12 problems

Okubo et al.'s conjecture. The minimal speed $c^*$ is linearly selected, that is,

Reproduction-function proof conjecture. A unified proof for the correspondence between the competitive exclusion principle (CEP) and the LCP, which has only been established in par…

Let $\boldsymbol\lambda_+(\kappa,\alpha)$ denote the quantity appearing in the preceding theorem, and let the limits refer to the corresponding normalized free-energy sequence…

Consider the diffusive Lotka–Volterra strongly competitive system on an interval $(0,L)$ with boundary-constrained controls, parameters $a>0$ and $b>1$ satisfying $b>a$, initial da…

Consider the diffusive Lotka–Volterra system with boundary controls and admissible states in $[0,1]\times[0,1]$, and let $(w_1^*,w_2^*)$ be an equilibrium lying strictly in the int…

A tree is a hypergraph whose hyperedges all have size two, and an LV-system is a Lotka–Volterra system associated with such a hypergraph. Two associated systems are LV-equivalent w…

Let a tree-system be a Lotka–Volterra system associated with a tree, and say that two such systems are equivalent when they are related by the equivalence relation induced by linea…

Peak-configuration conjecture. The possible configuration of peaks is

Hofbauer–Sigmund conjecture. If $A$ is $D$-stable, then an equilibrium state involving all species globally attracts all solutions in $\mathbb{R}^N_+$.

Let $f=x^p y^p(1-x-y)$, with $p\geq 1$, and let $V=H_1(f^{-1}(t),\mathbb Z)/V_0$ be the reduced first homology of a regular fiber, where $V_0$ is the kernel of the intersection for…

Let $(1.1)$ be the competitive Lotka–Volterra system, let $I_n$ denote the set of species indices, let $\pi_i$ be the relevant coordinate hyperplane, and let $\gamma_i$ be the thre…

Consider the Lotka–Volterra system $(2.1)$ and its listed cases ${\rm (rlv1)}$–${\rm (rlv6)}$ and ${\rm (lv1)}$–${\rm (lv5)}$, whose centers are surrounded by period annuli. The cy…