Lotka Volterra Competition System Research Articles

Global dynamics of 3D competitive Lotka-Volterra equations with the identical intrinsic growth rate

This paper focuses on investigating complete global topological classification of the three dimensional competitive Lotka-Volterra system with the identical intrinsic growth rate in the compactification of the positive octant of R3 including its infinity. The system is divided into 37 topological classes among 33 stable nullcline classes by the relation on the parameters of competitive system.

A way to model stochastic perturbations in population dynamics models with bounded realizations

In this paper, we analyze the use of the Ornstein–Uhlenbeck process to model dynamical systems subjected to bounded noisy perturbations. In order to discuss the main characteristics of this new approach we consider some basic models in population dynamics such as the logistic equations and competitive Lotka–Volterra systems. The key is the fact that these perturbations can be ensured to keep inside some interval that can be previously fixed, for instance, by practitioners, even though the resulting model does not generate a random dynamical system. However, one can still analyze the forwards asymptotic behavior of these random differential systems. Moreover, to illustrate the advantages of this type of modeling, we exhibit an example testing the theoretical results with real data, and consequently one can see this method as a realistic one, which can be very useful and helpful for scientists.

Global well-posedness of advective Lotka–Volterra competition systems with nonlinear diffusion

AbstractThis paper investigates the global well-posedness of a class of reaction–advection–diffusion models with nonlinear diffusion and Lotka–Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic–elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka–Volterra competition systems.

Regularization of a backward problem for a Lotka–Volterra competition system

Lotka–Volterra systems are used to describe the dynamics of biological systems. We study the backward problem for the Lotka–Volterra system to determine the population density of species at preceding times. The problem is ill-posed in the sense that if the solution exists it does not depend continuously on the given data. We propose two stable regularization methods to regularize the system. Furthermore, under some a priori assumptions on the sought solution, we show that the corresponding regularized solutions converge to the exact solutions in L2-norm. Numerical simulations are presented to demonstrate the theoretical results.

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.

Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in $ n $–dimensional space. More precisely, we prove that all planar traveling waves with speed $ c>c^* $ are exponentially stable in $ L^{\infty}(\mathbb{R}^n ) $ in the form of $ t^{-\frac{n}{2\alpha }}{\rm{e}}^{-\varepsilon_{\tau} \sigma t} $ for some constants $ \sigma >0 $ and $ \varepsilon_{\tau} \in (0,1) $, where $ \varepsilon_{\tau} = \varepsilon(\tau) $ is a decreasing function refer to the time delay $ \tau>0 $. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed $ c = c^* $, we show that they are algebraically stable in the form of $ t^{-\frac{n}{2\alpha}} $. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

Permanence and asymptotic behaviors of stochastic competitive Lotka-Volterra system with Markov switching and Lévy noise

This paper concerns the dynamics of a stochastic competitive Lotka-Volterra system with Markov switching and L?vy noise. The results show that stochastic permanence and extinction are characterized by two parameters B1 and B2: if B1B2 ? 0, then the system is either stochastically permanent or extinctive. That is, it is extinctive if and only if B1 < 0 and B2 < 0; otherwise, it is stochastically permanent. Some existing results are included as special cases.

A concrete example with three limit cycles in Zeeman’s class 29 for three dimensional Lotka–Volterra competitive systems

The number of limit cycles for three dimensional Lotka–Volterra competitive systems is an open problem. Recently, we have presented a concrete example with three limit cycles in Zeeman’s class 27 [6].In this paper, we present a concrete example with three limit cycles which belongs to Zeeman’s class 29. We explicitly give the critical parameter values such that the interior equilibrium is an exact unstable weak focus of order two. Also we verify that the system is permanent. This implies that there can exist three limit cycles around the interior equilibrium under suitable perturbations. We actually generate multiple limit cycles, and confirm them by numerical simulation. In addition, we present some other examples with three limit cycles in Zeeman’s class 27.

Existence and uniform asymptotic stability of positive almost periodic solutions for three-species Lotka–Volterra competitive system on time scales

In this paper, we establish existence and uniform asymptotic stability of unique positive almost periodic solutions for three-species Lotka–Volterra competitive system on time scales by using Lyapunov functional method.

Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case

In this paper, we will prove the uniqueness of traveling front solutions with critical and noncritical speeds, connecting the origin and the positive equilibrium, for the classical competitive Lotka-Volterra system with diffusion in the weak competition, which partially answers the open problem presented by Tang and Fife in [ 17 ]. In fact, once these traveling front solutions have the same wave speed and the same asymptotic behavior at $ξ = ±∞$, they are unique up to translation.

Stability in distribution of stochastic Lotka–Volterra delay system under regime switching

In this paper, we consider a class of stochastic competitive Lotka–Volterra system with time delay and Markovian switching. We prove that there exists a global positive solution under the random perturbation. Some sufficient conditions for the stability in distribution of the system are established which improved the classical case. An example is given to illustrate theoretical results.

Feedback Control and Parameter Invasion for a Discrete Competitive Lotka–Volterra System

State feedback is used to stabilize the Turing instability at the unstable equilibrium point of a discrete competitive Lotka–Volterra system. In addition, a regularization method is applied to parameter inversion for the given Turing system and numerical simulation can verify the effectiveness of the algorithm. Furthermore, how less or more sample data and dependence on the initial state affect estimation procedure are tested.

Dynamic Behaviors of a Nonautonomous Impulsive Competitive System with the Effect of Toxic Substance

We firstly propose a nonautonomous impulsive Lotka-Volterra competitive system with the effect of toxic substance. Only one of the two species could produce toxic substance. Sufficient condition which guarantees the extinction of one of the species and the global attractivity of the other species is obtained. We also present an example to verify our main results, which show that species still is possibly driven to extinction when only one of the two species produces toxic substances. The results of this paper supplement the existing results.

Extinction in a Nonautonomous Discrete Lotka-Volterra Competitive System with Time Delay

This paper is concerned with a nonautonomous discrete Lotka-Volterra competitive system with time delay. By using some analytical techniques, we prove that, under certain conditions, one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.

Turing Patterns of a Lotka–Volterra Competitive System with Nonlocal Delay

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

Uncertain impulsive Lotka–Volterra competitive systems: Robust stability of almost periodic solutions

Models with uncertain values of their parameters are of a practical significance in different fields of science and technology. Uncertainty in the competitive models can greatly affect their dynamical behavior and, therefore, the analysis of the effects of uncertain terms in such models is very important in both theory and application. In this paper, we extend the existing N-species impulsive competitive models to the uncertainty case. The existence of a unique strictly positive and robustly exponentially stable almost periodic solution is investigated for the model. The main results are obtained by using Lyapunov-type functions and a comparison principle.

Spreading of two competing species governed by a free boundary model in a shifting environment

We investigate the long-time spreading behavior of two competing species in a shifting environment. The evolution of the population densities of the species is governed by a one space dimension diffusive Lotka-Volterra competition system with two different free boundaries, describing a dynamical process of two competitors invading into the new habitat in the same direction. It is assumed that the unfavourable region of the environment moves into the otherwise favourable homogeneous environment with a given speed c>0 in the spreading direction of the species. By close examination of certain simple cases, we show that such a shifting environment could reverse the fates of the species. A complete classification of the long-time dynamical behavior of the system is obtained for such cases.

Hopf bifurcation in time-delayed Lotka–Volterra competition systems with advection

ABSTRACTIn this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s).

Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.

Stability in Distribution of a Stochastic Competitive Lotka-Volterra System with S-type Distributed Time Delays

This paper concerns the dynamics of a stochastic competitive Lotka-Volterra system with S-type distributed time delays. First, sufficient conditions for stability in mean and extinction of each population are obtained. Then, sufficient conditions for the stability in distribution of the solutions (SDS) to the system are established. Finally, some numerical simulations are provided to support the theoretical results.