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
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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 |
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.
Keywords:
Assortative mating,
phenotypic evolution,
polymorphic population,
nonlinear operator of trait inheritance,
convergence to multimodal distribution.
Mathematics Subject Classification:
Primary: 47H30; Secondary: 92D15.
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
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References:
| [1] |
N. H Barton, A. 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.
Google Scholar
|
| [3] |
M. G. Bulmer, The Mathematical Theory of Quantitative Genetics, Clarendon Press, Oxford, 1980.
Google Scholar
|
| [4] |
J. A. Cañizo, J. 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. Desvillettes, P. E. Jabin, S. 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. Diekmann, P. E. Jabin, S. 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. Doebeli, H. J. Blok, O. 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. Matessi, A. 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. Google Scholar |
| [19] |
O. Puebla, E. 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. Google Scholar |
| [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 Barton, A. 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.
Google Scholar
|
| [3] |
M. G. Bulmer, The Mathematical Theory of Quantitative Genetics, Clarendon Press, Oxford, 1980.
Google Scholar
|
| [4] |
J. A. Cañizo, J. 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. Desvillettes, P. E. Jabin, S. 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. Diekmann, P. E. Jabin, S. 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. Doebeli, H. J. Blok, O. 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. Matessi, A. 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. Google Scholar |
| [19] |
O. Puebla, E. 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. Google Scholar |
| [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 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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