[Paper Review] The Taylor-Frank method cannot be applied to some biologically important, continuous tness functions

Roberto H. Schonmann, Robert Boyd+1 more|arXiv (Cornell University)|Sep 6, 2012

Evolutionary Game Theory and Cooperation20 references3 citations

TL;DR

This paper demonstrates that the Taylor-Frank method, a widely used approach for modeling kin selection with continuous phenotypes, fails when fitness functions are non-differentiable, even if they are smooth and continuous. The authors show that in biologically relevant cases—such as microbial systems and repeated n-person games—a generalized direct fitness approach is required to properly account for kin selection under weak selection.

ABSTRACT

The Taylor-Frank method for making kin selection models when tness is a nonlinear function of a continuous phenotype requires this function to be dierentiable. This assumption sometimes fails for biologically important tness functions, for instance in microbial data and the theory of repeated n-person games, even when tness functions are smooth and continuous. In these cases, the Taylor-Frank methodology cannot be used, and a more general form of direct tness must replace the standard one to account for kin selection, even under weak selection.

Motivation & Objective

• To identify limitations of the Taylor-Frank method in modeling kin selection when fitness is a nonlinear, continuous function.
• To examine cases in evolutionary biology where fitness functions are smooth and continuous but not differentiable, rendering the Taylor-Frank method inapplicable.
• To propose a generalized direct fitness framework that can handle non-differentiable fitness functions in kin selection models.
• To demonstrate that standard Taylor-Frank assumptions break down in biologically important systems such as microbial cooperation and repeated n-person games.

Proposed method

• Analyzing fitness functions in microbial cooperation and repeated n-person games to identify non-differentiable points despite smoothness.
• Applying the standard Taylor-Frank method to test its applicability under non-differentiability conditions.
• Deriving a generalized direct fitness formulation that does not require differentiability of the fitness function.
• Extending the direct fitness approach to incorporate kin selection effects even when the fitness function lacks a derivative at certain points.
• Using weak selection approximations to maintain analytical tractability in the generalized framework.
• Comparing the standard Taylor-Frank approach with the generalized method to highlight failures and advantages in non-differentiable cases.

Experimental results

Research questions

• RQ1 In what biologically relevant systems does the Taylor-Frank method fail due to non-differentiable fitness functions?
• RQ2 Can a generalized direct fitness approach replace the Taylor-Frank method when fitness functions are continuous but not differentiable?
• RQ3 How does non-differentiability in fitness functions affect the prediction of evolutionary stability in kin selection models?
• RQ4 What modifications to the direct fitness framework are necessary to maintain validity in systems like microbial interactions and repeated games?

Key findings

• The Taylor-Frank method cannot be applied to certain biologically important fitness functions, such as those arising in microbial cooperation and repeated n-person games.
• Even when fitness functions are smooth and continuous, non-differentiability at critical points invalidates the Taylor-Frank approach.
• A generalized direct fitness formulation is required to properly account for kin selection in these non-differentiable cases.
• The failure of the Taylor-Frank method in these systems necessitates a shift from derivative-based to more general fitness decomposition methods.
• The generalized framework maintains analytical validity under weak selection, even when derivatives do not exist.
• The results imply that standard kin selection models based on differentiability may overlook key evolutionary dynamics in real biological systems.