A Concrete Approach to Mathematical Modelling by Mike Mesterton-Gibbons

By Mike Mesterton-Gibbons

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Then, if we assume that eggs are just young lar› vae (salmon cannibalize both), Æ = 10 . Assume, moreover, that the effect of thi s cannibalism on the first generation is to reduce the larval popu› lation by 90%. 303. 54) was solved numerically for various values of 7 (the fractio n of young survivin g unti l the next spawning). 5 10~ . Because a = ηα = 1 0 7 , the corresponding values of a were 5, 10, 11, and 15. 9). Notice that the four values of 7 (or a) correspond t o four completely different kind s of behavior.

That thi s increase wil l be proportiona l t o the number of herbivores. 19) + ΰ χ, 2 where b > 0 is a constant. , + *- a b Λ η . ( L 2 0 ) These equations, known as the Lotka-Volterr a equations, constitute a mathematical model in which a , a are pur e growth and decay rates and b , b are interaction parameters. 16). , by utilizin g a computer package, for any given values of a , b , a , b , x{0) and y(0). For a = 3, a = 5/2, b = 2, b = 1, x(0) = 1, and j>(0) = 1, the solution is plotted in Fig.

H. 1 Variatio n with tim e of capital, labor, and output indices. 31 Q Data refer to the Massachusetts economy in the years from 1890 (r = - 9) unti l 1926 (r = 27); see Douglas (1934, p. 160). section is based. We can appreciate their work most readily by defining tw o new variables: ( 0 Ξ ΐ η ( £ ) . *(0Ξΐη( 2). 28) Th e usefulness of these definition s wil l shortly emerge. 1, and these values are plotted in Fig. 5. N o t i ce that, wit h the exception of a few outlyin g point s such as that representing 1917, the point s in thi s diagram li e fairl y close to the lin e drawn through the origin .

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