Mah 36. Rumbos Spring 1 1 Soluions o Assignmen #6 1. Suppose he growh of a populaion is governed by he differenial equaion where k is a posiive consan. d d = k (a Explain why his model predics ha he populaion will decrease as ime increases. Soluion: Since k >, ( = k < for all and all >. I hen follows ha ( will always decrease. (b If he populaion a = is o, find he ime, in erms of k, a which he populaion will be reduced by half. Soluion: Solving he equaion subjec o he iniial condiion ( = o yields ( = o e k. We wan such ha ( = o /; ha is Solving for k yields o e k = o = 1 k ln ( 1 or e k = 1. = ln( k.. Consider a bacerial populaion whose relaive growh rae is given by 1 d d = K where K = K( is a coninuous funcion of ime,. (a Suppose ha o = ( is he iniial populaion densiy. Verify ha ( ( = o exp K(τ dτ solves he differenial equaion and saisfies he iniial condiion. Soluion: Firs observe ha ( ( = o exp K(τ dτ = o exp( = o,
Mah 36. Rumbos Spring 1 and so ( saisfies he iniial condiion. To show ha ( solves he differenial equaion, use he Chain Rule and he Fundamenal Theorem of Calculus o ge ( ( ( = o exp K(τ dτ d K(τ dτ d ( = o exp K(τ dτ K( = K((; ha is, d d = K. { 1 if 1 (b Find ( if K( =. Skech he graph of (. if > 1 { / if 1 Soluion: Compue K(τdτ =. Then, 1/ if > 1 ( = { o e / if 1 o e 1/ if > 1 Figure 1 shows a skech of his soluion for he case o = 1.. 1.8 Plo of versus for o = 1 1.6 1.4 1. 1.8.6.4..5 1 1.5.5 3 Time Figure 1: Graph of ( in Problem ((b
Mah 36. Rumbos Spring 1 3 3. For any populaion (ignoring migraion, harvesing, or predaion one can model he relaive growh rae by he following conservaion principle 1 d d = birh rae (per capia deah rae (per capia = b d, where b and d could be funcions of ime and he populaion densiy. (a Suppose ha b and d are linear funcions of given by b = b o α and d = d o + β where b o, d o, α and β are posiive consans. Assume ha b o > d o. Skech he graphs of b and d as funcions of. Give a possible inerpreaion for hese graphs. Soluion: This model assumes ha he per capia birh and deah raes are linear funcions o he populaion densiy,. The birh rae decreases wih increasing, while he deah rae increases wih increasing. The populaion size, K, for which boh raes are he same gives an equilibrium poin. (b Find he poin where he wo lines skeched in par (a inersec. Le K denoe he firs coordinae of he poin of inersecion. Show ha K = b o d o α + β. K is he carrying capaciy of he populaion. (c Show ha d ( d = r 1 where r = b o d o is he inrinsic growh K rae.
Mah 36. Rumbos Spring 1 4 4. The following equaion models he evoluion of a populaion ha is being harvesed a a consan rae: d d =.1 75. Find equilibrium soluions and skech a few possible soluion curves. According o model, wha will happen if a ime = he iniial populaion densiies are 4, 6, 15, or 17? Soluion: The equilibrium poins are 5 and 15. The firs one is unsable, while he second one is sable. If he iniial populaion densiy is below 5 he populaion will go exinc in finie ime. If he iniial populaion is above 5, soluions will end owards he sable equilibrium poin a 15. Figure shows a skech generaed by he MATLAB R program dfield.m. 18 =.1 75 16 14 1 1 8 6 4.5 1 1.5.5 3 3.5 4 4.5 5 Figure : Soluion Curves for Problem (4
Mah 36. Rumbos Spring 1 5 5. Consider he modified logisic model d d = r ( 1 K ( T 1 where ( denoes he populaion densiy a ime, and < T < K. (a Find he equilibrium soluions and deermine he naure of heir sabiliy. Soluion: Equilibrium soluions are 1 =, = T and = K. is unsable, while 1 and 3 are asympoically sable. (b Skech oher possible soluions o he equaion. Soluion: Figure 3 shows a skech generaed by he MATLAB R program dfield.m for he case r = 1, K = and T = 1. = r (1 /K (/T 1 r = 1 T = 1 K = 3.5 1.5 1.5 1 3 4 5 6 Figure 3: Soluion Curves for Problem (5(b (c Describe wha he model predics abou he populaion and give a possible explanaion. Soluion: The model predics ha if he iniial populaion size is below he hreshold value T, he populaion will evenually go exinc. If he iniial populaion value is above he hreshold value, he populaion will end owards he carrying capaciy value of K.