Diffusive Lotka-Volterra System Research Articles

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.

Steady states of a diffusive Lotka–Volterra system with fear effects

Steady states of a diffusive Lotka–Volterra system with fear effects

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

AbstractNew classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

The Lotka-Volterra models with non-local reaction terms

<p style='text-indent:20px;'>In this paper we consider a diffusive Lotka-Volterra system including nonlocal terms in the reaction functions. We analyze the main types of interactions between species: competition, predator-prey and cooperation. We provide existence and non-existence of positive solutions results. For that, we employ mainly bifurcation method and a priori bounds.</p>

Traveling wave solutions for the diffusive Lotka–Volterra equations with boundary problems

The main purpose of this paper is to study the traveling wave solutions of the diffusive Lotka–Volterra systems with boundary conditions. With the help of Gröbner bases elimination method, a series of new traveling wave solutions have been obtained through symbolic computation. In particular, it is worth noting that these traveling wave solutions may inspire us to explore new phenomena in this system. The obtained results in this paper substantially improve the corresponding results in the known literatures. Finally, we summarize the current study and give the future work.

New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System

The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well.

On Some Dynamics of a Diffusive Lotka-Volterra Competition-Advection System with Lethal Boundary Conditions

In this paper, we mainly study a diffusive Lotka-Volterra competition-advection system with lethal boundary conditions in a general heterogeneous environment. By using the basic theory of partial differential equations and some nonlinear analysis techniques, we investigate the existence, uniqueness and global asymptotic behavior of steady-state solutions of the system equations. The existence, uniqueness and global asymptotic behavior of steady-state solutions are proved by upper and lower solutions, maximum principle and other methods. In theory, the methods and skills to deal with this kind of nonlinear problem are further developed, which provides a theoretical basis for understanding some practical problems.

Entire solutions of diffusive Lotka-Volterra system

This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system on the real line. We prove the existence of some entire solutions that are asymptotic, as t→−∞, to a traveling wave consisting of a single species. Depending on system parameters, these entire solutions evolve into two ore more stacked invasion waves as t→+∞. Our results cover both the weak and strong competition case. In the weak competition case, the existence of a class of entire solutions that forms a 4-dimensional manifold is proved.

Existence and stability of non-monotone travelling wave solutions for the diffusive Lotka–Volterra system of three competing species

This paper considers the problem: if coexistence occurs in the long run when a third species w invades an ecosystem consisting of two species u and v on , where u, v and w compete with one another. Under the assumption that the influence of w on u and v is small and other suitable conditions, we show that the three species can coexist as a non-monotone travelling wave. Such type of non-monotone waves plays an important role in the study of three-species phenomena. However, fewer results are known for the existence of such waves in the literature. Our approach, based on the method of sub- and super-solutions and bifurcation theory, provides a new approach to construct non-monotone waves of this type. Moreover, we show that the waves we construct are stable. To the best of our knowledge, this is the first rigorous result of stability for such type of waves.

Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

<p style='text-indent:20px;'>This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue.

Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats

We are concerned with the coexistence states of the diffusive Lotka-Volterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the one-dimensional problem, we employ the generalized Jacobi elliptic function method to find semi-exact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the above-mentioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semi-trivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included.

Asymptotic spreading speed for the weak competition system with a free boundary

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.

An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([ 6 ]), we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. To this end, this gives rise to the N-barrier maximum principle for a second-order elliptic equation involving two distinct unknown functions and a quadratic nonlinearity. An immediate consequence of the N-barrier maximum principle is an a priori estimate for the total populations of the two species. As an application of this maximum principle, we show under certain conditions the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new \begin{document}$(1, 0, 0)$\end{document} - \begin{document}$(u^{*}, v^{*}, 0)$\end{document} waves are given in terms of the tanh function, provided that the system's parameters satisfy certain conditions.

A maximum principle for diffusive Lotka–Volterra systems of two competing species

Using an elementary approach, we establish a new maximum principle for the diffusive Lotka–Volterra system of two competing species, which involves pointwise estimates of an elliptic equation consisting of the second derivative of one function, the first derivative of another function, and a quadratic nonlinearity. This maximum principle gives a priori estimates for the total mass of the two species. Moreover, applying it to the system of three competing species leads to a nonexistence theorem of traveling wave solutions.

Stability and bifurcation in a diffusive Lotka–Volterra system with delay

In this paper, we investigate the dynamics of a class of diffusive Lotka–Volterra equation with time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady state solution is investigated by applying Lyapunov–Schmidt reduction. The stability and nonexistence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.

Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species

In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed.

Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system

This paper is concerned with a three dimensional diffusive Lotka-Volterra system which is combined with cooperative-competitive interactions between the three species. By using the method of super-sub solutions and comparison principle with cross iteration, some results on the asymptotic spreading speed of the system are established under certain assumptions on the parameters appearingin the system.

Stability of a three-dimensional diffusive Lotka–Volterra system of type-K with delays

A three-dimensional diffusive Lotka–Volterra system of type-K with delays is investigated. We give a stability analysis in detail for all equilibria of the system and obtain some threshold conditions for linear instability and linear asymptotic stability of each equilibrium. We develop the analytical method for stability analysis of reaction–diffusion equations with multi-delays.

Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system

Lie and Q-conditional symmetries of the classical three-component diffusive Lotka–Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component nonlinear reaction–diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka–Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed.

Traveling wave solutions in [formula omitted]-dimensional delayed reaction–diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions for n-components delayed reaction–diffusion systems with mixed monotonicity. Based on a certain kind of mixed-quasimonotonicity reaction terms of higher dimension, we propose new conditions on the reaction terms and new definitions of upper and lower solutions. By using Schauder’s fixed point theorem and a cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained will be applied to type-K monotone diffusive Lotka–Volterra systems of higher dimension.