January 2018, 23(1): 459-472. doi: 10.3934/dcdsb.2018031

Does assortative mating lead to a polymorphic population? A toy model justification

1. 

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

2. 

Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

* Corresponding author

Received  October 2016 Revised  February 2017 Published  January 2018

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

Citation: Ryszard Rudnicki, Radoslaw Wieczorek. Does assortative mating lead to a polymorphic population? A toy model justification. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 459-472. doi: 10.3934/dcdsb.2018031
References:
[1]

N. H BartonA. M. Etheridge and A. Véber, The infinitesimal model: Definition, derivation, and implications, Theor. Popul Biol., (2007). doi: 10.1016/j.tpb.2017.06.001.

[2] P. Billingsley, Probability and Measure, John Wiley and Sons, New York, 1986.
[3] M. G. Bulmer, The Mathematical Theory of Quantitative Genetics, Clarendon Press, Oxford, 1980.
[4]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156. doi: 10.1007/s10440-012-9758-3.

[5]

L. DesvillettesP. E. JabinS. Mischler and G. Raoul, On selection dynamics for continuous populations, Commun. Math. Sci., 6 (2008), 729-747. doi: 10.4310/CMS.2008.v6.n3.a10.

[6]

O. DiekmannP. E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul Biol., 67 (2005), 257-271. doi: 10.1016/j.tpb.2004.12.003.

[7]

U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation, Nature, 400 (1999), 354-357. doi: 10.1038/22521.

[8]

M. DoebeliH. J. BlokO. Leimar and U. Dieckmann, Multimodal pattern formation in phenotype distributions of sexual populations, Proc. R. Soc. B, 274 (2007), 347-357. doi: 10.1098/rspb.2006.3725.

[9]

R. A. Fisher, The correlations between relatives on the supposition of Mendelian inheritance, Trans. R. Soc. Edinburgh, 52 (1919), 399-433. doi: 10.1017/S0080456800012163.

[10]

S. Gavrilets and C. R. B. Boake, On the evolution of premating isolation after a founder event, The American Naturalist, 152 (1998), 706-716. doi: 10.1086/286201.

[11]

P. Hinow, Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math., 70 (2009), 40-62. doi: 10.1137/080732420.

[12]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions, J. Math. Biol., 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.

[13]

F. A. Kondrashov and A. S. Kondrashov, Interactions among quantitative traits in the course of sympatric speciation, Nature, 400 (1999), 351-354. doi: 10.1038/22514.

[14]

C. MatessiA. Gimelfarb and S. Gavrilets, Long-term buildup of reproductive isolation promoted by disruptive selection: How far does it go?, Selection, 2 (2001), 41-64. doi: 10.1556/Select.2.2001.1-2.4.

[15]

J. Maynard Smith, Sympatric speciation, Am. Nat., 100 (1966), 637-650. doi: 10.1086/282457.

[16]

P. Michel, A singular asymptotic behavior of a transport equation, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 155-159. doi: 10.1016/j.crma.2007.12.010.

[17]

B. Perthame, Transport Equations in Biology Frontiers in Mathematics, Birkhäuser, Basel, 2007.

[18]

J. Polechová and N. H. Barton, Speciation through competition: A critical review, Evolution, 59 (2005), 1194-1210.

[19]

O. PueblaE. Bermingham and F. Guichard, Pairing dynamics and the origin of species, Proc. Biol. Sci., 279 (2012), 1085-1092. doi: 10.1098/rspb.2011.1549.

[20]

R. Rudnicki and R. Wieczorek, On a nonlinear age-structured model of semelparous species, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2641-2656. doi: 10.3934/dcdsb.2014.19.2641.

[21]

R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol., 70 (2015), 1295-1321. doi: 10.1007/s00285-014-0798-3.

[22]

K. A. Schneider and R. Bürger, Does competitive divergence occur if assortative mating is costly?, Journal of Evolutionary Biology, 19 (2006), 570-588.

[23]

K. A. Schneider and S. Peischl, Evolution of assortative mating in a population expressing dominance, PLoS ONE, 6 (2011), e16821. doi: 10.1371/journal.pone.0016821.

[24]

P. Zwoleński, Trait evolution in two-sex populations, Math. Mod. Nat. Phenom., 10 (2015), 163-181. doi: 10.1051/mmnp/20150611.

show all references

References:
[1]

N. H BartonA. M. Etheridge and A. Véber, The infinitesimal model: Definition, derivation, and implications, Theor. Popul Biol., (2007). doi: 10.1016/j.tpb.2017.06.001.

[2] P. Billingsley, Probability and Measure, John Wiley and Sons, New York, 1986.
[3] M. G. Bulmer, The Mathematical Theory of Quantitative Genetics, Clarendon Press, Oxford, 1980.
[4]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Appl. Math., 123 (2013), 141-156. doi: 10.1007/s10440-012-9758-3.

[5]

L. DesvillettesP. E. JabinS. Mischler and G. Raoul, On selection dynamics for continuous populations, Commun. Math. Sci., 6 (2008), 729-747. doi: 10.4310/CMS.2008.v6.n3.a10.

[6]

O. DiekmannP. E. JabinS. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul Biol., 67 (2005), 257-271. doi: 10.1016/j.tpb.2004.12.003.

[7]

U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation, Nature, 400 (1999), 354-357. doi: 10.1038/22521.

[8]

M. DoebeliH. J. BlokO. Leimar and U. Dieckmann, Multimodal pattern formation in phenotype distributions of sexual populations, Proc. R. Soc. B, 274 (2007), 347-357. doi: 10.1098/rspb.2006.3725.

[9]

R. A. Fisher, The correlations between relatives on the supposition of Mendelian inheritance, Trans. R. Soc. Edinburgh, 52 (1919), 399-433. doi: 10.1017/S0080456800012163.

[10]

S. Gavrilets and C. R. B. Boake, On the evolution of premating isolation after a founder event, The American Naturalist, 152 (1998), 706-716. doi: 10.1086/286201.

[11]

P. Hinow, Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math., 70 (2009), 40-62. doi: 10.1137/080732420.

[12]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions, J. Math. Biol., 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.

[13]

F. A. Kondrashov and A. S. Kondrashov, Interactions among quantitative traits in the course of sympatric speciation, Nature, 400 (1999), 351-354. doi: 10.1038/22514.

[14]

C. MatessiA. Gimelfarb and S. Gavrilets, Long-term buildup of reproductive isolation promoted by disruptive selection: How far does it go?, Selection, 2 (2001), 41-64. doi: 10.1556/Select.2.2001.1-2.4.

[15]

J. Maynard Smith, Sympatric speciation, Am. Nat., 100 (1966), 637-650. doi: 10.1086/282457.

[16]

P. Michel, A singular asymptotic behavior of a transport equation, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), 155-159. doi: 10.1016/j.crma.2007.12.010.

[17]

B. Perthame, Transport Equations in Biology Frontiers in Mathematics, Birkhäuser, Basel, 2007.

[18]

J. Polechová and N. H. Barton, Speciation through competition: A critical review, Evolution, 59 (2005), 1194-1210.

[19]

O. PueblaE. Bermingham and F. Guichard, Pairing dynamics and the origin of species, Proc. Biol. Sci., 279 (2012), 1085-1092. doi: 10.1098/rspb.2011.1549.

[20]

R. Rudnicki and R. Wieczorek, On a nonlinear age-structured model of semelparous species, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2641-2656. doi: 10.3934/dcdsb.2014.19.2641.

[21]

R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol., 70 (2015), 1295-1321. doi: 10.1007/s00285-014-0798-3.

[22]

K. A. Schneider and R. Bürger, Does competitive divergence occur if assortative mating is costly?, Journal of Evolutionary Biology, 19 (2006), 570-588.

[23]

K. A. Schneider and S. Peischl, Evolution of assortative mating in a population expressing dominance, PLoS ONE, 6 (2011), e16821. doi: 10.1371/journal.pone.0016821.

[24]

P. Zwoleński, Trait evolution in two-sex populations, Math. Mod. Nat. Phenom., 10 (2015), 163-181. doi: 10.1051/mmnp/20150611.

Figure 1.  Evolution of trait distribution with $K(x, y, dz)=\delta_{\frac{x+y}{2}}(dz)$ and the preference function $\varphi(r)=1$, for $r < 1$ and $0$ otherwise. The initial function is similar to $f_0$ in Example
Figure 4.  A "bifurcation graph" of positions of the limit Dirac measures with respect to the size of the support of initial function
Figure 2.  Evolution of trait distribution with $K(x, y, dz)=\delta_{\frac{x+y}{2}}(dz)$ and different preference functions $\psi(x, y)=\varphi_i(|x-y|)$ with $\varphi_1(r)=1$, $\varphi_2(r)=(r-1)^2(r+1)^2$, and $\varphi_3(r)=(1-r)^3$, respectively, for $r < 1$ and $0$ otherwise. In all three cases the initial population's trait is uniformly distributed on the interval $[1.5, 1.5]$
Figure 3.  Evolution of trait distribution with $K(x, y, dz)=\delta_{\frac{x+y}{2}}(dz)$ and $\varphi(r)=(r-1)^2(r+1)^2$, for $r < 1$ and $0$ otherwise. The initial trait is uniformly distributed on the intervals of length $2.5$, $3$, $4.3$, and $6$ in subsequent rows
Figure 5.  Evolution of trait distribution with $K(x, y, dz)=\kappa\left(z-\frac{x+y}{2}\right)dz$ with probability distribution $\kappa(r)=C_{a}(r-a)^2(r+a)^2, \text{ for } |r|\le a$, where $a=0.125$, $a=0.25$, $a=0.5$ in subsequent rows
Figure 6.  Evolution of trait distribution when all offspring is distributed with probability distribution $\kappa(r)=C_{a}(r-a)^2(r+a)^2, \text{ for } |r|\le a$, where $a=0.1$ and $a=0.2$ in the first and second row, respectively
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