Sôbre um sistema de equações relativas a um modêlo matemático de populações de Himenópteros endogâmicos

Gomes, Frederico Pimentel

doi:10.1590/S0071-12761951000100005

Resumo

This paper deals with the solution of a system of equations relating with a mathematical model of populations of endogamic Hymenoptera. The Author proves that, unless inequality (5.1) 4R5 + 8R R4 - 4R R³ + 8R² (R -1) R² - A a A a A a a A - R² (4R² + 4R - 1) R +2R³ < 0 a a A a is satisfied, one of the genes is eliminated from the population. He shows that the relative frequencies of different kinds of matings in the population can be obtained when the root R between zero and VRa of equation 2R4 + 2R³ -2R² (RA + Ra) - R(RA +Ra) + 2RA Ra =0 is known. In special, if we let b = RA / Ra > 1 , inequation (5.1) shows that we must have __________________ b³ + 2b² + b + V2b4 + 2b³ - 2b² + 2b Ra < __________________________________ = f(b) 2 (b4 + 2b³ + 2b - 1) The greatest value of f (b) is 0,75 and is obtained for b = 1, that is for RA = Ra.

Escola Superior de Agricultura "Luiz de Queiroz", Universidade de São Paulo

ABSTRACT

This paper deals with the solution of a system of equations relating with a mathematical model of populations of endogamic Hymenoptera. The Author proves that, unless inequality

(5.1) 4R⁵ + 8R R⁴ - 4R R³ + 8R² (R -1) R² -

A a A a A a a A

- R² (4R² + 4R - 1) R +2R³ < 0

a a A a

is satisfied, one of the genes is eliminated from the population. He shows that the relative frequencies of different kinds of matings in the population can be obtained when the root R between zero and VRa of equation

2R⁴ + 2R³ -2R² (RA + Ra) - R(RA +Ra) + 2RA Ra =0 is known.

In special, if we let b = RA / Ra > 1 ,

inequation (5.1) shows that we must have

__________________

b³ + 2b² + b + V2b⁴ + 2b³ - 2b² + 2b

Ra < ^{__________________________________} = f(b)

2 (b⁴ + 2b³ + 2b - 1)

The greatest value of f (b) is 0,75 and is obtained for b = 1, that is for RA = Ra.

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AGRADECIMENTO

Agradecemos ao Sr. Dr. Warwick Estevam Kerr a revisão dêste trabalho sob o ponto de vista biológico.

BIBLIOGRAFIA

KERR, Warwick Estevam - (1950) - Estudos Sôbre a Genética de Populações de Himenópteros em Geral e dos Apineos Sociais em Particular. I - III, 1 - 100, 10 figs. Piracicaba.