Skip to main content

The recovery of a recessive allele in a Mendelian diploid model


We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. Neukirch and Bovier (J Math Biol 75:145–198, 2017) proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they would in populations reproducing asexually. In the present paper we prove that this indeed opens the possibility that individuals with a pure genotype can reinvade in the population after the appearance of further mutations. We thus expose a rigorous description of a mechanism by which a recessive allele can re-emerge in a population. This can be seen as a statement of genetic robustness exhibited by diploid populations performing sexual reproduction.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. The article Neukirch and Bovier (2017) only state that survival occurs up to time \(K^{1/4-{\alpha }}\). However, taking into account that it is really only the survival of the aA population that needs to be ensured, one can easily improve this to \(K^{1/2-{\alpha }} \).


  • Baar M, Bovier A, Champagnat N (2017) From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. Ann Appl Probab 27(2):1093–1170

    Article  MathSciNet  Google Scholar 

  • Billiard S, Smadi C (2017) The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference. Stoch Process Appl 127(3):701–748

    Article  MathSciNet  Google Scholar 

  • Bürger R (2000) The mathematical theory of selection, recombination, and mutation, vol 228. Wiely, Chichester

    MATH  Google Scholar 

  • Champagnat N (2006) A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch Process Appl 116(8):1127–1160

    Article  MathSciNet  Google Scholar 

  • Champagnat N, Ferrière R, Ben Arous G (2001) The canonical equation of adaptive dynamics: a mathematical view. Selection 2:73–83

    Article  Google Scholar 

  • Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stochastic Models 24(suppl. 1):2–44

    Article  MathSciNet  Google Scholar 

  • Champagnat N, Méléard S (2011) Polymorphic evolution sequence and evolutionary branching. Probab Theory Relat Fields 151(1–2):45–94

    Article  MathSciNet  Google Scholar 

  • Collet P, Méléard S, Metz JAJ (2013) A rigorous model study of the adaptive dynamics of Mendelian diploids. J Math Biol 67(3):569–607

    Article  MathSciNet  Google Scholar 

  • Coron C (2014) Stochastic modeling of density-dependent diploid populations and the extinction vortex. Adv Appl Probab 46(2):446–477

    Article  MathSciNet  Google Scholar 

  • Coron C (2016) Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size. J Math Biol 72(1–2):171–202

    Article  MathSciNet  Google Scholar 

  • Coron C, Méléard S, Porcher E, Robert A (2013) Quantifying the mutational meltdown in diploid populations. Am Nat 181(5):623–636

    Article  Google Scholar 

  • Crow JF, Kimura M (1970) An introduction to population genetic theory. Harper and Row, New York, p 656

  • Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5–6):579–612

    Article  MathSciNet  Google Scholar 

  • Ethier SN, Kurtz TG (1986) Markov processes. Wiley Series in Probability and Statistics. Wiley, Hoboken

    Google Scholar 

  • Ewens WJ (2012) Mathematical population genetics 1: theoretical introduction, vol 27. Springer, Berlin

    MATH  Google Scholar 

  • Fisher R (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans. Roy. Soc. Edinb. 42:399–433

    Google Scholar 

  • Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919

    Article  MathSciNet  Google Scholar 

  • Haldane J (1924) A mathematical theory of natural and artificial selection. Part I. Trans Camb Philos Soc 23:19–41

    Google Scholar 

  • Haldane J (1924) A mathematical theory of natural and artificial selection. Part II. Trans Camb Philos Soc Biol Sci 1:158–163

    Article  Google Scholar 

  • Hirsch MW, Pugh CC, Shub M (1977) Invariant manifolds, vol 583. Lecture notes in mathematics. Springer, Berlin

    Chapter  Google Scholar 

  • Hofbauer J, Sigmund K (1990) Adaptive dynamics and evolutionary stability. Appl Math Lett 3(4):75–79

    Article  MathSciNet  Google Scholar 

  • Kisdi E, Geritz SAH (1999) Adaptive dynamics in allele space: evolution of genetic polymorphism by small mutations in a heterogeneous environment. Evolution 53:993–1008

    Article  Google Scholar 

  • Marrow P, Law R, Cannings C (1992) The coevolution of predator-prey interactions: ESSS and Red Queen dynamics. Proc R Soc Lond B: Biol Sci 250(1328):133–141

    Article  Google Scholar 

  • Metz J, Nisbet R, Geritz S (1992) How should we define “fitness” for general ecological scenarios? Trends Ecol Evol 7(6):198–202

    Article  Google Scholar 

  • Metz JAJ, Geritz SAH, Meszena G, Jacobs FJA, van Heerwaarden JS (1995) Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. Iiasa working paper, IIASA, Laxenburg, Austria

  • Nagylaki T et al (1992) Introduction to theoretical population genetics, vol 142. Springer, Berlin

    Book  Google Scholar 

  • Neukirch R, Bovier A (2017) Survival of the recessive allele in the Mendelian diploid model. J. Math. Biol. 75:145–198

    Article  MathSciNet  Google Scholar 

  • Perko L (2013) Differential equations and dynamical systems, vol 7. Springer, Berlin

    MATH  Google Scholar 

  • Wright S (1931) Evolution in Mendelian populations. Genetics 16:97–157

    Google Scholar 

  • Yule GU (1907) On the theory of inheritance of quantitative compound characters on the basis of Mendel’s laws: a preliminary note. Spottiswoode & Company Limited, London

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Loren Coquille.

Additional information

We acknowledge financial support from the German Research Foundation (DFG) through the Hausdorff Center for Mathematics, the Cluster of Excellence ImmunoSensation, and the Priority Programme SPP1590 Probabilistic Structures in Evolution. L.C. has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) through the Exploratory Project CanDyPop, as well as by the Swiss National Science Foundation through the Grant No. P300P2_161031, and by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement - École Polytechnique - Muséum National d’Histoire Naturelle - Fondation X. We would like to thank Pierre Collet and Vincent Beffara for their help on the theory of dynamical systems and fruitful discussions.



We collect in this appendix all the important definitions of times separating phases or subphases of the process, and provide a proof summary with all the implications. Recall Definition 4.1, and Figs. 5 and 6. We write \(N_\delta (x)\) for the \(\delta \)-neighbourhood of \(x\in {\mathbb R}^6\). We consider

$$\begin{aligned} {\Delta }>\delta>{\varepsilon }_0>{\varepsilon }>0. \end{aligned}$$


$$\begin{aligned} T_1&:=T_{{\varepsilon }_0}^{aa+aA+aB+AB+BB},\\ T_=&:=T^{aA=aB},\\ T_2&:=T^{aA=\delta AA}\wedge T^{aB=\delta AB}\wedge T^{aa= aA\wedge aB},\\ T_3&:=T^{aa}_{{\bar{n}}_a-{\varepsilon }^{{\gamma }/2}}. \end{aligned}$$

where \(\gamma :=2/(1+\eta {\bar{n}}_B-\Delta )\) is the order of magnitude of \(n_{AA}\) at \(T_2\) : \(n_{AA}(T_2)={\Theta }({\varepsilon }^\gamma )\).

We summarise below the detailed structure of the proof (we abbreviate Lemma, Proposition and Theorem by L,P, and T respectively):

  1. Phase 1:

    \(t\in [0,T_1]\) Initial conditions: \((n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})=({\Theta }({\varepsilon }^2),{\varepsilon },{\bar{n}}_A\pm {\Theta }({\varepsilon }), 0,{\varepsilon }^3, 0)\). With those initial conditions the following bounds hold:

    $$\begin{aligned} n_{AA}\le {\bar{n}}_A,\quad n_{BB}\le {\bar{n}}_B, \quad n_{aB}\le n_{AB}\quad \text {(Proposition 4.3) } \end{aligned}$$

    Here are the main steps of the proof of Propostion 4.4 and Corollary 4.5. Until the time \(T_1\),

    $$\begin{aligned} \text {(P4.3)}\Rightarrow \quad&n_{BB}\le n_{AB}^2 \quad \text {(1)}\\ \text {(1)}\Rightarrow \quad&n_{aB}\le n_{aA}n_{AB} \quad \text {(2)}\\ \text {(1)(2)}\Rightarrow \quad&n_{aa}\le n_{aA}^2 \quad \text {(3)}\\ \text {(1)(2)(3)}\Rightarrow \quad&T_1= T^{AB}_{{\varepsilon }_0}={\Theta }(\log ({\varepsilon }_0/{\varepsilon }^3)^{1/(\Delta -{\Theta }({\varepsilon }_0))})\quad \text {(Proposition 4.4)}\\ \text {(P4.4)}\Rightarrow \quad&{\dot{n}}_{AB}={\Theta }(\Delta ) n_{AB} \quad \text {(Corollary 4.5(1))}\\ \text {(P4.4)}\Rightarrow \quad&{\bar{n}}_A-{\Theta }({\varepsilon }) \le \Sigma _5\le {\bar{n}}_A+2\Delta {\varepsilon }_0 \quad \text {(4)}\\ \text {(4)(P4.4)}\Rightarrow \quad&n_{aB}\le {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}{\varepsilon }_0), n_{aA}\le {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}),\\&n_{aa}\le {\Theta }({\varepsilon }^{2-{\Theta }({\varepsilon }_0)}), n_{AA}={\bar{n}}_A\pm {\Theta }({\varepsilon }_0) \quad \text {(Corollary 4.5(3)))}\\ \end{aligned}$$
  2. Phase 2:

    \(t\in [T_1,T_2]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&=({\Theta }({\varepsilon }^{2-{\Theta }({\varepsilon }_0)}),{\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}),\\&{\bar{n}}_A\pm {\Theta }({\varepsilon }_0 ), {\Theta }({\varepsilon }^{1-{\Theta }({\varepsilon }_0)}{\varepsilon }_0),{\varepsilon }_0, {\Theta }({\varepsilon }_0^2)). \end{aligned}$$

    As the process stays uniformly bounded in time, until \(T_2\) we have

    $$\begin{aligned} n_{aa},n_{aA},n_{aB}\le {\Theta }(\delta )\qquad (*) \end{aligned}$$
    $$\begin{aligned} (*)&\Rightarrow \quad (n_{AA},n_{AB},n_{BB})=(n^{up}_{AA},n^{up}_{AB},n^{up}_{BB})+{\Theta }(\delta ) \quad \text {(Lemma 4.6)}\\ \text {L4.6,}(*)&\Rightarrow \quad \Sigma _5={\bar{n}}_B-{\Delta }n_{AA}/{c{\bar{n}}_B}+ {{\Theta }({\Delta }^2n_{AA})} \quad \text {(Proposition 4.7)}\\ \text {P4.7,}(*)&\Rightarrow \quad {\dot{\Sigma }}_{aA,aB}\ge -{\Theta }({\Delta })\Sigma _{aA,aB} \quad \text {(Lemma 4.8)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{aa}={\Theta }(\Sigma _{aA,aB}^2) \Rightarrow T_2=T^{aA=\delta AA}\wedge T^{aB=\delta AB} \quad \text {(Lemma 4.9)}\\ \text {P4.7,L4.9}&\Rightarrow \quad \text {Until }T_=,\quad n_{aB}\le n_{aA}={\Theta }({\varepsilon }) \quad \text {(Proposition 4.10)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{aB}\le \frac{n_{AB}+2n_{BB}+{\textstyle {2\Delta \over c}}}{n_{AB}+2n_{AA}}n_{aA} \quad \text {(Lemma 4.11)}\\ \text {P4.7,L4.9,P4.10,}&\Rightarrow \quad T_=<T_2 \quad \text {(Lemma 4.12)}\\ \text {L4.11}&\Rightarrow \quad n_{AA}(T_=)\le n_{BB}(T_=)+{\Theta }(\Delta ) \quad \text {(Lemma 4.12)}\\ \text {P4.7,}(*)&\Rightarrow \quad \max _{t\in [T_1,T_2]}n_{AB}=:n_{AB}(T^{max}_{AB})={\bar{n}}_B/2+{\Theta }(\Delta ) \quad \text {(Proposition 4.14)}\\ \text {P4.7,}(*)&\Rightarrow \quad n_{AA}(T^{max}_{AB}),n_{BB}(T^{max}_{AB})={\bar{n}}_B/4+{\Theta }(\Delta ) \quad \text {(Proposition 4.14)}\\ \text {L4.9,P4.7,}(*)&\Rightarrow \quad n_{aA}\le n_{aB}\vee {\Theta }({\varepsilon }) \quad \text {(Proposition 4.15)}\\ \text {L4.8,4.9,P4.7,}(*)&\Rightarrow \quad n_{aA}= \Sigma _{aA,aB} {\Theta }(n_{AB}+2n_{AA}) \quad \text {(Lemma 4.16)}\\ \text {L4.9,P4.7,}(*)&\Rightarrow \quad {\dot{\Sigma }}_{aA,aB}=n_{aA}(\eta n_{BB}-{\Theta }(\Delta ))\quad \text {(Proposition 4.17)}\\ \text {L4.16,P4.14,4.17,}&\Rightarrow \quad T_2={\Theta }\left( {\varepsilon }^{1/(1+\eta {\bar{n}}_B-\Delta )}\right) \quad \text {(Theorem 4.19)}\\ \text {L4.9,T4.19}&\Rightarrow \quad T_2=T^{aA=\delta AA}\quad \text {(Propostion 4.20)}\\ \end{aligned}$$
  3. Phase 3:

    \(t\in [T_2,T_3]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&=({\Theta }({\varepsilon }^2),{\Theta }({\delta }{\varepsilon }^{\gamma }),{\Theta }({\varepsilon }^\gamma ),\\&{\Theta }(\delta {\varepsilon }^{\gamma /2} ),{\Theta }({\varepsilon }^{\gamma /2}),{\bar{n}}_B-{\Theta }({\varepsilon }^{\gamma /2})). \end{aligned}$$
    $$\begin{aligned} \text {P4.22}&\Rightarrow \quad \Sigma _5={\bar{n}}_B\pm {\Theta }(n_{AB}) \quad \text {(Lemma 4.23)}\\ \text {L4.23}&\Rightarrow \quad n_{AA},n_{aA}\le {\Theta }(n_{AB}^2) \quad \text {(Lemma 4.24)}\\ \text {L4.23,4.24,4.26}&\Rightarrow \quad n_{AA},n_{aA}\ge {\Theta }(n_{AB}^2) \quad \text {(Lemmas 4.25 and 4.27)}\\ \text {L4.23,4.24,4.25,4.27}&\Rightarrow \quad n_{AB}\le {\varepsilon }^{\gamma /10} \quad \text {(Lemmas 4.26 and 4.28)}\\ \text {L4.24,4.27}&\Rightarrow \quad {\dot{n}}_{aa}\ge cn_{aa}\text { with } c>0 \quad \text {(Lemma 4.28)} \end{aligned}$$
  4. Phase 4:

    \(t\in [T_3,\infty ]\) Initial conditions:

    $$\begin{aligned} (n_{aa},n_{aA},n_{AA},n_{aB},n_{AB},n_{BB})&={\bar{n}}_a-{\varepsilon }_0,{\Theta }({\varepsilon }^{\gamma /5}), {\Theta }({\varepsilon }^{\gamma /5}), \\&{\Theta }({\varepsilon }^{\gamma /10}), {\Theta }({\varepsilon }^{\gamma /10}),{\bar{n}}_B\pm {\Theta }({\varepsilon }^{\gamma /10}) ) . \end{aligned}$$

    T4.29 \(\Rightarrow \) For \(\eta <c\cdot 0.593644\), the fixed point \(p_{aB}=({\bar{n}}_a,0,0,0,0,{\bar{n}}_B)\) is stable and for \({\varepsilon },{\varepsilon }_0\) small enough, the system converges to it at speed 1 / t (Theorem 4.30).

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bovier, A., Coquille, L. & Neukirch, R. The recovery of a recessive allele in a Mendelian diploid model. J. Math. Biol. 77, 971–1033 (2018).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification