A microbiologist is studying the cell division process in two types of mysteriou

Question

A microbiologist is studying the cell division process in two types of mysterious microorganisms. We shall refer to these microorganisms as being type X or type Y. The microbiologist has discovered that type X microorganisms never die! Every day, each type X microorganism produces three type Y microorganisms. Every day, each type Y microorganism produces two type X microorganisms, and then dies. Write down a set of equations to describe the evolution of both types of microorganisms. How fast does the overall population of microorganisms grow? What will be the limiting ratio of type X to type Y microorganisms? Suppose that there are 11 type X microorganisms and 1 type Y microorganism on day zero. Find the number of microorganisms of each type on any day k.

Explanation / Answer

X produces 3 Y

Y produces 2X and dies;

we can use two apporaches 1 calculus; 2 number system;

1)

dX/dt = 2Y;

dY/dt = 3X-Y; Y dies after production;

we get Y''=6Y-Y';

solution to above equation is;

Y = c1 e^-3t+c2e^2t;

similarly X = -c1/3 e^-3t + c2/2 e^2t +C;

here growth is exponential;

also limiting ratio which is Y/X for very large t; is c2/c2/2=2;

d) given X=11 and Y=1;

we know Y dies every day after production;

we get

Y=3Xk-1;

Xk=Xk-1 + 2Yk-1;

Xk=Xk-1+6Xk-2;

assume Xk=Ax^k;

we get x^2-x-6=0; (x+2)(x-3)=0; x=-2 and x=3;

we get Xk=A*3^k + B(-2)^k;

We have X1 =11;

X2 = 13;

so A = 7/2; and B = -2;

We get Xk = 7/2 * 3^k + (-2)^(k+1);

Yk = 7/2*3^k+3(-2)^k;

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.