) reproductive interference, that is, negative interspecific interactions due to incomplete species recognition in mating trials, can promote positive frequency‐dependence and (2) density‐dependent intraspecific adaptation load, that is, reduced population growth rates due to adaptation to intraspecific sexual (or social) interactions, produces negative frequency‐dependence. We show how reproductive interference and density‐dependent intraspecific adaptation load can decrease and increase niche differences in the framework of modern coexistence theory, respectively. Finally, we discuss future empirical and theoretical approaches for studying how sexual interactions and related phenomena (e.g., reproductive interference, intraspecific adaptation load, and sexual dimorphism) driven by sexual selection and conflict can affect community dynamics.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Dynamics of a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics.

Temperature-Dependent Growth Characteristics and Competition of Pseudanabaena and Microcystis

Global warming has been considered to accelerate the expansion of cyanobacterial blooms, which are frequently composed of the bloom-forming genera, Microcystis and Pseudanabaena, in freshwater ecosystems worldwide. Nonetheless, the impact of changes due to toxin production or lack thereof on the growth of co-existing strains, both arising from increasing temperature, has remained unknown to date. We conducted competition experiments involving toxic Microcystis PCC7806, a non-toxic mcyB− mutant, and two strains of Pseudanabaena (TH-1 and DC-1) identified as P. mucicola. In mono-culture, the specific growth ratio of Pseudanabaena increased; nevertheless, the maximum cell density declined with increasing temperature. The maximum growth ratios of Pseudanabaena TH-1 and Pseudanabaena DC-1 were 0.512 day−1 in the 30 °C group and 0.440 day−1 in the 35 °C group, respectively. The optimum temperature for the growth of Pseudanabaena was 25 °C. Remarkably, there was no significant disparity in the maximum cell density of Microcystis PCC7806 and the mcyB− mutant across varied temperature groups, even though their maximum growth rates differed marginally, reaching 0.280 day−1 and 0.306 day−1 in the 30 °C group, respectively. In co-cultures, the growth of Pseudanabaena TH-1 was uniformly inhibited, whereas that of DC-1 was somewhat influenced by co-culturing with toxic and non-toxic Microcystis, except for the 35 °C group, where inhibition was absent amongst DC-1 and Microcystis. Moreover, the growth of Microcystis was promoted with a co-culture of TH-1 in the 20 °C groups. Conversely, the co-culture of Microcystis with Pseudanabaena DC-1 led to the inhibition of the former in the 30 °C and 35 °C groups. With a Lotka–Volterra competition model, the results showed that Microcystis dominated or co-existed with Pseudanabaena, conforming to expectations from the wild.