Lotka-Volterra Competition Research Articles

Speed determinacy of the traveling waves for a three species time‐periodic Lotka–Volterra competition system

Some physical environmental conditions changing in time create a great difficulty in studying time‐periodic traveling wave solutions. In this paper, we investigate speed selection of time‐periodic traveling waves for a three species time‐periodic Lotka–Volterra competition system. By transforming the competitive system into a cooperative system, the existence of the traveling wave solutions and speed determinacy of the spread speed are studied via the upper–lower solution method as well as the comparison principle. Through constructing specific types of upper and lower solutions to the system, some sets of sufficient conditions composed of the parameters in the system are established to justify the linear and nonlinear selection of the spread speed. As the positive periodic functions in system reduces to the constant coefficients, parts of our results could cover the previous works, and the other results are novel.

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.

Propagation Dynamics of Bistable Traveling Wave to a Time-Periodic Lotka-Volterra Competition Model: Effect of Seasonality

Propagation Dynamics of Bistable Traveling Wave to a Time-Periodic Lotka-Volterra Competition Model: Effect of Seasonality

Propagation Phenomena for a Nonlocal Dispersal Lotka–Volterra Competition Model in Shifting Habitats

Propagation Phenomena for a Nonlocal Dispersal Lotka–Volterra Competition Model in Shifting Habitats

Estimates of Population Size for Traveling Wave Solutions of Spatially Non-local Lotka-Volterra Competition System

Estimates of Population Size for Traveling Wave Solutions of Spatially Non-local Lotka-Volterra Competition System

Two species nonlocal diffusion systems with free boundaries

<p style='text-indent:20px;'>We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [<xref ref-type="bibr" rid="b7">7</xref>], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.</p>

The minimal wave speed of the Lotka-Volterra competition model with seasonal succession

<p style='text-indent:20px;'>This paper focuses on the minimal wave speed of time-periodic traveling waves to a Lotka-Volterra competition model with seasonal succession. It is the first time the general conditions of linear selection and nonlinear selection have been derived by the comparison principle and the upper-lower solution method. Based on the decay characteristics of traveling waves, we obtain some explicit conditions for determining the selection mechanism of the minimal wave speed by constructing upper/lower solutions, which include the first explicit condition for the nonlinear selection and the explicit conditions for the linear selection that greatly improve the result in the reference.</p>

A REVIEW ON THE DYNAMICS OF TWO SPECIES COMPETITIVE ODE AND PARABOLIC SYSTEMS

This paper is devoted to a review on the dynamics of two species competition systems including the classical ODE, reaction-diffusion as well as reaction-diffusion-advection models. The primary purpose is to illustrate the effect of competition intensity, movement (diffusion and/or advection) and spatial variation on the population dynamics. Specific topics include LotkaVolterra competition models in heterogeneous environments and in advective environments, linear second order eigenvalue problems, and the evolution of movement strategy. Several fundamental tools such as the monotone theory, the principal eigenvalue theory (for single equations or systems) and some technical approaches are introduced. Some recent developments are discussed and also several problems that deserve future investigation are proposed.

Time-Periodic Traveling Waves for A Periodic Lotka-Volterra Competition System with Nonlocal Dispersal

Time-Periodic Traveling Waves for A Periodic Lotka-Volterra Competition System with Nonlocal Dispersal

Global dynamics of a Lotka–Volterra competition patch model* *S Chen is supported by National Natural Science Foundation of China (Nos. 12171117 and 11771109) and Shandong Provincial Natural Science Foundation of China (No. ZR2020YQ01), J Shi is supported by US-NSF Grant DMS-1715651 and DMS-1853598, and Z Shuai is supported by US-NSF Grant DMS-1716445.

The global dynamics of the two-species Lotka–Volterra competition patch model with asymmetric dispersal is classified under the assumptions that the competition is weak and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. We show that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs follow the theory of monotone dynamical systems and a graph-theoretic approach based on the tree-cycle identity.

Microbial mutualism promoting the coexistence of competing species: Double-layer model for two competing hosts and one microbial species

Gause's law of competitive exclusion holds that the coexistence of competing species is extremely unlikely when niches are not differentiated. This law is supported by many mathematical studies, yet the coexistence of competing species is nearly ubiquitous in real ecosystems. We pay attention to the fact that plants and animals usually contact with microbial species as mutualistic partners. The activity spaces of host species are different from those of micro-organisms. In the present study, we apply double-layer model to the association of two competing hosts and a microorganism. Two lattices are prepared: one is for hosts, and the other is for microorganism. The basic equation obtained by mean-field theory is an extension of Lotka-Volterra competition model. Both mathematical analysis and numerical simulations reveal that a shared microbial mutualist can permit the coexistence of competing hosts. From the derived condition of coexistence, we believe the microbial mutualism promotes biodiversity in many ecological systems.

Coexistence states of a Lotka Volterra cooperative system with cross diffusion

In this work we study the steady state cross diffusive Lotka Volterra cooperative system. We have incorporated different type of cross diffusive term at each trophic level. Using bifurcation theory, we derive sufficient conditions which ensures existence of positive solutions of the system. The analysis carried out allows us to give an affirmative answer to the question that whether cooperative species under the influence of cross diffusion can co exist or not. We have also compared the results obtained for the competitive Lotka Volterra system with our results and an ecological explanation is provided to understand the difference. • To study a cooperative Lotka Volterra prey predator system with different type of cross diffusion at each trophic level. • To show using bifurcation theory that these positive solution branches tend to have an extension that is globally defined. • To find out through the analysis carried out whether a cross diffusive cooperative system admits coexistence states. • Comparison of the results with the corresponding predator prey system.

Dynamical behavior of a Lotka–Volterra competition system in open advective environments

In this paper, we are concerned with a general reaction–diffusion–advection model which characterizes the interactions between two competing species in river environments. By regarding advection rates of two species as variable parameters, we provide an efficient way to completely understand the local dynamics. It turns out that there always exist two critical curves which may separate competition outcomes into competitive exclusion, bistability, and coexistence. As a further development, under the assumption that the ratio of advection and diffusion rates of one species is greater than or equal to that of the other species, we give a specific classification on the global dynamics, which shows that bistability does not happen, but coexistence and competitive exclusion may occur. These interesting results indicate that advective movements play a key role in determining the dynamical behaviors.

Spatial competition strategies: The case of two refuges

In this paper we simulate numerically different types of positive solutions of diffusive Lotka–Volterra competition systems: semitrivial positive solutions, semitrivial positive metasolutions, coexistence states and meta-coexistence states, in the case in which one of the species has two refuges. The existence of these solutions has been demonstrated in López-Gómez (2003). Once the location and extension of both refuges and the distribution of the resources in the territory are known, the competition strategies of each of the species can be designed and incorporated into the model. Or vice versa, if the competition strategies are known, the resources into the territory can be distributed in a way to obtain better results.

Propagation dynamics of Lotka-Volterra competition systems with asymmetric dispersal in periodic habitats

This paper is mainly concerned with the new types of entire solutions which are different from pulsating fronts of Lotka-Volterra competition systems with asymmetric dispersal in spatially periodic habitats. The asymmetry of kernel function leads to a great influence on the profile of pulsating fronts and the sign of wave speeds, which gives rise to the properties of the entire solution more complex and diverse. We first give a relationship between the critical speeds c ⁎ ( ξ ) and c ⁎ ( − ξ ) , corresponding to the minimal speeds of two pulsating fronts propagating in the direction of ξ and − ξ , respectively. Then, the exponential behavior of pulsating fronts as they approach their limiting states is obtained. Finally, by considering the interactions of two different pulsating fronts coming from two opposite/same directions and applying the super-/subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of some different types of entire solutions defined for all time and whole space. Moreover, we give some numerical simulations to describe intuitively these obtained entire solutions.

On a Lotka-Volterra Competition Diffusion Model with Advection

On a Lotka-Volterra Competition Diffusion Model with Advection

Existence of Front–Back-Pulse Solutions of a Three-Species Lotka–Volterra Competition–Diffusion System

Existence of Front–Back-Pulse Solutions of a Three-Species Lotka–Volterra Competition–Diffusion System

Modelling temperature-dependent dynamics of single and mixed infections in a plant virus

Multiple viral infection is an important issue in health and agriculture with strong impacts on society and the economy. Several investigations have dealt with the population dynamics of viruses with different dynamic properties, focusing on strain competition during multiple infections and the effects on viruses’ hosts. Recent interest has been on how multiple infections respond to abiotic factors such as temperature (T). This is especially important in the case of plant pathogens, whose dynamics could be affected significantly by global warming. However, few mathematical models incorporate the effect of T on parasite fitness, especially in mixed infections. Here, we investigate simple mathematical models incorporating thermal reaction norms (TRNs), which allow for quantitative analysis. A logistic model is considered for single infections, which is extended to a Lotka-Volterra competition model for mixed infections. The dynamics of these two models are investigated, focusing on the roles of T-dependent replication and competitive interactions in both transient and asymptotic dynamics. We determine the scenarios of co-existence and competitive exclusion, which are separated by a transcritical bifurcation. To illustrate the applicability of these models, we ran single- and mixed-infection experiments in plants growing at 20 ∘C and 30 ∘C using two strains of the plant RNA virus Pepino mosaic virus. Using a macroevolutionary algorithm, we fitted the models to the data by estimating the TRNs for both strains in single infections. Then, we used these TRNs to feed the mixed-infection model estimating the strength of competition. We found an asymmetrical pattern in which each strain dominated at different T values due to differences in their TRNs. We also identified that T can modify competition interference greatly for both isolates. The models proposed here can be useful for investigating the outcomes of multiple-infection dynamics under abiotic changes and have implications for the understanding of viral responses to global warming.

Traveling Wave Solutions of Partial Differential Equations Via Neural Networks

This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller–Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller–Segel equation, the Allen–Cahn model with relaxation, and the Lotka–Volterra competition model.

Singular limit for a reaction-diffusion-ODE system in a neolithic transition model

A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed in [7]. In this model, farmers are assumed to be divided into two subpopulations according to a mobility rule, namely, into sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of farmers and it is superimposed on the classical Lotka-Volterra competition model, so that it is described by a three-component reaction-diffusion-ODE system. In this article we consider a singular limit problem when the conversion rate tends to infinity and prove under appropriate conditions that solutions of the three component system converge to solutions of a two-component system with a linear diffusion and nonlinear degenerate diffusion.