Semi--exact equilibrium solutions for three-species competition--diffusion systems

Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system

“…In particular, since the moving frame is centred on the leading front, in the entire solution the spatial profile will tend pointwise to the profile of the stable non-trivial wave as the time goes to ±∞. In other words, for r 3 = r HC it seems that we have a homoclinic bifurcation in the system (8), for the equilibrium given by the stable non-trivial travelling wave, the homoclinic orbit at r 3 = r HC given by the entire solution described above and the periodic orbit for r 3 < r HC given by the breathing travelling wave.…”

“…Recently, it has been shown that non-trivial solutions never exist if r 3 is sufficiently small [9]. On the other hand, there are only a few analytical results on the existence of non-trivial solutions [8]. Numerically, the difficulty lies in finding the right parameter values for which non-trivial solutions exists.…”

An N-barrier maximum principle for autonomous systems of <inline-formula><tex-math id="M1">\begin{document}$n$\end{document}</tex-math></inline-formula> species and its application to problems arising from population dynamics

“…NBMP for n species (Theorem 1.1) is proved in Section 2. As an application of Corollary 1.2, we establish in Section 4 a nonexistence result for traveling wave solutions of the Lotka-Volterra system for four competing species 3,4) are positive constants; θ ∈ R is the propagation speed of the traveling wave.…”

Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats

“…We remark that when k 0 = 0 and k 3 = 2, k 1 = −2, the generalized Jacobi elliptic function φ = φ(x) which solves (9) is reduced to the hyperbolic function tanh z. In this case, exact traveling wave solutions for Lotka-Volterra competitive systems of two-species ( [20]) and three-species ( [4,5]) have been found by the hyperbolic function method. However, exact solutions in [4,5,20] are given only for the case where the growth rates are constants.…”

“…Instead, we shall make it by using semi-exact solutions as well as employing the method of upper and lower solutions. In [4], the authors show that under certain conditions, traveling wave solutions for Lotka-Volterra competitive systems of three-species exist by giving semi-exact solutions. By a semi-exact solution which seems, to the best of our knowledge, to be the first in the literature, we mean that each component of the solution (u, v, w) can be expressed in terms of a polynomial of an implicit function that satisfies an autonomous differential equation.…”