Diffusive Competition Model Research Articles

A competition model with impulsive interventions and environmental perturbations in moving environments

In order to understand how impulsive interventions and environmental perturbations affect dynamics of competitors, we focus on a diffusive competition model with free boundaries and periodic pulses in a temporally heterogeneous environment with upward or downward advection. The dependence of the principal eigenvalue of corresponding periodic impulsive eigenvalue problem on advection rates, habitat sizes and pulses is investigated, which gives precise conditions that classify the dynamics into four types of competition outcomes including coexistence, co-extinction, two different competition exclusions for small or negative advection rates. Some sufficient conditions on pulses or initial habitats for species spreading or vanishing, and spreading speeds are then established. Our results not only extend the existing ones to the case with pulses, but also reveal the effects of human and natural factors, that is, impulsive interventions factors including positive or negative impulsive effect, pulse intensity and timing can significantly affect and alter the competition outcomes. The different performances of the superior and inferior affected by environmental perturbations are also reflected in the simulations.

Global dynamics of two-species reaction–diffusion competition model with Gompertz growth

ABSTRACT In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that ‘slower diffuser prevails’, which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the “weak competition” case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the “strong–weak competition” case, we give the expression of critical competition intensity and the weak competitor will be wiped out.

Modeling Multi-Generation Product Diffusion in the Context of Dual-Brand Competition and Sustainable Improvement

The diffusion of competition under the coexistence of multi-generation products has become one of the important challenges faced by enterprises in their daily and sustainable operations. At the same time, the competition between different brands has intensified the difficulty and complexity of decision making in the process of multi-generation product operations. Therefore, based on the Norton–Bass model diffusion process, this paper introduces two marketing variables: dynamic price and quality level. Then, this paper builds a multi-generation product diffusion model under dual-brand competition and analyzes the impact of the company’s revenue on launch time to market, pricing, quality, and technical levels. By using the system dynamics (SD) method (from the perspective of strong brand and weak brand enterprises), the competition diffusion model is built and simulated. The simulation indicates the following: (i) When enterprises have the same brand competitiveness, reducing the pricing level cannot obtain more revenue and instead diminishes the overall revenue of the industry. Raising the pricing level can obtain more revenue and also improve the revenue of competitors. (ii) When the competitive strengths of enterprises are different, strong brands tend to maintain stable pricing on the basis of improving the quality level (or slightly raising the price). Weak brands tend to raise the pricing of new products significantly on the basis of improving the quality level. (iii) The launch-time-to-market decision of new products is influenced by the degree of the product quality upgrade. Therefore, the frequency of releasing new products should trade off against the degree of technological upgrading of the product quality. This research provides a theoretical basis and new insights for new product launches and operation decisions of enterprises.

Semi-waves and spreading speeds of stage-structured diffusive competition model with a free boundary

A stage-structured diffusive competition model with a free boundary is considered. The main aim of this paper is to study the evolutions of mature invasive species with competitive ability and determine whether it disrupts the stable ecosystem provided by native species. According to the long-time behaviors under the case of weak-strong competition, strong-weak competition, and weak competition, we obtain respectively the exact asymptotic spreading speed by delay-induced semi-wave problems if the spreading of invasive species occurs. And we demonstrate that, as the moving coefficient goes to infinite, the corresponding Cauchy problem can be viewed as the limiting profile of the model, and the asymptotic spreading speed approaches the minimal speed of the corresponding traveling wavefront solutions. Moreover, under the strong-weak competition situation, the asymptotic spreading speed of invasive species will decrease when the maturation time is longer.

Decoupled algorithms for non-linearly coupled reaction–diffusion competition model with harvesting and stocking

We propose, analyze and test two novel fully discrete decoupled linearized algorithms for a nonlinearly coupled reaction–diffusion N-species competition model with harvesting or stocking effort. The time-stepping algorithms are first and second order accurate in time and optimally accurate in space. Stability and optimal convergence theorems of the decoupled schemes are proved rigorously. We verify the predicted convergence rates of our analysis and efficacy of the algorithms using numerical experiments and synthetic data for analytical test problems. We also study the effect of harvesting or stocking and diffusion parameters on the evolution of species population density numerically, and observe the co-existence scenario subject to optimal harvesting or stocking.

The Diffusion of Competitive Platform-Based Products with Network Effects

The existence of network effects not only changes traditional product diffusion patterns but also has significant impacts on individuals’ decision behaviors. Previous studies on competitive product diffusion have focused on macro-level diffusion speed and effects while neglecting the micro-level impacts of individual heterogeneity and social interaction on product diffusion. This paper introduces the individual heterogeneities and social interactions of consumers into the competitive product diffusion model on the basis of a two-sided market framework and complex network theory. We proposed a small world network model and behavioral game theory. Specifically, the small world network model was used to build interactions between users, and behavioral game theory was utilized to describe the interactions between users. In our model, the direct network effect was distinguished from the indirect network effect on the basis of the synergy of multiple complementary products. The results show that the final distribution often presented a situation wherein two products coexisted, but the market share was very different. In an asymmetric first-mover situation, the direct network effect dominated the indirect network effect. Moreover, the dominant position of one product at present can be changed by the other under certain conditions. Finally, when the switching and learning costs were high, the market maintained its concentration, and the prior platform was unable to dominate the market. A decrease in the costs raised the prior platform’s market share and the speed of market occupation.

The Partially Degenerate Nonlocal Diffusion Competition Model with a Free Boundary: Invasion of an Inferior or Superior Competitor

The Partially Degenerate Nonlocal Diffusion Competition Model with a Free Boundary: Invasion of an Inferior or Superior Competitor

A nonlocal diffusion competition model with seasonal succession and free boundaries

In this paper, we investigate a nonlocal diffusion competition model with seasonal succession and free boundaries, which can be used to describe the dynamics of two species in an environment alternating between bad and good seasons. We first show the existence and uniqueness of global solution, and then focus on the long time behaviors of two species, namely, if h∞−g∞<∞ (the two species cannot spread over R), then they must die out in the long run; If h∞=−g∞=∞ (the two species spread to infinity as t→∞ at both fronts and survives in the new environment), then they must tend to the unique positive time periodic solution. Moreover, some sufficient conditions for spreading and vanishing are also given.

Dynamics of a diffusive competition model with memory effect and spatial heterogeneity

In this paper, a diffusive competition model is formulated by considering the memory-based diffusions of species in their movement and spatial heterogeneity of the resource. The global existence of solution for the proposed model is proved. Under weak competition, the existence and stability of the positive steady state are obtained, when the random diffusion rates are large enough, by using implicit function theorem and some priori estimations. However, when memory-based diffusions dominate the dynamics, spatial-temporal patterns can be observed. We also consider the case that both competition coefficients equal to one, showing that the stabilities of semi-trivial steady states, existence of positive steady state by the method of upper and lower solutions, and bistable dynamics of the model. The results reveal that the memory-based diffusion plays an important role on the competition outcomes. Specifically, the species with slower random diffusion may not help itself to wipe out its competitor because of disadvantageous memory-based diffusion, while the species with faster random diffusion but with advantageous memory-based diffusion may win the competition.

A hot topic diffusion approach based on the independent cascade model and trending search lists in online social networks.

In online social networks, users can quickly get hot topic information from trending search lists where publishers and participants may not have neighbor relationships. This paper aims to predict the diffusion trend of a hot topic in networks. For this purpose, this paper first proposes user diffusion willingness, doubt degree, topic contribution, topic popularity and the number of new users. Then, it proposes a hot topic diffusion approach based on the independent cascade (IC) model and trending search lists, named the ICTSL model. The experimental results on three hot topics show that the predictive results of the proposed ICTSL model are consistent with the actual topic data to a great extent. Compared with the IC, independent cascade with propagation background (ICPB), competitive complementary independent cascade diffusion (CCIC) and second-order IC models, the Mean Square Error of the proposed ICTSL model is decreased by approximately 0.78%-3.71% on three real topics.

Free boundary problem for a nonlocal time-periodic diffusive competition model.

In this paper we consider a free boundary problem for a nonlocal time-periodic competition model. One species is assumed to adopt nonlocal dispersal, and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We first prove the global well-posedness of solutions to the free boundary problem with more general growth functions, and then discuss the spreading and vanishing phenomena. Moreover, under the weak competition condition, we study the long-time behaviors of solutions for the spreading case.

On the Conjecture of the Role of Advection in a Two-Species Competition-Diffusion Model

We consider a reaction-diffusion-advection model for two competing species in a heterogeneous environment where the two species are ecologically identical except that they adopt different dispersal strategies: one is assumed to disperse randomly while the other is “smarter,” dispersing by random diffusion together with advection upward along the resource gradient. In the work by Averill, Lam, and Lou [Mem. Amer. Math. Soc., 245 (2017), no. 1161], among other things, the authors conjectured the following: (i) if the species without advection is a slower diffuser, then it will exclude its competitor when the advection rate is sufficiently small and lose competitive advantage when the advection rate passes some critical value; (ii) the species without advection will always be invaded by its competitor if it adopts a faster diffusion rate. In this paper, we partially solve this conjecture under mild assumptions on the resource function and the diffusion rates of the two competing species.

Competitive Influence Minimization in Multi-Group Social Networks: An Opinion-Based Solution

Misinformation control has been a vibrant subject of research in online social networks (OSNs). With the diversity of OSNs, we observe that the emergence of group has also notably increased the exposure rate of misinformation. In the process of misinformation dissemination, the opinions adopted by an individual not only depend on the choices made by individuals’ peers, but also highly depend on the individuals’ own knowledge. However, we find that individuals’ opinions are not consistent in the misinformation dissemination, and they are transferred dynamically. Motivated by these facts, we do some novel works on the problem of competitive influence minimization in multi-group OSNs. Firstly, we propose a novel dynamic competitive diffusion model. Secondly, a spontaneous mechanism and a contact mechanism are introduced to analyze users’ opinion transfer processes, based on probabilistic discrete-time Markov chains. Thirdly, in order to make negative opinions be minimized, we use this model to work on a new Opinion Minimization (OM) problem. To quantitatively analyze this problem, a greedy algorithm of Equilibrium Opinion Minimization (EOM) is performed to select seed nodes. Furthermore, the experiments show that our proposed EOM algorithm outperforms the state of the arts, e.g., classical heuristic algorithms and local greedy algorithms of Influence Minimization in terms of minimizing opinion in steady-state opinion distribution.

Global dynamics of a diffusive competition model with habitat degradation.

In this paper, we propose a diffusive competition model with habitat degradation and homogeneous Neumann boundary conditions in a bounded domain that is partitioned into the healthy region (undisturbed habitat) and the degraded region (due to anthropogenic habitat disturbance). Species follow the Lotka-Volterra competition in the healthy region while in the degraded region species experience only exponential decay (not necessarily at the same rate). This setup is novel in that it requires no positivity assumption on the environmental heterogeneity, either absolute or on average, which would be far too restrictive for the study of the effects of habitat degradation. We rigorously show competitive exclusion and coexistence via global stability analysis. A remarkable finding is that the quality heterogeneity of landscapes can lead to the competitive exclusion of the slower species by the faster species. This result is robust as long as the degraded region has positive area, and moreover is at odds with classical results predicting the deterministic extinction of the stronger species. On the other hand, if the degraded region has intermediate negative effect on the faster competitor, species can coexist. Differing from comparable existing results, coexistence does not rely on a limit as the diffusion coefficients tend to zero or infinity. Together, these results imply that coexistence is always a possibility under this basic, yet general, configuration, providing insights into the varying impacts found through empirical study of habitat loss and fragmentation on species.

Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries

&lt;p style='text-indent:20px;'&gt;In this paper we give a classification of the global asymptotic stability for a nonlocal diffusion competition model with free boundaries consisting of an invasive species with density &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and a native species with density &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We not only prove that such nonlocal diffusion problem has a unique global solution and also determine the long-time asymptotic behavior of the solution for three competition cases : (&lt;b&gt;I&lt;/b&gt;) &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is an inferior competitor, (&lt;b&gt;II&lt;/b&gt;) &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a superior competitor and (&lt;b&gt;III&lt;/b&gt;) the weak competition case. Especially, in case (&lt;b&gt;II&lt;/b&gt;), under some additional conditions, we determine the long-time asymptotic behavior of the solution when vanishing happens. Moreover, the criteria for spreading and vanishing are obtained.&lt;/p&gt;

Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession* *M Wang was supported by NSFC Grant 11771110, Q Zhang was supported by the China Scholarship Council, and X-Q Zhao was supported by the NSERC of Canada.

In this paper, we investigate the propagation dynamics of a reaction–diffusion competition model with seasonal succession in the whole space. Under the weak competition condition, the corresponding kinetic system admits a globally stable positive periodic solution . By the method of upper and lower solutions and the Schauder fixed point theorem, we first obtain the existence and nonexistence of traveling wave solutions connecting (0, 0) to . Then we use the comparison arguments to establish the spreading properties for a large class of solutions.

A Free Boundary Problem of the Diffusive Competition Model with Different Habitats

A Free Boundary Problem of the Diffusive Competition Model with Different Habitats

On a Lotka-Volterra Competition Diffusion Model with Advection

On a Lotka-Volterra Competition Diffusion Model with Advection

Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Intervening Coupling Diffusion of Competitive Information in Online Social Networks

The vigorously rising of social media brings a new opportunity for information diffusion in online social networks. However, the existing models of information diffusion only consider the single information, such as rumor. What's more, most of intervention frameworks are modeled under the ideal circumstances without reality constraints. In this article, we propose a novel model of competitive information coupling diffusion to describe the complex process of information diffusion in online social networks. Especially, in order to intervene the process of competitive information coupling diffusion, we introduce three intervention strategies and propose an intervention framework. More importantly, we take the dynamic constraints into consideration such as the budget of intervention and current state of the system, and further propose the constrained intervention model. To reduce the system loss, we establish an optimal control problem with constraints to achieve the optimal allocation of intervention strategies over time and minimize the total loss. We theoretically prove the existence and uniqueness of the optimal solution of the problem, and derive the optimal control solution. Through the experiments, we verify the effectiveness of the model and analyze the efficiency of different intervention strategies about competitive information coupling diffusion with or without constraints, respectively. The results show that the collaborative intervention strategies can effectively impact the process of diffusion and get the minimum system loss. This article provides high realistic significance to the commercial marketing in online social networks.