1. (a) The sizes of two interacting populations, x1(t) and x2(t), are modelled over a short time frame (with time measured in years) by the linear system of differential equations
~x 0 = A~x, where A = 3 −2 4 −1 and ~x = x1(t) x2(t) .
(i) Solve this system of differential equations along with the initial conditions
~x(0) = 4000 1000 .
(ii) If this model is only valid up to the first positive time tend such that x1(tend) = 1 000, find the value of tend rounded to two decimal places. HINT: it might be helpful to use technology (e.g. MATLAB/Octave or Python) to help answer this question.
(b) The weight, w(t) in grams (g), of a starving animal t days after it stops eating is modelled by the differential equation
dw dt = − 1 3 w 4 5 .
(i) Solve this differential equation for w as an explicit function of t in the special case of an animal whose weight at the start of starving is 100 000 g, then calculate the weight of that animal after 10 days of starving, giving your answer rounded to the nearest whole gram.
(ii) Chossat’s law says that an animal dies after it has lost 50% of it body weight. Under the assumption that Chossat’s law applies, how long would it take for the animal described in part (i) to die? Do not answer this question by trial-and-error. Show all calculations and round your answer to the nearest whole day.
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