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

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• 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.

Mathematics Subject Classification: Primary: 47H30; Secondary: 92D15.

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• 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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