129 problems

Let $L$ be a constant migration matrix that is nonsymmetric, and let $\Lambda(m,T)$ denote the growth rate of the corresponding time-periodic system, where $m$ is the migration rat…

Ni's conjecture. In the one-dimensional case, the supremum of the total population over all $d>0$ and $m\in\mathcal{M}_1$ is $3$.

Sublinear-threshold conjecture. For every $0<\varepsilon<1$,

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

Consider the GMS, random population deaths (RPD), and random location deaths (RLD) processes, and define … where $f_c=1$ by convention if no $f$ satisfies $|\mathcal{B}(f)|<\infty$…

Consider the random location deaths (RLD) process with birth probability $p$, death probability $q=1-p$, and parameter $r$. Let $L_n(a)$ be the number of species alive at time $n$…

Consider the random location deaths (RLD) process: species are born with probability $p$, die with probability $q=1-p$, and, on a death event, the least fit species is removed with…

In the random population deaths (RPD) process, at each time step a species is born with probability $p$ and dies with probability $q=1-p$; with probability $r$ the least fit specie…

Let $M_n$ be the right boundary of the population at time $n$, defined by … Suppose the population model is one-dimensional, so $d=1$, and let $v$ be the almost-sure linear spreadi…

Let $T$ and $x_{0}>0$ be fixed, and let $(\mu^{*}_{t})_{t\in[0,T]}$ be the deterministic solution considered in the allometric setting. Write $\mu^{*}_{t}=u_{t}(x)\,dx$ when it has…

Consider the limiting model in the setting of the paper, with birth and death rates and growth rate satisfying the stated assumptions, including the possibility of a death rate giv…

Stability and convergence conjecture. (A) If

Let $K$ and $P$ be the carrying-capacity and dispersal-weight functions, respectively, and suppose that $r=f(K/P)$, where $f$ is an increasing function. Let $M(d)$ be the total sta…

Let $K$, $P$, and $r=\alpha\left(\frac{K}{P}\right)^\lambda$ satisfy hypothesis (A), and let $M_\lambda(d)$ denote the total stationary population for diffusion coefficient $d$. Cr…

Let $u_d$ be a stationary solution of the logistic diffusion problem on a domain $\Omega$, and define the total stationary population by … Assume $r(x)=\alpha K(x)$ for some consta…

Let $M_+$ be the interior state space of the stochastic Rosenzweig–MacArthur model, let $x_1^x(t)$ denote the first component of the solution started from $x\in M_+$, and let…

Let the noisy $\lfloor N^\gamma\rfloor$-BRW be a branching-selection particle system, with the selection regime determined by whether selection favors the fittest or the luckiest i…

Consider the model defined by with maturation-dependent survival $m(\tau,x)$ as in, under the initial conditions in. When the existence condition holds, there is a unique positive…

Consider the delayed population model with fixed delay $\tau\in\mathbb{N}$ and positive equilibrium $N^*_+$. Theorem-based analysis gives conditions under which $N^*_+$ is locally…

Saddle-point conjecture. The fixed point $E_2$ is always a saddle point if it exists, that is,

Consider a single-species Beverton–Holt population model in which only the carrying capacity varies environmentally, and compare the resulting stationary population size with the c…

Consider the deterministic prey–predator system … with $d<\overline\gamma K<1+2d$, and let … be its fixed point. The source states that this fixed point is locally asymptotically s…

Consider a single-species Beverton–Holt difference equation in a periodic environment, and compare its long-term average population size with the corresponding nonperiodic or avera…

Let $r(\mu)$ denote the low-density consumer growth rate as a function of the reproductive delay $\tau$. Suppose there are two critical values $0<\tau_*\leq\tau^*$ such that…

Let $\alpha$ and $\beta$ be constant parameters, and let $r>0$ denote the growth-rate parameter. For each $r$, consider the long-term behavior of the model as the domain length $L$…