Abstract
We study the fixation probability of a mutant type when introduced into a resident population. We implement a stochastic competitive Lotka–Volterra model with two types and intra- and interspecific competition. The model further allows for stochastically varying population sizes. The competition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same net growth rate for both types. In this general framework, we derive a formula for the fixation probability varphi of the mutant type under weak selection. We find that the most important parameter deciding over the invasion success of the mutant is its death rate due to competition with the resident. Furthermore, we compare our approximation to results obtained by implementing population size changes deterministically in order to explore the parameter regime of validity of our method. Finally, we put our formula in the context of classical evolutionary game theory and observe similarities and differences to the results obtained in that constant population size setting.
Highlights
The evolutionary dynamics of a mutant strain in a resident population is a well-studied topic in the field of population dynamics
Results concerning the fixation probability, the average fixation time or coexistence behavior can be applied in various biological fields, e.g. population genetics, bacterial evolution, viral dynamics or cancer initiation (Nowak 2006; Altrock et al 2015; Ewens 2004)
In population genetics and theoretical ecology, studies focused more on the effect that population dynamics have on the fixation probability rather than the concrete interaction mechanisms between the mutant and wild-type individuals (Ewens 1967; Kimura and Ohta 1974; Otto and Whitlock 1997)
Summary
The evolutionary dynamics of a mutant strain in a resident population is a well-studied topic in the field of population dynamics. While the first theoretical analysis of such processes relied on deterministic differential equations, over the course of time more detailed models were studied describing the stochasticity of microscopic processes on the individual level. These individual based models can be approximated and studied by the replicator equation (in the large population size limit), by branching processes (in case of a small number of invading mutants) or by multi-type birth-death processes (Nowak 2006; Hofbauer and Sigmund 1998; Sandholm 2010; Haccou et al 2005; Ewens 2004). For a historical overview on the calculation of fixation probabilities, see Patwa and Wahl (2008)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
