Diffusive Competition Model Research Articles

Dynamics for a nonlocal competition system with a free boundary

ABSTRACTIn this paper, we investigate a nonlocal reaction–diffusion competition model with a free boundary and discuss the long time behavior of species. The main objective is to understand the effect of the nonlocal term in the form of an integral convolution on the dynamics of competing species. Specially, for the weak competition case, when spreading occurs, we provide some sufficient conditions to prove that two competing species stabilize at a positive constant equilibrium state. Furthermore, for the case of successful spreading, we estimate the asymptotic spreading speed of the free boundary.

Numerical Methods for a Two-Species Competition-Diffusion Model with Free Boundaries

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches.

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.

Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity

This paper is concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global attractivity of the semi-trivial solution. In addition, an attracting region was obtained by means of the method of upper and lower solutions.

Note on a two-species competition-diffusion model with two free boundaries

In Guo and Wu (2015) and Wu (2015), the authors studied a two-species competition-diffusion model with two free boundaries. These two free boundaries describing the spreading fronts of two competing species, respectively, may intersect each other as time evolves. The existence, uniqueness and long time behavior of global solution have been established. In this note we discuss the conditions for spreading and vanishing, and more accurate limits of (u,v) as t→∞ when spreading occurs. Some new results and simpler proofs will be provided.

Customer social network affects marketing strategy: A simulation analysis based on competitive diffusion model

Abstract To explain the competition phenomenon and results between QQ and MSN (China) in the Chinese instant messaging software market, this paper developed a new population competition model based on customer social network. The simulation results show that the firm whose product with greater network externality effect will gain more market share than its rival when the same marketing strategy is used. The firm with the advantage of time, derived from the initial scale effect will become more competitive than its rival when facing a group of common penguin customers within a social network, verifying the winner-take-all phenomenon in this case.

Competitive seeds-selection in complex networks

Abstract This paper investigates a competitive diffusion model where two competitors simultaneously select a set of nodes (seeds) in the network to influence. We focus on the problem of how to select these seeds such that, when the diffusion process terminates, a competitor can obtain more supports than its opponent. Instead of studying this problem in the game-theoretic framework as in the existing work, in this paper we design several heuristic seed-selection strategies inspired by commonly used centrality measures—Betweenness Centrality (BC), Closeness Centrality (CC), Degree Centrality (DC), Eigenvector Centrality (EC), and K-shell Centrality (KS). We mainly compare three centrality-based strategies, which have better performances in competing with the random selection strategy, through simulations on both real and artificial networks. Even though network structure varies across different networks, we find certain common trend appearing in all of these networks. Roughly speaking, BC-based strategy and DC-based strategy are better than CC-based strategy. Moreover, if a competitor adopts CC-based strategy, then BC-based strategy is a better strategy than DC-based strategy for his opponent, and the superiority of BC-based strategy decreases as the heterogeneity of the network decreases.

Technology and demand forecasting for carbon capture and storage technology in South Korea

Among the various alternatives available to reduce greenhouse gas (GHG) emissions, carbon capture and storage (CCS) is considered to be a prospective technology that could both improve economic growth and meet GHG emission reduction targets. Despite the importance of CCS, however, studies of technology and demand forecasting for CCS are scarce. This study bridges this gap in the body of knowledge on this topic by forecasting CCS technology and demand based on an integrated model. For technology forecasting, a logistic model and patent network analysis are used to compare the competitiveness of CCS technology for selected countries. For demand forecasting, a competition diffusion model is adopted to consider competition among renewable energies and forecast demand. The results show that the number of patent applications for CCS technology will increase to 16,156 worldwide and to 4,790 in Korea by 2025. We also find that the United States has the most competitive CCS technology followed by Korea and France. Moreover, about 5 million tCO2e of GHG will be reduced by 2040 if CCS technology is adopted in Korea after 2020.

The diffusion process, competitive relationship, and equilibrium analysis of smartphone operating systems

ABSTRACTThis study started from the perspective of predicting science and technology competition, building on this to discuss the dynamic competition between Android and iOS by sales volume in each quarter. The Lotka–Volterra system of equations, one well-known competitive diffusion models, has been adopted to represent the competitive situations of the smartphone operating system market, with the Lyapunov function used to carry out equilibrium and stability analysis. The results showed that Android and iOS have a predator–prey relationship, in which iOS serves as the predator, and Android as a prey. The equilibrium and stability analysis show that both products are able to coexist in the market, and thus, each system has its own stable sales volume.

Semi-wave and spreading speed for the diffusive competition model with a free boundary

We determine the asymptotic spreading speed of an invasive species, which invades the territory of a native competitor, governed by a diffusive competition model with a free boundary in a spherically symmetric setting. This free boundary problem was studied recently in [8], but only rough bounds of the spreading speed were obtained there. We show in this paper that there exists an asymptotic spreading speed, which is determined by a certain traveling wave type system of one space dimension, called a semi-wave. This appears to be the first result that gives the precise asymptotic spreading speed for a two species system with free boundaries.

Long time behavior of a diffusive competition model

In this paper we investigate the long time behavior of a diffusive competition model in a bounded domain Ω ⊂ R n with no-flux boundary condition. This model comes from the study of the effect of migration (dispersal) (Dockery et al., 1998; Lou, 2006). We prove that lim t → ∞ u ( x , t ) = s , lim t → ∞ v ( x , t ) = 1 − s uniformly on Ω ̄ for some s ∈ [ 0 , 1 ] provided that either d 1 = d 2 , or d 1 and d 2 are suitably large.

The selection for dispersal: a diffusive competition model with a free boundary

This paper considers the population dynamics of an invasive species and a resident species, using a diffusive competition model in a radially symmetric heterogeneous environment with a free boundary. We assume that the resident species diffuses and expands in \({\mathbb{R}^n}\) , and the invasive species initially resides in a finite ball, but invades the environment with a spreading front that evolves as the free boundary. Our investigation aims to understand how the model dynamics are affected by the dispersal rate \({d_u}\) , expansion capacity \({\mu}\) and initial number u 0 of the invasive species. We show that a spreading–vanishing dichotomy exists and obtain the sharp criteria for spreading and vanishing by varying the parameters d u , \({\mu}\) and u 0. For the invasive species, we found an unconditional selection for slow dispersal rate, but a conditional selection for fast dispersal rate, that is, the selection for fast dispersal depends on the expansion capacity and initial number of the invasive species.

Leader in a diffusion competition model

A one-dimensional Cauchy problem is considered for a system of reaction-diffusion equations that, in the point version, generalizes the Volterra competition model. It is proved that the number of the leader in the propagation velocity of nonvanishing solution values at the periphery is independent of nonnegative finite initial distributions.

Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

This paper is devoted to the study of propagation phenomena for a Lotka–Volterra reaction–advection–diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction–diffusion model.

Dynamic Analysis of Competitive Technology Innovation Diffusion Model

Technological innovation diffusion theory and models play an important position in the forecasting field. Many researchers use these models to predict in different areas. Although the basic Bass model easy to use, but there are many limitations, the effects of Bass diffusion model for innovation diffusion factors (such as price, advertising, income, and technology supply) cannot measure. In order to find the principles of the relationships between innovations and the importance of the affection on the active of an innovation, this paper reviews the application of innovation diffusion models in the field of demand forecasting, through summarizing and analyzing a variety of predication models. Competitive innovation diffusion model include of monopolistic competition in a technology diffusion models and the evolution of the coexistence of two technologies competing diffusion evolution model. Under the aim of this paper, it uses mathematical methods in-depth analysis of competition, technological innovation diffusion models. This mathematical method includes nonlinear equilibrium point, stability analysis. Through the asymmetric evolutionary stability analysis, the paper derived the long-term evolutionary stable equilibrium of the innovation. Finally, the paper gives some suggestions on how to strength the competitive technology innovation in diffusion model.

Dynamics for a two-species competition–diffusion model with two free boundaries

To understand the spreading and interaction of two-competing species, we study the dynamics for a two-species competition–diffusion model with two free boundaries. Here, the two free boundaries which describe the spreading fronts of two competing species, respectively, may intersect each other. Our result shows there exists a critical value such that the superior competitor always spreads successfully if its territory size is above this constant at some time. Otherwise, the superior competitor can be wiped out by the inferior competitor. Moreover, if the inferior competitor does not spread fast enough such that the superior competitor can catch up with it, the inferior competitor will be wiped out eventually and then a spreading–vanishing trichotomy is established. We also provide some characterization of the spreading–vanishing trichotomy via some parameters of the model. On the other hand, when the superior competitor spreads successfully but with a sufficiently low speed, the inferior competitor can also spread successfully even the superior species is much stronger than the weaker one. It means that the inferior competitor can survive if the superior species cannot catch up with it.

Can climate change lead to gap formation?

Consider a situation where spatial heterogeneity leads to a cline, a gradual transition in dominance of two competing species. We first prove, in the context of a simplified competition–diffusion model, that there exists a stationary solution showing that the two species coexist in a transition zone. What happens then if, owing to climate change, the environmental profile moves with constant speed in space? We show here that, when the speed with which the environmental condition shifts exceeds the Fisher invasion speed of the advancing species, an expanding gap will form. We raise the question of whether such a phenomenon has been or can be observed.

Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment

In this study, we consider the population dynamics of an invasive species and a resident species, which are modeled as a diffusive competition process in a radially symmetric setting with a free boundary. We assume that the resident species undergoes diffusion and growth in Rn, while the invasive species initially exists in a finite ball, but invades the environment with a spreading front evolving according to a free boundary. When the invasive species is inferior, we show that if the resident species is already well established initially, then the invader can never invade deep into the underlying habitat, thus it dies out before its invading front reaches a certain finite limiting position. When the invasive species is superior, a spreading–vanishing dichotomy holds, and sharp criteria for spreading and vanishing with d1, μ, and u0 as variable factors are obtained, where d1, μ, and u0 are the dispersal rate, expansion capacity, and initial number of invaders, respectively. In particular, we obtain some rough estimates of the asymptotic spreading speed when spreading occurs.

Parameter analysis and identification of the multiple-advanced manufacturing mode diffusion model

To better understand the development of advanced manufacturing modes in the competitive diffusion, this article analyzes and identifies the parameters of the multiple-advanced manufacturing modes diffusion model. First, considering the diffusion characteristics, the competitive multiple-advanced manufacturing modes diffusion model is described. Second, various parameters of the diffusion model are analyzed to reveal their influence on the diffusion process. Third, a parameter identification model and a further parameter identification algorithm based on an improved genetic algorithm are proposed. Finally, based on a practical application of the diffusion model, the parameter identification model is constructed, and optimized parameters are obtained for the multiple-advanced manufacturing mode diffusion model. And the resulting model output is compared and contrasted with real data. The proposed model and algorithm provide a new approach to the identification of parameters in the diffusion model, which allows us to better understand the diffusion rules, help enterprises to predict the diffusion status of advanced manufacturing modes, and provide a decision-making basis for enterprises and governments.

The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor

In this paper we consider the diffusive competition model consisting of an invasive species with density $u$ and a native species with density $v$, in a radially symmetric setting with free boundary. We assume that $v$ undergoes diffusion and growth in $\mathbb{R}^N$, and $u$ exists initially in a ball ${r < h(0)}$, but invades into the environment with spreading front ${r = h(t)}$, with $h(t)$ evolving according to the free boundary condition $h'(t) = -\mu u_r(t, h(t))$, where $\mu>0$ is a given constant and $u(t,h(t))=0$. Thus the population range of $u$ is the expanding ball ${r < h(t)}$, while that for $v$ is $\mathbb{R}^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, where $(u^*,0)$ and $(0, v^*)$ are the semitrivial steady-states of the system. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given. When $u$ is an inferior competitor, we show that $(u,v)\to (0,v^*)$ as $t\to\infty$, so the invasive species $u$ always vanishes in the long run.