Two-species Lotka-Volterra Model Research Articles

Bifurcation and stability of a two-species diffusive Lotka-Volterra model

This paper is devoted to a two-species Lotka-Volterra model with general functional response. The existence, local and global stability of boundary (including trivial and semi-trivial) steady-state solutions are analyzed by means of the signs of the associated principal eigenvalues. Moreover, the nonexistence and steady-state bifurcation of coexistence steady-state solutions at each of the boundary steady states are investigated. In particular, the coincidence of bifurcating coexistence steady-state solution branches is also described. It should be pointed out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov-Schmidt reduction, and bifurcation theory.

A global asymptotic stability condition for a Lotka–Volterra model with indirect interactions

ABSTRACTA two-species Lotka–Volterra model extended with an arbitrary number of indirect interactions through diffusible and renewable compounds is presented in view of its considerable interest to the microbial community modelling. After the determination of the system’s fixed points and a short discussion over their local asymptotic stability, Lyapunov’s second method is applied to derive a sufficient condition of global asymptotic stability. Biologically, this condition indicates the necessity for one microbial type to show strong self-inhibition and the compounds to be quickly replaced.

Invasion waves in the presence of a mutualist

This paper studies invasion waves in the diffusive Competitor-Competitor-Mutualist model generalizing the 2-species Lotka-Volterra model studied by Weinberger et al. The mutualist may benefit the invading or the resident species producing two different types of invasions. Sufficient conditions for linear determinacy are derived in both cases, and when they hold, explicit formulas for linear spreading speeds of the invasions are obtained by linearizing the model. While in the first case the linear speed is increased by the mutualist, it is unaffected in the second case. Mathematical methods are based on converting the model into a cooperative reaction-diffusion system.

Environmental versus demographic variability in two-species predator-prey models.

We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.

Kinetic Monte Carlo simulations of travelling pulses and spiral waves in the lattice Lotka-Volterra model

Kinetic Monte Carlo simulations are used to study the stochastic two-species Lotka-Volterra model on a square lattice. For certain values of the model parameters, the system constitutes an excitable medium: travelling pulses and rotating spiral waves can be excited. Stable solitary pulses travel with constant (modulo stochastic fluctuations) shape and speed along a periodic lattice. The spiral waves observed persist sometimes for hundreds of rotations, but they are ultimately unstable and break-up (because of fluctuations and interactions between neighboring fronts) giving rise to complex dynamic behavior in which numerous small spiral waves rotate and interact with each other. It is interesting that travelling pulses and spiral waves can be exhibited by the model even for completely immobile species, due to the non-local reaction kinetics.

On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model

We aim to clarify the relationship between interacting three-species models and the two-species Lotka-Volterra (LV) model. We utilize mean-field theory and Monte Carlo simulations on two-dimensional square lattices to explore the temporal evolution characteristics of two different interacting three-species predator-prey systems: (1) a cyclic rock-paper-scissors (RPS) model with conserved total particle number but strongly asymmetric reaction rates that lets the system evolve towards one corner of configuration space; (2) a hierarchical food chain where an additional intermediate species is inserted between the predator and prey in the LV model. For model variant (1), we demonstrate that the evolutionary properties of both minority species in the steady state of this stochastic spatial three-species corner RPS model are well approximated by the LV system, with its emerging characteristic features of localized population clustering, persistent oscillatory dynamics, correlated spatio-temporal patterns, and fitness enhancement through quenched spatial disorder in the predation rates. In contrast, we could not identify any regime where the hierarchical model (2) would reduce to the two-species LV system. In the presence of pair exchange processes, the system remains essentially well-mixed, and we generally find the Monte Carlo simulation results for the spatially extended model (2) to be consistent with the predictions from the corresponding mean-field rate equations. If spreading occurs only through nearest-neighbor hopping, small population clusters emerge; yet the requirement of an intermediate species cluster obviously disrupts spatio-temporal correlations between predator and prey, and correspondingly eliminates many of the intriguing fluctuation phenomena that characterize the stochastic spatial LV system.

Global existence and nonexistence for some degenerate and strongly coupled quasilinear parabolic systems

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic systems u t = v α Δ u + u ( a 1 − b 1 u + c 1 v ) , v t = u β Δ v + v ( a 2 + b 2 u − c 2 v ) with null Dirichlet boundary condition and positive initial conditions describing a cooperating two-species Lotka–Volterra model with cross-diffusion, where the constants a i , b i , c i > 0 for i = 1 , 2 and α, β are non-negative. The local existence of positive classical solutions is proved. Moreover, the authors proved that the solutions are global if intra-specific competition of the species are strong, whereas the solutions may blow up if the inter-specific cooperation are strong and α , β ⩽ 1 .

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positi...

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.

Influence of evolution on the stability of ecological communities

In randomly assembled communities, diversity is known to have a destabilizing effect. Evolution may affect this result, but our theoretical knowledge of its role is mostly limited to models of small food webs. In the present article, I introduce evolution in a two-species Lotka-Volterra model in which I vary the interaction type and the cost constraining evolution. Regardless of the cost type, evolution tends to stabilize the dynamics more often in trophic interactions than for mutualism or competition. I then use simulations to study the effect of evolution in larger communities that contain all interaction types. Results suggest that evolution usually stabilizes the dynamics. This stabilizing effect is stronger when evolution affects trophic interactions, but happens for all interaction types. Stabilization decreases with diversity and evolution becomes destabilizing in very diverse communities. This suggests that evolution may not counteract the destabilizing effect of diversity observed in random communities.

Periodicity and blowup in a two-species cooperating model

In this paper, the cooperating two-species Lotka–Volterra model is discussed. The existence and asymptotic behavior of T -periodic solutions for the periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are first investigated. The blowup properties of solutions for the same system are then given. It is shown that periodic solutions exist if the intra-specific competitions are strong whereas blowup solutions exist under certain conditions if the intra-specific competitions are weak. Numerical simulations and a brief discussion are also presented in the last section.

Global Existence and Convergence of Solutions to a Cross-Diffusion Cubic Predator-Prey System with Stage Structure for the Prey

We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

COEXISTENCE AND ASYMPTOTIC PERIODICITY IN A COOPERATING MODEL

In this paper, the cooperating two-species Lotka–Volterra model is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability, and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions are weak. The existence of the positive T-periodic solutions, the local stability, and the global attractivity for the parabolic system are also given.

Spatial Variability Enhances Species Fitness in Stochastic Predator-Prey Interactions

We study the influence of spatially varying reaction rates on a spatial stochastic two-species Lotka-Volterra lattice model for predator-prey interactions using two-dimensional Monte Carlo simulations. The effects of this quenched randomness on population densities, transient oscillations, spatial correlations, and invasion fronts are investigated. We find that spatial variability in the predation rate results in more localized activity patches, which in turn causes a remarkable increase in the asymptotic population densities of both predators and prey and accelerated front propagation.

Coexistence in a strongly coupled system describing a two-species cooperative model

This paper is concerned with a cooperative two-species Lotka–Volterra model. Using the fixed point theorem, the existence results of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions are obtained. Our results show that this model possesses at least one coexistence state if the birth rates are big and cross-diffusions are suitably weak.

Coexistence of two species in a strongly coupled cooperative model

In this paper, the cooperative two-species Lotka–Volterra model is discussed. We study the existence of solutions to a elliptic system with homogeneous Dirichlet boundary conditions. Our results show that this problem possesses at least one coexistence state if the birth rates are big and self-diffusions and the intra-specific competitions are strong.

A degenerate parabolic system with self-diffusion for a mutualistic model in ecology

This paper deals with the behavior of positive solution for a degenerate parabolic system with homogeneous Dirichlet boundary conditions describing a cooperating two-species Lotka–Volterra model. The local existence and uniqueness of a classical solution are given. Some comparison principles and positivity lemmas are also presented. Further, we show that the solution is global if the intra-specific competitions of the species are strong, whereas the solution may blow up if the intra-specific competitions are weak.

Blowup estimates for a mutualistic model in ecology

The cooperating two-species Lotka-Volterra model is dis- cussed. We study the blowup properties of the solution to a parabolic system with homogeneous Dirichlet boundary conditions. The upper and lower bounds of blowup rate are obtained. ut d1 u = u(a1 b1u c1v); x 2 ; t > 0; vt d2 v = v(a2 b2u c2v); x 2 ; t > 0; u(x) = v(x) = 0; x 2 @ ; t > 0; u(x; 0) = u0(x); v(x; 0) = v0(x); x 2 ;

Rock-scissors-paper and the survival of the weakest.

In the children's game of rock-scissors-paper, players each choose one of three strategies. A rock beats a pair of scissors, scissors beat a sheet of paper and paper beats a rock, so the strategies form a competitive cycle. Although cycles in competitive ability appear to be reasonably rare among terrestrial plants, they are common among marine sessile organisms and have been reported in other contexts. Here we consider a system with three species in a competitive loop and show that this simple ecology exhibits two counter-intuitive phenomena. First, the species that is least competitive is expected to have the largest population and, where there are oscillations in a finite population, to be the least likely to die out. As a consequence an apparent weakening of a species leads to an increase in its population. Second, evolution favours the most competitive individuals within a species, which leads to a decline in its population. This is analogous to the tragedy of the commons, but here, rather than leading to a collapse, the 'tragedy' acts to maintain diversity.