SM-4312: Mathematical Modelling, Semester I, Session 2020-2021 Page 5

(ii) Show that the amounts of pollutants in the lakes at time t > 0 are

x1(t) = r21

r12 + r21

(x1(0) + x2(0)) +

r12 x1(0) − r21 x2(0)

r12 + r21

e−(r12+r21)t

,

x2(t) = r12

r12 + r21

(x1(0) + x2(0)) − r12 x1(0) − r21 x2(0)

r12 + r21

e−(r12+r21)t

.

(iii) Make a suggestion with appropriate mathematical analysis in order to achieve the eventual ratio

x1/x2 of the pollutants greater than one, that is, x1/x2 > 1.

Question 8

Tank 1

x1(t) x2(t)

Tank 2

r12 = r

r21 = r

In the figure, both Tank 1 and Tank 2 contain V1 l and V2 l

of the same chemical solution, respectively. The solution

flows from Tank 1 to Tank 2 and back to Tank 1 with flow

rates r12 = r l/min and r21 = r l/min, as indicated. The

solution in each tank is mixed uniformly and the amounts

of chemical in the tanks are denoted by x1 = x1(t) g and

x2 = x2(t) g at time t.

(i) Use the small change principle over a time period [t, t + ∆t], to formulate and derive the

compartmental system governing the chemical concentrations

Q1 = Q1(t) g/l and Q2 = Q2(t) g/l

for time t > 0 with Q1(0) and Q2(0) being the initial concentrations in the respective tanks.

(ii) Determine the chemical concentrations Q1(t) and Q2(t) at any time t > 0. Hence, show that the

limiting concentrations Q1(∞) and Q2(∞) satisfy

Q1(∞) = Q2(∞) = Q1(0) α1 + Q2(0) α2

α1 + α2

g/l,

where α1 = r/V1 and α2 = r/V2.

Question 9

Consider a competition model of two species X and Y whose population sizes x = x(t) and y = y(t)

are governed by

Species X :

dx

dt = r1 x − αy

Species Y :

dy

dt = r2 y − βx,

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