40 problems

Let $r(x,t)$ be a fixed resource environment satisfying (H0), and consider the corresponding time-periodic forager-exploiter system. A non-constant positive periodic solution is a…

No-surfing conjecture. In the presence of a strong Allee effect, a result analogous to the theorem asserting no gene surfing of deleterious mutations should hold; in particular, de…

Foutel-Rodier–Etheridge conjecture. Under an appropriate scaling regime, the particle system should converge to the solution of the introduced system of PDEs.

Logarithmic grow-up conjecture. In the critical case $\sigma=-2$, solutions to Eq. (1) exhibit logarithmic grow-up as $t\to\infty$.

Let $\psi$ be a selection function such that $\psi(x)\geq\delta$ for all $x\in[0,1]$ and $\psi(x)=\delta$ for all $x\in[1-p,1]$ for some $p\in(0,1)$. Asymptotic-velocity conjecture…

Consider a particle system of branching Brownian motion with rank-based branching function $b:[0,1]\to[0,\infty)$ and increasing deletion function $D:[0,1]\to[0,1]$. Let $u$ denote…

Consider coupled systems of reaction-diffusion equations of activator-inhibitor type in which the diffusivities are widely separated. The associated spatial ordinary differential e…

Lambda-profile conjecture. $\Lambda$-shaped profiles should appear in $\Omega_{(0,1)}(u)$, so the inclusion in Theorem and Corollary might be strict.

The paper considers reaction-diffusion equations of the form … where $A$ and $f$ are the operator and reaction term from the deterministic equation, and $N$ is a Hölder continuous…

For each direction $e\in\mathbb{S}^{N-1}$, let $c_e$ denote an admissible speed of a pulsating front. A function of $e$ is sign-unchanging if it is continuous and does not change s…

Linear conjecture. Nonlinear differential equations governing population spread always have the same propagation velocity as their linear approximation.

Monotonicity conjecture. The quantity $|d_T|$ is nonincreasing as a function of $T>0$. Consequently, $d_T$ has a fixed sign independent of $T>0$, equal to the sign of

For any $T>0$ and $b1\in (0,1)$, let $X_T^{\alpha}(t)$ be the $\alpha$-level set of the almost periodic traveling wave solution $u_T(t,x)$, defined by … For each $s\in\mathbb{R}$,…

Consider the Cauchy problem studied in the paper, with parameters $m$, $p$, and $\sigma$, and let $L$ be the quantity defined by the equation referenced as $(\mathrm{const})$. A so…

Consider the Cauchy problem studied in the paper, with parameter $L$ defined by the equation referenced as $(\mathrm{const})$. A solution is non-trivial if it is not identically ze…

Let $f$ satisfy the hypotheses defining a propagating terrace connecting $1$ and $0$, with terrace solution $Phi(t,x;\xi)$ obtained from the corresponding traveling waves and suita…

Let $u$ be the solution considered in Theorem, and let $\Omega(u)$ denote its omega-limit set. Write $\mathcal{E}$ for the set defined in the paper, and let $\varphi$ be the minima…

Localised dihedral bifurcation conjecture. For any $$ taken sufficiently small, true localised dihedral solutions of the system, arising from the limit $N\to\infty$, should exist a…

Small-noise speed-limit conjecture. The limit

Consider a semilinear reaction-diffusion system with state constraints, in a setting where the nonlinearity is globally Lipschitz but controlled trajectories may move away from tar…

Let $\theta_{\textup{min}}$ and $\theta_{\textup{max}}$ be the minimum and maximum Allee thresholds, let $\alpha$ be the mutation parameter, and consider the critical case … Extinc…

Let $u=u(t,x,\mathbf y)$ be a population density structured by a phenotypic trait $\mathbf y\in\Omega\subset\mathbb R^k$, with $k\geq1$, satisfying … Here $\rho$ is the total popul…

Let $\theta_{\textup{min}}$ and $\theta_{\textup{max}}$ satisfy … and let $\alpha$ be the mutation parameter. The outcomes are classified as “E. or P.” when both extinction and per…

Let $\mathbfcal{A}$ denote the set of speed functions admissible for spatially periodic bistable reaction-diffusion equations, with each speed function assigning a propagation spee…

Let $V_0^{(1)}(t)$ and $V_0^{(2)}(t)$ be the first- and second-order perturbation processes in the stochastic travelling-wave expansion, with values in…