15 problems

Consider the two-species competition system in the parameter regime $D>d$, with critical advection value $\alpha_1>0$, semi-trivial equilibrium $(u^*,0)$, and a positive steady sta…

Let $u(t,x)$ and $v(t,x)$ denote the densities of species $u$ and $v$ at time $t$ and location $x\in\Omega$, and let $u_0,v_0$ be initial conditions satisfying $u_0+v_0\leq 1$. Wri…

Let $K$ be the carrying-capacity function and let the admissible fishing strategies be constrained by requiring that the $L^1$ distance from $K$ to the admissible controls is small…

Consider the two-strain SIS spatial model on a domain $\Omega$ with susceptible density $S$, infected densities $I_1,I_2$, and strains that differ only in their recovery-rate funct…

Consider the two-strain SIS spatial model on a domain $\Omega$ with susceptible density $S$, infected densities $I_1,I_2$, diffusion rates $d_{I,1},d_{I,2}$, transmission rates…

Consider the two-species phytoplankton competition model with equal advection rates $\alpha_1=\alpha_2=\alpha$, equal growth functions $g_1=g_2$, equal additional diffusion paramet…

Consider the two-species river competition system with diffusion rates $d,D>0$, common advection rate $\alpha>0$, growth rate $r(x)$, upstream flux boundary conditions, and zero Di…

Consider the two-species river competition model on $(0,L)$ with positive decreasing growth rate $r(x)$, common advection rate $\alpha>0$, diffusion rates $d,D>0$, and semi-trivial…

Consider the Speirs–Gurney model on $(0,L)$ with diffusion rate $d$, advection rate $\alpha>0$, growth rate $r>0$, and lethal downstream boundary condition, whose persistence condi…

Let $u^*(x,d)$ be the unique positive solution of … where $m$ is a positive non-constant environmental carrying-capacity function and $d>0$ is the diffusion rate. Monotonicity conj…

Ni's biomass-ratio conjecture. There exists a constant $C=C(N)$ depending only on $N$ such that

Assume $1\le k<n$, assumptions $\mathrm{(A2)}$–$\mathrm{(A4)}$ hold, $(d_{ij})$ is ideal-free on patches $1\le i\le k$ and satisfies $d_{ij}=0$ whenever $i>k$ or $j>k$, and…

Consider a source-sink landscape with $1\le k<n$, where $g_i(0)>1$ exactly for $1\le i\le k$, $g_i(0)<1$ for at least one $i>k$, and $u_i^*>0$ is the unique solution of…

An ideal-free dispersal strategy is a strategy leading to an ideal free distribution, in which individuals are distributed so that they cannot increase their fitness by moving to a…

Let $\Omega$ be a bi-dimensional rectangle, and let $D\subset\Omega$ denote the favorable region in an optimal environment. Suppose that the ratio $|D|/|\Omega\setminus D|$ is suff…