11 problems

Consider a reaction-network dynamical system with a positive complex-balanced equilibrium, meaning an equilibrium at which the total flow into and out of each complex agrees. The a…

Global stability conjecture. The equilibrium $\mathcal E^*$ is globally asymptotically stable for positive initial conditions, regardless of the dimension of $\Gamma_{\Omega}$.

Let $f\in C^3(\mathbb R)$ satisfy the hypotheses $(\mathbf H)$: $xf(x)<0$ for $x\ne0$ with $f'(0)<0$; $f$ has at most one critical point $x^*$, with $f(x)>C$ for some…

Global stability conjecture. There exists a positive number $\varepsilon$ such that $E_*$ is globally asymptotically stable for

Consider the scalar delay differential equation … under conditions (W), with the trivial solution and its local asymptotic stability understood as in the paper. Smith and Wright's…

Consider the Clark equation, the case $\alpha=1$ of the delay difference equation … where $k$ is a fixed integer delay and $f:\mathbb R_+\to\mathbb R_+$ is smooth, decreasing, and…

Consider the two-species competition patch model with equal drift rates $q_1=q_2=q>0$, and let $(\bm{\tilde u},\bm 0)$ and $(\bm 0,\bm{\tilde v})$ denote the two semi-trivial equil…

Let an epidemic patch model have an endemic equilibrium (EE) solution, and suppose that this EE solution is unique. Global stability conjecture. If the EE solution is unique, then…

Let $\Lambda$ be a differentiable vector field with Jacobian matrix $J(X)$, and consider the differential system … Suppose that, at any point, the eigenvalues of $J(X)$ have negati…

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

Let $u_*\equiv 0$ denote the trivial steady state of the nonlinear problem \, let $\mathcal{Y}$ be the parameter space, and let $\mathcal{R}:\mathcal{Y}\to\mathbb{R}$ be the net…