From crater functions to partial differential equations: a new approach to ion bombardment induced nonequilibrium pattern formation

2009 ◽  
Vol 21 (22) ◽  
pp. 224017 ◽  
Author(s):  
Scott A Norris ◽  
Michael P Brenner ◽  
Michael J Aziz
Keyword(s):  
Partial Differential Equations ◽  
Pattern Formation ◽  
Differential Equations ◽  
Ion Bombardment ◽  
Partial Differential ◽  
New Approach

Adaptive Diversification and Speciation as Pattern Formation in Partial Differential Equation Models

2011 ◽  
Author(s):  
Michael Doebeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Pattern Formation ◽  
Differential Equations ◽  
Adaptive Dynamics ◽  
Partial Differential ◽  
Adaptive Diversification ◽  
Differential Equation Models ◽  
Phenotype Distribution

This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a mathematically concise formulation from which some analytical insights can be obtained. It has been argued that because partial differential equations can describe polymorphic populations, results from such models are fundamentally different from those obtained using adaptive dynamics. In partial differential equation models, diversification manifests itself as pattern formation in phenotype distribution. More precisely, diversification occurs when phenotype distributions become multimodal, with the different modes corresponding to phenotypic clusters, or to species in sexual models. Such pattern formation occurs in partial differential equation models for competitive as well as for predator–prey interactions.

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