27 problems

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…

Stability and convergence conjecture. (A) If

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 $L$ be an irreducible dispersal matrix, let $A_i=R_i+L$ be the matrices in the differential inclusion model, and let $x_i$ be eigenvectors of $A_i$. Assume that the eigenvector…

Let $t\geq 0$ and let $\psi_1,\psi_2\in\mathcal{D}(\mathbb{T}^d)$. Write $\eta_N$ for the scaling factor, $P_t^N(\psi_1,\psi_2)$ for the isolation-by-distance quantity, $Z_t$ for t…

Assume $a>0>b$ and $a+2b<0$. For the seasonal system denoted by (47ab1), let $\Lambda(m,T)$ be its growth rate, with migration rate $m$ and period $T$. Asymptotic growth conjecture…

Assume $a>0>b$ and $a+2b<0$. For the seasonal system denoted by (47ab), let $\Lambda(m,T)$ be its growth rate, with migration rate $m$ and period $T$. Asymptotic growth conjecture.…

Assume $a>0>b$ and $a+2b<0$, and let $\Lambda(m,T)$ be the growth rate of the three-patch seasonal system under consideration, with migration rate $m$ and period $T$. Upper-bound c…

Let $\Lambda(m,T)$ denote the growth rate for migration intensity $m$ and period parameter $T$, and suppose that the migration matrix is time independent. Let $r_i(\tau)$ denote th…

Let $(u,v)$ solve the reaction diffusion system referenced as … . Suppose … and the parameters satisfy the restrictions in … . Uniform-convergence conjecture. The solution $(u,v)$…

Consider the predator-prey model given by Equation (HarvestingEqn), and let $q^*$ denote the harvesting threshold appearing in the conjecture. Here $q$ is the predator-harvesting e…

Consider the competition system … with positive initial data, basic reproduction numbers $\mathcal{R}_0^{(1)}>1$ and $\mathcal{R}_0^{(2)}>1$, boundary equilibria $E_1,E_2$, and coe…

Carrying-capacity conjecture. If

Let $\{(Z_n,\widetilde{Z}_n)\}_{n\in\mathbb{N}_0}$ be a PP-2SDDBPO, with total population variables $T_n$ and $\widetilde{T}_n$, parameter $\mu$, and positive initial values…

Consider the competition model given by equations, with $p=q=1$, and assume … The competitors coexist. Critical-window conjecture. There is a critical window…

Persistence threshold conjecture. Persistence of the system is possible if and only if the number of environments is at least the number of species.

Exponential convergence-rate conjecture. For every initial population state $x$ with $x^{i_0}\neq 0$ and every $s\in E$,

Consider the nonlocal competition model described above, with two population densities and a locally stable steady state. Global stability conjecture. The locally stable steady sta…

Let $n$ be the population-size scaling parameter, let $I_h^n(t)$ and $I_v^n(t)$ denote infected hosts and infected vectors, and let $\mathcal{R}_0$ be the basic reproduction number…

Consider the system of interacting Wright–Fisher diffusions on $\mathbb Z^d$ with $N=1$ and parameter $\mu>1$, where $s$ is the selection parameter and coexistence refers to long-t…

Consider the population process with unbounded harvesting governed by … where $(Z_t)_{t\geq0}$ is a non-negative, increasing, right-continuous process adapted to…

Extinction and convergence conjecture. The predators $(X_{j^*+1},\dots,X_n)$ go extinct, in the sense that

Let $\{x_n\}$ be a positive solution of equation (4ab), and assume that condition (fxp0) holds. Consider its even subsequence $\{x_{2n}\}$ and odd subsequence $\{x_{2n-1}\}$. Parit…

Smaller-refuge-with-overcrowding conjecture. There exist a patch $\Omega_2$ and a coefficient $d_4$ such that every single refuge patch of measure at least $|\Omega_2|$, with the s…