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Search: MSC category 34L30 ( Nonlinear ordinary differential operators )

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1. CJM 2003 (vol 55 pp. 3)

Baake, Michael; Baake, Ellen
An Exactly Solved Model for Mutation, Recombination and Selection
It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of M\"obius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, Möbius inversion
Categories:92D10, 34L30, 37N30, 06A07, 60J25

2. CJM 2000 (vol 52 pp. 248)

Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
The nonlinear Sturm-Liouville equation $$ -(py')' + qy = \lambda(1 - f)ry \text{ on } [0,1] $$ is considered subject to the boundary conditions $$ (a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j = 0,1. $$ Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending on the independent variable $x$ alone, while $f$ depends on $x$, $y$ and $y'$. Results are given on existence and location of sets of $(\lambda,y)$ bifurcating from the linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.

Categories:34B24, 34C23, 34L30

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