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Register a new userTraveling Waves in a Three Species Competition-cooperation System
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This paper studies the traveling wave solutions to a three species competition cooperation system. The existence of the traveling waves is investigated via monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions. The results are then applied to a Lotka Volterra system with spatially averaged and temporally delayed competition.
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We investigate the spreading properties of a three-species competition-diffusion system, which is non-cooperative. We apply the Hamilton-Jacobi approach, due to Freidlin, Evans and Souganidis, to establish upper and lower estimates of spreading speed of the slowest species, in terms of the spreading speed of two faster species, and show that the estimates are sharp in some situations. The spreading speed will first be characterized as the free boundary point of the viscosity solution for certain variational inequality cast in the space of speeds. Its exact formulas will then be derived by solving the variational inequality explicitly. To the best of our knowledge, this is the first theoretical result on three-species competition system in unbounded domains.
We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling $1$-pulse solutions with undetermined width and propagating speed. Next, we compute the associated action functional for this profile from which we derive the conditions for existence and a saddle-node bifurcation as the zeros of the action functional and its derivatives. We obtain the same conditions by using a different analytical approach that exploits the singular limit of the problem. We also apply this methodology of the action functional to the problem for traveling $2$-pulse solutions and derive the explicit conditions for existence and a saddle-node bifurcation. From these we deduce a necessary condition for the existence of traveling $2$-pulse solutions. We end this article with a discussion related to Hopf bifurcations near the saddle-node bifurcation.
We study the (hydro-)dynamics of multi-species driven by alignment. What distinguishes the different species is the protocol of their interaction with the rest of the crowd: the collective motion is described by different communication kernels, $phi_{alphabeta}$, between the crowds in species $alpha$ and $beta$. We show that flocking of the overall crowd emerges provided the communication array between species forms a connected graph. In particular, the crowd within each species need not interact with its own kind, i.e., $phi_{alphaalpha}=0$; different species which are engaged in such `game of alignment require a connecting path for propagation of information which will lead to the flocking of overall crowd. The same methodology applies to multi-species aggregation dynamics governed by first-order alignment: connectivity implies concentration around an emerging consensus.
In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders $Omegatimesmathbb Rsubseteqmathbb R^{d+1}$ have only real discrete spectrum located to the right of the right most point of the essential spectrum. We describe the essential spectrum using the limiting properties of the potential. To track the discrete spectrum we use spatial dynamics and bi-semigroups of linear operators to estimate the decay rate of eigenfunctions associated to isolated eigenvalues.
For any $N geq 2$, we show that there are choices of diffusion rates ${d_i}_{i=1}^N$ such that for $N$ competing species which are ecologically identical and having distinct diffusion rates, the slowest diffuser is able to competitive exclude the remainder of the species. In fact, the choices of such diffusion rates is open in the Hausdorff topology. Our result provides some evidence in the affirmative direction regarding the conjecture by Dockery et al. in cite{Dockery1998}. The main tools include Morse decomposition of the semiflow, as well as the theory of normalized principal bundle for linear parabolic equations.
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