i don't know how to solve C, D, and E. Please show me your work and what answer you got.

PS8 Due Thursday March 28, 2024 Student Name: Student #: report all answers to 3 significant digits Problem 1: (textbook problem 7.2.16) BACTERIA POPULATION GENETICS Sup- pose there are two bacterial strains 1 and 2, each undergoing growth according to the differential equations dN1 dt dN2 2 = 12 N2. dt with 71 > r2. The variable M is the size of the strain 1 population, and the variable N2 is the size of the strain 2 population. The parameter r, is the growth rate for strain 1, and r2 is the growth rate for strain 2. The independent variable t is time. Define N1 p(t) = N1 + N2 to be the frequency of strain 1 at time t. A. Calculate dp/dt, and use the definition of p(t) above to show that p obeys the differential equation dt dp = sp(1 - p), with s = 1 - 12. (Hint: start with the quotient rule on p(t).) of - f(t ) 19 de = g(t) at 7P (9 (+ 1 ) 2 de = (NitN2) CN_ N(dN de NIN25 dt ( NI + N2 ) 2 ( NITNZ) P= NI N2 NITNZ I-P= _ de = (Nit N2) (rini) - Ni (riNitrzN2) Nit NZ dt ( Nit N2) 2 de de = Sp (1-P ) de ring - NirzNZ dt ( NitNZ ) 2 de = NIN2 (ri- 12) de ( NIt N 2 ) 2 B. What are the equilibria of the differential equation of part A? O= SP( 1 -P ) P=0 ( for so ) 1PS8 Due Thursday March 28, 2024 C. Carry out a phase-line analysis for the case s = 1 - 12 = 1, and characterize the stability of the equilibria. D. Carry out a phase-line analysis for the case s = 1 - 12 = -1, and characterize the stability of the equilibria. E. Based off of what you have found so far, do you think it is possible for two strains of bacteria to coexist if one grows faster than the other? Why or why not? 2