Uniqueness Of Coexistence States Research Articles

On the existence of coexistence states for an advection-cooperative system with spatial heterogeneities

This paper is devoted to the analysis of the existence of coexistence states for cooperative systems under homogeneous Dirichlet boundary conditions. We assume rather general spatial heterogeneities for the coefficients of the non-linearities and the cooperative terms. In order to obtain sharp existence results, we add advection terms related to the cooperative terms so that the system can be transformed into a variational one. Applying then several variational methods and bifurcation theory, we arrive at necessary and sufficient condition on the parameter λ for the existence and the uniqueness of coexistence states. We also compare the conditions for the heterogeneous case and the constant case.

Steady states of a non-cooperative model arising in nuclear engineering

This paper studies the existence and the uniqueness of coexistence states in a spatially heterogeneous reaction diffusion model originated by the theory of nuclear reactors. The mathematical relevance of our analysis relies on the fact that the underlying variational systems are of non-cooperative type and, consequently, the nice comparison techniques available for cooperative systems fail to be true, making the problem of ascertaining whether or not the model admits a unique coexistence state an extremely challenging task. In this paper we are giving a series of new uniqueness results complementing those available in the literature.

Coexistence states for a diffusive one-prey and two-predators model with B–D functional response

In this paper, we study a diffusive one-prey and two-predators system with Beddington–DeAngelis functional response. The sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory. In addition, the stability and uniqueness of coexistence states are investigated. Finally, we give the sufficient conditions for extinction and permanence of the time-dependent system.

Uniqueness and asymptotic behavior of coexistence states for a non-cooperative model of nuclear reactors

In this paper, we analyze the asymptotic behavior of coexistence states for a non-cooperative model of nuclear reactors. In addition, we also present some remarks on the uniqueness of coexistence states in a high dimensional case. Our results complement the work of López-Gómez [J. López-Gómez, The steady states of a non-cooperative model of nuclear reactors, J. Differential Equations 246 (2009), 358–372].

Analysis of diffusive two-competing-prey and one-predator systems with Beddington–Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington–DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.

Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each pointx∊ Ω we have a pair of ODE's: thekinetic. If for somex∊ Ω kinetic has a unique stable coexistence state, we show that there existsuch that for everythe RD-model ispersistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there existxu,xv∊ Ω such that the semitrivial coexistence states (u, 0) and (0,v) of thekineticare globally asymptotically stable atx=xuandx=xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.Singular perturbation techniques, monotone schemes, fixed point index, global analysis ofpersistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model hasthree or morecoexistence states.