Understand the relationship
The reasoning behind the result
Neutral fixation probability equals initial frequency
u(p₀)=p₀
In the ideal neutral absorbing model, each initial gene copy is exchangeable, so the focal allele's eventual fixation probability equals its starting copy share.
A finite set of simulations fixed by an entered horizon is a different random quantity.
Selection changes the diffusion hitting probability
u(p₀)=[1−e^(−4Nₑsp₀)]/[1−e^(−4Nₑs)]
The displayed form is a constant diploid effective-size diffusion approximation with genic selection and no dominance. The neutral result is its continuous limit as s approaches zero.
Stable exponential-difference arithmetic preserves weak selection and reports a log probability when a disfavored result falls below direct finite display.
Effective size is a model parameter
Entered Nₑ controls the drift scale and need not equal census size or the gene-copy denominator used to describe p₀. This calculator does not estimate Nₑ.
Changing size, age structure, linked selection or reproductive skew requires a different model.
Eventual fixation is not timing
The equation supplies a hitting probability with absorbing boundaries. It does not calculate time to fixation, probability by a finite generation or the trajectory conditional on fixation.
Mutation, migration, dominance, spatial structure and competing alleles are absent.
Follow the numbers
Compare weak genic selection with neutrality
- The initial focal frequency is p₀ = 0.01.
- With Nₑ = 1000 and s = 0.001, the scaled coefficient is 4Nₑs = 4.
- The numerator is 1 − exp(−4 × 0.01) = 0.0392106.
- The denominator is 1 − exp(−4) = 0.981684.
- The model fixation probability is 0.039942, compared with the neutral baseline 0.01.
The difference belongs to the entered constant-size genic diffusion model and is not a measured population probability.
Quick guide
How to use this calculator
- Choose neutral probability or the entered genic-selection diffusion approximation.
- Enter initial frequency directly or derive it from explicit focal and total gene-copy counts.
- Treat the result as an eventual probability under the stated ideal model, not a finite-horizon observation or forecast for a real population.
Calculation method
Calculation and interpretation
Keep model probability, observed frequency and finite-horizon simulation outcomes distinct.
Neutral: u(p₀)=p₀. Genic diffusion: u(p₀)=[1−exp(−4Nₑsp₀)]/[1−exp(−4Nₑs)], with the s→0 limit p₀
Worked example
Compare weak genic selection with neutrality
The difference belongs to the entered constant-size genic diffusion model and is not a measured population probability.
Neutral: u(p₀)=p₀. Genic diffusion: u(p₀)=[1−exp(−4Nₑsp₀)]/[1−exp(−4Nₑs)], with the s→0 limit p₀
Supported inputs
Precision and limits
Ideal absorbing model
Constant effective size and absorbing loss/fixation boundaries are assumed.
Genic diffusion scope
Selection mode assumes constant genic s, no dominance and changes small enough for the diffusion approximation.
No time horizon
No fixation time, finite-generation probability or path distribution is calculated.
No other forces
Mutation, migration, linkage, spatial structure, age structure and changing size or selection are excluded.
Numerical support
Initial frequencies span 0–1; copy inputs are safe whole counts; Nₑ is a safe whole diploid size through 10⁹; s spans −1 through 1. Extremely small probabilities retain a natural-log result.
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