Summary: The existence of a travelling wave solution to a nonlinear differential-integral equation is established. This equation arises from the mathematical modelling of an asexual reproduction process. The biological background and the mathematical formulation are described.
A time independent function \(\psi(x,t)=\varphi (x+ct,t)\), \(c>0\) \((\varphi(x,t)\) represents the probability density of genotypic values) is denoted \(\psi(x)\); \(\psi (x)\) is a travelling wave solution. For \(\psi(x)\) are obtained the following conditions: \[ \psi(x)= \delta P\int^\infty_x \int^\infty_{-\infty} e^{ \bigl((1- \delta)P- \alpha\bigr)(z-x)- \beta (z^3-x^3)} f(z-y)\psi(y) dydz, \]
\[ P=\int^\infty_{-\infty} (\alpha+3\beta x^2)\psi(x) dx,\quad \int^\infty_{- \infty} \psi(x)dx=1,\quad \psi(x)\geq 0,\;(\forall)x\in R \] where: \(P\) is the probability of producing of offspring per unit time, all constants involved are positive and \(f\) is a known function.