A Concrete Approach to Mathematical Modelling by Mike Mesterton-Gibbons

By Mike Mesterton-Gibbons

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Similarly , in the absence of herbivores, the carnivor e population would di e off exponentially (assuming that carnivores don’t eat grass). Th e appro› priat e model for thi s would be μ = -a , where a > 0 is constant. But ther e are some herbivores, whose effect wil l be to enlarge the growth rat e of th e carnivor e population. , 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.

Then we have Rate of Increase of Product = Rate of Conversion of Reactant (1-65) k[A)[B) where A: is a constant. , ξ(ί) moles of A plus ( ) moles of Β become ξ(ί) moles of Ρ which is perfectly consistent, because the molecules of Ρ are heavier than those of A or B. 65) becomes 0 0 £ = *(Æ - (0 & - (0) . 66) Thi s may, at first , appear to be a new mathematical equation. But you can easily check that the substitutions χ = max(fl . *o) ~ £. 0 R = (o k a ~ *o). 62). 10, and we leave that to you. You should check that, as t oo i n your solution, th e product concentration £ tends t o m i n ( a , b ); whereas th e reactant concentrations tend to 0 and m a x ( a , b )-min(a , b ).

3. The unit of tim e is a decade, and x(t) denotes magnitude at tim e t. Notice that population increases wit h time. The simplest hypothesis that might explain thi s observation is that χ is proportiona l to t; then dxldt would be constant. 55) D(t)= -{x«+l)-x(t-l)}. 12). Because D also increases wit h t, we reject the hypothesis that χ is proportiona l to time. 3 shows that Dlx is almost constant. 5Q is correct to one significant figure. 7). 9e ’ . S. population in millions, 1790-1850. 3 Data for x(i) are taken from Historical Statistics of the United States, Colonial Times to 1970, Bicenten› nial Edition, Part I , p.

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