Lotka Volterra Cooperative System Research Articles

Forwards Attractor Structures in a Planar Cooperative Non-autonomous Lotka–Volterra System

The global attractor of a dissipative dynamical system provides the necessary information to understand the asymptotic dynamics of all the system’s solutions. A crucial question consists in finding the structure of this set. In this paper we provide a full characterization of the structure of attractors for a planar non-autonomous Lotka–Volterra cooperative system. We show sufficient conditions for the existence of forward attractors and give a full description of them by proving the existence of such bounded global solutions that all bounded global solutions join them, i.e. converge towards them when time tends to plus and minus infinity. These results generalize those known in an autonomous framework. The case of particular interest in our work is the situation where globally forward-stable non-autonomous solutions have both coordinates strictly positive. We study this case in detail and obtain sufficient conditions that the problem parameters must satisfy in order to obtain various structures of non-autonomous attractors. This allows us to understand different paths of the solutions towards the unique globally stable one.

Spatiotemporal Dynamics of a General Two-Species System with Taxis Term

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.

Global dynamics of 3D cooperative Lotka-Volterra system with the identical intrinsic growth rate

In this paper, we complete the classification on global topological structures of the three-dimensional cooperative Lotka-Volterra system with the identical intrinsic growth rate inside the Poincaré compactification of the positive octant of R3. Precisely, with the help of the replicator equations it is proved that this kind of system can have exactly 8 topologically different phase portraits. As a consequence, we obtain the necessary and sufficient conditions for the system to be bounded in the positive octant.

Propagation Dynamics of Forced Pulsating Waves for a Time Periodic Lotka-Volterra Cooperative System with Nonlocal Effects in Shifting Habitats

Propagation Dynamics of Forced Pulsating Waves for a Time Periodic Lotka-Volterra Cooperative System with Nonlocal Effects in Shifting Habitats

Poisson stable solutions and solution maps for stochastic functional differential equations

We observe Poisson stable solutions for nonlinear stochastic functional differential equations (SFDEs) with finite delay. Firstly, we prove the existence and uniqueness of bounded (in square-mean sense) solutions and solution maps for SFDEs with finite delay by remoting start (or dissipative method) and classical pull-back attraction method. Then, based on the relationship between the solution and coefficients, we obtain such Poisson stable solutions by using Shcherbakov’s comparability method in character of recurrence. Because the solutions of the delay equations are not-Markov, we employ the solution maps in the appropriate phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability. For illustration of our results, we give the application arising from stochastic Lotka–Volterra cooperative systems with distributed delay.

Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results.

Spreading speeds determinacy for a cooperative Lotka–Volterra system with stacked fronts

Spreading speeds determinacy for a cooperative Lotka–Volterra system with stacked fronts

Asymptotic stability of a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps

In this article, we study a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps. Applying the M-matrix theory, we prove the existence and uniqueness of a global solution for the system. Then we use an optimized Euler-Maruyama numerical scheme to approximate the solution. We obtain second-moment boundedness and convergence rate of the numerical solutions. The numerical solutions illustrate the theoretical results. For more information see https://ejde.math.txstate.edu/Volumes/2023/02/abstr.html

Nonlinear Stability of Traveling Wavefronts for a Discrete Cooperative Lotka-Volterra System With Delays

Nonlinear Stability of Traveling Wavefronts for a Discrete Cooperative Lotka-Volterra System With Delays

Spreading properties for a time-space periodic Lotka–Volterra cooperative system in multi-dimensional media

Spreading properties for a time-space periodic Lotka–Volterra cooperative system in multi-dimensional media

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.

Turing Patterns for a Nonlocal Lotka\u2013Volterra Cooperative System

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

The asymptotic stability of numerical analysis for stochastic age-dependent cooperative Lotka-Volterra system.

In this study, we explore a stochastic age-dependent cooperative Lotka-Volterra (LV) system with an environmental noise. By applying the theory of M-matrix, we prove the existence and uniqueness of the global solution for the system. Since the stochastic age-dependent cooperative LV system cannot be solved explicitly, we then construct an Euler-Maruyama (EM) numerical solution to approach the exact solution of the system. The convergence rate and the pth-moment boundedness of the scheme have also been obtained. Additionally, numerical experiments have been conducted to verify our theoretical results.

On a three species ratio-dependent Lotka-Volterra cooperative system with delays

On a three species ratio-dependent Lotka-Volterra cooperative system with delays

Distinct Spreading Speeds in a Lotka–Volterra Cooperative System

This paper deals with the spreading speeds in the classical Lotka–Volterra cooperative system, of which the bounds have been studied earlier. By introducing an auxiliary cooperative system and constructing an upper solution, we obtain a sufficient condition to confirm two distinct spreading speeds of unknown functions. Due to the different average moving speeds of different level sets, we find the existence of propagation terraces in such a cooperative system with constant coefficients. We also present some numerical results to illustrate our results.

Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat.

Dynamic Models for Exploring the Resilience in Territorial Scenarios

The present paper focuses on the role covered by dynamic models as support for the decision-making process in the evaluation of policies and actions for increasing the resilience of cities and territories. In recent decades, urban resilience has been recognized as a dynamic and multidimensional phenomenon that characterizes urban and metropolitan area dynamics. Therefore, it may be considered a fundamental aspect of urban and territorial planning. The employment of quantitative methods, such as dynamic models, is useful for the prediction of the dynamic behavior of territories and of their resilience. The present work discusses the system dynamics model and the Lotka–Volterra cooperative systems and shows how these models can aid technicians in resilience assessment and also decision makers in the definition of policies and actions, especially if integrated in wide evaluation frameworks for urban resilience achievements. This paper aims to provide an epistemological perspective of the application of dynamic models in resilience assessment, underlying the possible contribution to this issue through the analysis of a real case study and methodological framework. The main objective of this work is to lay the basis for future compared applications of these two models to the same case study.

Forced waves and their asymptotics in a Lotka–Volterra cooperative model under climate change

The existence and asymptotics of forced traveling wave front in a Lotka–Volterra cooperative system under climate change are concerned in this paper. By constructing appropriate upper and lower solutions combined with the monotone iteration scheme, we show that for any given positive speed of the shifting habitat edge, there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed, which indicates that no matter how moderate of the habitat shifts, two species will still be driven to extinction as the mutualistic effects exist but weak. Numerical simulations are also given to illustrate our results.

Pattern formation and parameter inversion for a discrete Lotka–Volterra cooperative system

In this paper, stability analysis is applied to a discrete Lotka–Volterra cooperative system with the periodic boundary conditions, then Turing pattern formation conditions can be derived, theory analysis and numerical simulation show that turing patterns can be realized. In addition, we also pay attention on what reason or what system environment to result into the current state patterns, which can be reduced to estimate or identify the system parameter. A regularization method is applied to parameter inversion, and numerical simulation can verify the effectiveness of the algorithm.

Long-time behaviors of a stochastic cooperative Lotka–Volterra system with distributed delay

This article focuses on a stochastic cooperative Lotka–Volterra system with distributed delay. We first transfer the stochastic system with weak kernel into a degenerate stochastic system made up of four equations. For the deterministic system, global stability of the positive equilibrium is investigated. For the stochastic system with distributed delay, sharp sufficient conditions for the persistence of two species are established. What is more, we obtain the existence and uniqueness of the stationary distribution by constructing suitable Lyapunov function and proving the global attraction of the positive solution. The results show that, the weaker white noises can ensure the existence of a unique stationary distribution and the stronger white noises can result in the extinction of one or two species, though the positive equilibrium is globally stable without white noises.