An integral equation describing an asexual population in a changing environment. (English) Zbl 1021.45010
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.
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.
Reviewer: Ion Onciulescu (Iaşi)
MSC:
45K05 | Integro-partial differential equations |
45G10 | Other nonlinear integral equations |
92D25 | Population dynamics (general) |
References:
[1] | Hastings, A., Population Biology (1997), Springer: Springer Berlin · Zbl 1225.92058 |
[2] | Hochstadt, H., Integral Equations (1973), Wiley: Wiley New York · Zbl 0137.08601 |
[3] | Kimura, M., A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Nat. Acad. Sci. USA, 54, 731-736 (1965) · Zbl 0137.14404 |
[4] | M. Ridley, Evolution, Boston: Blackwell, 1996.; M. Ridley, Evolution, Boston: Blackwell, 1996. |
[5] | Waxman, D.; Peck, J. R., Sex and adaptation in a changing environment, Genetics, 153, 1041-1053 (1999) |
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