Question

(1 point) Assume that two colonies each have 500 members at time t = 0 and that each evolves with a constant relative birth rate k=ro-rd- For colony 1, assume that individuals migrate into the colony at a rate of 70 individuals per unit time. Assume that this immigration occurs for 0<t<1 and ceases thereafter. For colony 2, assume that a similar migration pattern occurs but is delayed by one unit of time; that is, individuals immigrate at a rate of 70 individuals per unit time, 1<t < 2. Suppose we are interested in comparing the evolution of these two populations over the time interval 0<t < 2. The initial value problems governing the two populations are d = kP + M(+), (0= 500, M (t) ={ 70, 10, 0<t < 1, 1<t< 2. = kP2 + M (t), P2(0) = 500, M2(t) = { S 0, - 05 1< IAA a. Solve both problems to find P, and P, at time t = 2. P (2) = P (2) =

Answer 1

Solection Page no: 0 het us Consider given Equation dpi Kp, + M.lt) S 70 Osts! where as , 8, 10=500, M. (t)=) 30 \<t<2. dt Now ofi - KP, = M, 10 • there Equation © is leibritze linear Equatico form To Solve this Equation, we need D.F - Integralion factol is predefined for libritz Equation. Slot - DF Let Idf = ett Here the standard Solution will be obtained o the for the Equation is : novo Multiply both sides by I.F de CIF )-KP: (1.f) = M.(5:)

honda (P. (J.f) )= M(1.6)+ @ By The integrating on both sides of Egy" ☺ We obtains (Piet)s ment » Salpetty Imettat = P, (2) 22K _ p,16) = {to ett att for Pilase yon = 401 Pilaje* 50 (-1) 19 (2)e ak 180 + 7o (1-) P, C). 50022* + 70 cdx (1-e47 = 701 - Equation 3 is Solution fol picas. O 20. Now we need to Solve pa de kp2+ m2 (t) co, oste!! P_CO) = 400, m = 3=0, 1sts2.

where as If = eht D . K Solve Similarly den - KP2=ma het al die (P2. e*t) - mekt Sa (p. ext) = sment at Peca). et 2.60) - 0 + (90646) Po(a).@@k_500 = - Ceket Paa) = Something (ef cak) BQ) 500 elit sed ett Blo) = Sore * (ex)10 Here Equation G is Solutions for fa(2) I to 2k 10 X -