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# (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.

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,