Table 1 Description of model parameters and variables

R _m

Intrinsic growth rate of the population, λ/μ

λ

Density-independent birth rate

μ

Density-independent death rate

γ

Density-dependent rate constant per generation time

m

Proportion of progeny of driving Y males that inherit the driving Y

u

The fraction of female progeny of a driving Y male that inherit an X chromosome with a resistant mutation (Model I)

v

The chance of a suppressor mutation arising on an autosome (per individual per autosome, for all births, male or female) (Model II)

w

Fitness parameter for mutations (relative to fitness one for wild-types and driving Y males without the resistant gene). Model I: w for heterozygote females, \(w^{2}\) for homozygous females and hemizygous males; Model II: w for heterozygotes, w ² for homozygotes

a

Amplitude of seasonal variation in parameter for density-dependence, \({\upgamma }\left( t \right)\)

N ₀

Equilibrium wild-type population size (before release of driving Y and in absence of mutation)

h ₀

Release amount of driving Y males

\(H\left( t \right), M\left( t \right),F(t)\)

Driving Y, wild-type male and wild-type female population sizes (non-mutant)

N(t)

Total population size

P ₁

Probability that at least one mutation arises and establishes, preventing population elimination

P _Mut

Probability that at least one mutation arises (regardless of its fate)

P _Con

Conditional probability that if one or more mutations arise, at least one establishes

p _est (t _a )

Probability that a single mutation arising at time t _a after release of the driving Y will establish (in the absence of other mutations)