25 problems

Adaptive mutation-strength conjecture. This adaptive procedure should drive the system toward a state that is not worse than the current state with respect to the Nash condition, w…

Let $n_A(t,p)$ and $n_B(t,p)$ denote the population densities of two populations structured by cooperation probabilities $p$, with mean cooperation probabilities…

Let $H$ denote the model's horizontal gene-transfer parameter, let $\tau$ be the model parameter appearing in the ESS condition, and let an $m$-morphic ESS mean an evolutionarily s…

Let $f(X,Y)$ be the performance difference function for random traits $X$ and $Y$, and let $\boldsymbol{\nabla} f$ denote its gradient. Suppose that the gradient is nonzero almost…

Formal front-propagation conjecture. There exists an interval of trait values $J_0$ centered at $1$ such that, for every open interval $J\subset J_0$ centered at $1$, the density…

Let $N$ be the finite population size, let $E(\theta)$ be the cost function, and let $\beta^*$ be the threshold from the optimization theorem. For $\beta>\beta^*$, the theorem esta…

Quasi-stationary distribution conjecture. Then:

Generalized extinction conjecture. With a suitable modification of condition $$ , the conclusions of Theorem should hold for a general positive symmetric matrix ${\bf A}$.

Let $s>0$ and let $r$ be any bounded function. Write $\rho_\alpha(\gamma)$ for the extinction rate associated with the polymorphic quasi-stationary distribution as a function of th…

Hölder-exponent conservation conjecture. The Hölder exponent is conserved in time by the multilevel system given by the paper's equation, so that

Extension conjecture. The limit theorem proved for binary random variables $M_i$ should also hold for non-binary $M_i$ in this scaling regime, at least when the variances of the…

A nonlinear system has an equilibrium point $(x_0,u_0)$, and its linearization around that point determines whether the system is locally non-passive: the linearization is non-pass…

Consider a stage-structured population in which only one stage disperses. An ideal-free dispersal strategy is a strategy leading to an ideal-free distribution, and a dispersal stra…

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…

Global-infimum conjecture. For each fixed $N\geq4$ and $n\geq2$, the right-hand side of the displayed formula is a global infimum of all positive values of $(b/c)^*$ over all migra…

Let $N=N_1+N_2$ be the total population size, let $D=|N_1-N_2|$ be the difference in island sizes, and let $(b/c)^*$ denote the critical benefit-to-cost ratio for two islands with…

Yu–Etheridge–Cuthbertson conjecture. The average rate at which the mean fitness increases is

The Lenski experiment measures the relative fitness $F(B|A)$ of strain $B$ with respect to strain $A$ through the ratio of their exponential growth rates: … Over the course of the…

The Lenski experiment tracks the evolution of bacterial populations over generations and measures the relative fitness $F(B|A)$ of strain $B$ with respect to strain $A$ by comparin…

Consider a superstar population structure with death-birth updates with selection on survival, in which mutants can occupy the root node and subsequently spread through the reservo…

Let a finite population of size $N$ evolve according to the Moran process, with individuals placed on a graph whose nodes represent individuals and whose links define their neighbo…

Stationary stability conjecture. Analogous stationary-stability results should hold for the random graph and for $k$-regular graphs, and likely for other graph classes as well. The…

Feldman's conjecture. Whenever a parameter controlled by a gene enters linearly into the recursion on frequency dynamics, there exists a VAHW modifier polymorphism satisfying all t…

Asymptotic growth conjecture. As $t\to\infty$,

Let $n$ be the number of possible haplotypes in a deterministic model of viability selection and recombination, and consider its isolated fixed points, equivalently its equilibria.…