The approximate controllability of a model for mutant selection
Hans Weinberger
Evolution Equations & Control Theory 2013, 2(4): 741-747 doi: 10.3934/eect.2013.2.741
It is shown that the problem of eliminating a less-fit allele by allowing a mixture of genotypes whose densities satisfy a system of reaction-diffusion equations with population control to evolve in a reactor with impenetrable walls is approximately controllable.
keywords: Parabolic system approximate controllability. evolutionary selection population genetics
On sufficient conditions for a linearly determinate spreading speed
Hans Weinberger
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 2267-2280 doi: 10.3934/dcdsb.2012.17.2267
It is shown how to construct criteria of the form $f(u)\le f'(0)K(u)$ which guarantee that the spreading speed $c^*$ of a reaction-diffusion equation with the reaction term $f(u)$ is linearly determinate in the sense that $c^*=2\sqrt{f'(0)}$. Some of these criteria improve the classical condition $f(u)\le f'(0)u$, and permit the presence of sharp Allee effects. Inequalities which guarantee the failure of linear determinacy are also presented.
keywords: spreading speed. Reaction-diffusion systems
An extension of the formula for spreading speeds
Hans F. Weinberger Xiao-Qiang Zhao
Mathematical Biosciences & Engineering 2010, 7(1): 187-194 doi: 10.3934/mbe.2010.7.187
A well-known formula for the spreading speed of a discrete-time recursion model is extended to a class of problems for which its validity was previously unknown. These include migration models with moderately fat tails or fat tails. Examples of such models are given.
keywords: integrodifference equations linear determinacy. spreading speeds discrete-time recursions
Spreading speeds for a partially cooperative 2-species reaction-diffusion model
Hans F. Weinberger Kohkichi Kawasaki Nanako Shigesada
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 1087-1098 doi: 10.3934/dcds.2009.23.1087
It is shown that a trick introduced by H. R. Thieme [6] to study a one-species integral equation model with a nonmonotone operator can be used to show that some multispecies reaction-diffusion systems which are cooperative for small population densities but not for large ones have a spreading speed. The ideas are explained by considering a model for the interaction between ungulates and grassland.
keywords: spreading speed partially cooperative reaction-diffusion system

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