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Selection Theorem for Systems With Inheritance
"... Abstract. The problem of finitedimensional asymptotics of infinitedimensional dynamic systems is studied. A nonlinear kinetic system with conservation of supports for distributions has generically finitedimensional asymptotics. Such systems are apparent in many areas of biology, physics (the the ..."
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Abstract. The problem of finitedimensional asymptotics of infinitedimensional dynamic systems is studied. A nonlinear kinetic system with conservation of supports for distributions has generically finitedimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finitedimensional asymptotics demonstrates effects of “natural ” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t → ∞. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of selfsynchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinitedimensional space is discussed, and the notion of “completely thin” sets is introduced.
Ontogenetic niche shifts and evolutionary branching in sizestructured populations
"... There are many examples of sizestructured populations where individuals sequentially exploit several niches in the course of their life history. Efficient exploitation of such ontogenetic niches generally requires specific morphological adaptations. Here, we study the evolutionary implications of t ..."
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There are many examples of sizestructured populations where individuals sequentially exploit several niches in the course of their life history. Efficient exploitation of such ontogenetic niches generally requires specific morphological adaptations. Here, we study the evolutionary implications of the combination of an ontogenetic niche shift and environmental feedback. We present a mechanistic, sizestructured model in which we assume that predators exploit one niche when they are small and a second niche when they are big. The niche shift is assumed to be irreversible and determined genetically. Environmental feedback arises from the impact that predation has on the density of the prey populations. Our results show that, initially, the environmental feedback drives evolution towards a generalist strategy that exploits both niches equally. Subsequently, it depends on the sizescaling of the foraging rates on the two prey types whether the generalist is a continuously stable strategy or an evolutionary branching point. In the latter case, divergent selection results in a resource dimorphism, with two specialist subpopulations. We formulate the conditions for evolutionary branching in terms of parameters of the sizedependent functional response. We discuss our results in the context of observed resource polymorphisms and adaptive speciation in freshwater fish species.
Lifehistory evolution in harvested populations: the role of natural predation
"... Models and experiments of the evolution of age and/or sizeatmaturation in response to population harvesting have consistently shown that selective harvesting of older and larger individuals can cause earlier maturation. These predictions, however, are all based on singlespecies considerations and ..."
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Models and experiments of the evolution of age and/or sizeatmaturation in response to population harvesting have consistently shown that selective harvesting of older and larger individuals can cause earlier maturation. These predictions, however, are all based on singlespecies considerations and thus crucially neglect the selective forces caused or mediated by species interactions. Here we develop simple models of phenotypic evolution of ageatfirstreproduction in a prey population subject to different types of predation and harvesting. We show that, in the presence of natural predation, the potential evolutionary response of ageatfirstreproduction to population harvesting is ambiguous: harvesting can cause either earlier or later maturation depending on the type of predator interaction and its strength relative to the fishing pressure. The counterintuitive consequences of harvesting result from the indirect effects that harvesting of a prey population has on the selection pressure exerted by its natural predator, since this selection pressure itself typically depends on prey density. If harvest rates are high, the direct selection pressures considered in classical analyses prevail and harvesting decreases the ageatfirstreproduction, whereas at lower harvest rates the indirect, interspecifically mediated effects of harvesting can qualitatively overturn predictions based on simpler singlespecies models.