# American Institute of Mathematical Sciences

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
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##### References:
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
A "bifurcation graph" of positions of the limit Dirac measures with respect to the size of the support of initial function
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 $[1.5, 1.5]$
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
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
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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