Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
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1 1. A country had millions of inhabitants, and ten years later, it has grown to 33 millions of inhabitants. Assuming a continuous logistic growth, calculate the value of r for a carrying capacity K of 100 millions of inhabitants. Which population is expected in 10 years time? Plot a graph of the expected population for the considered values of K and r as a function of the starting population. Remove@"Global` "D DSolve@8P'@tD rp@td H1 P@tDê100L, P@0D <, P, td Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. à ::P FunctionB8t<, 0 rt 7+3 rtf>> ::P FunctionB8t<, 0 rt 1, 2T 7+3 rtf>>p1, FunctionB8t<, 0 rt 7+3 rtf f = % FunctionB8t<, 0 rt 7+3 rtf Solve@f@10D 33., r, RealsD Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. à 88r << r = `; f@20d Remove@PD; DSolve@8P'@tD rp@td H1 P@tDê100L, P@0D == P0<, P, td Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. à ::P FunctionB8t<, t t +1. J P0 N 1.F>> P0 ::P FunctionB8t<, t 1, 2T t +1. J P0 1.F>>P1, N P0
2 2 Exercise1(unit1).nb FunctionB8t<, t t +1. J P0 N 1.F P0 f = %; Plot@f@20D, 8P0, 0, 100<, AspectRatio AutomaticD Another way (with x=p/k and without clicking get solution on the suggestion bar): K =100; Remove@x, rd DSolve@8x'@tD rx@td H1 x@tdl, x@0d êk<, x, td; s = x ê. %@@1DD FunctionB8t<, 3 rt 7+3 rtf Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. à Clear@rD N@Solve@s@10D 33êK, r, RealsDD; r =rê. %@@1DD Which population is expected in 10 years time? x@20d K Plot a graph of the expected population for the considered values of K and r as a function of the starting population.
3 Exercise1(unit1).nb 3 Remove@x, PoD DSolve@8x'@tD rx@td H1 x@tdl, x@0d PoêK<, x, td; x = x ê. %@@1DD Plot@K x@20d, 8Po, 0, 100<, AspectRatio AutomaticD Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. à FunctionB8t<, t t +1. I Po M 1.F Po
4 Exercise 4 (unit 1): 4. Consider an initial population P 0 = 10 millions with a discrete logistic growth with carrying capacity K = and r = 3.7. Plot the population values until t= 20. Which is the expected population for t= 20? Do the same computations when P 0 = and compare the results. Remove@"Global` "D K = ; r = 3.7; f = Function@x, rxh1 xld Function@x, rxh1 xld NestList@f, 10êK, 20D A = ListPlot@%, Joined True, DataRange 80, 20<, PlotRange 80, 1<D %%@@21DDK : 1, , 0.584, , , , , , , , , , , , , , , , , , > Now P 0 =10.01.
5 2 Exercise4(unit1).nb ê K, 20D B = ListPlot@%, Joined True, DataRange 80, 20<, PlotRange 80, 1<D %%@@21DDK , , , , , , , , , , , , , , , , , , , , < Show@A, BD We obtain very different results.
6 Exercise 10 (unit 1): 10. Given the following demand and supply functions: DHpL=100-3p, SHpL=-20+2p, consider the following mechanism of adjustment of the price to excess demand or supply: d p d t =ahdhphtll-shphtlll, where a>0 is the reactivity of price to excess demand or supply. Describe the behavior of price over time for an initial price p 0 = 45 when a=0.1, a=0. and a=0.4. In[199]:= Remove@"Global` "D In[200]:= De =Function@8p<, 100 3pD; Su = Function@8p<, 20+2pD; P0 = 45; α = 0.1; DSolve@8P'@tD α HDe@P@tDD Su@P@tDDL, P@0D P0<, P, td; pr =Pê. %@@1DD Plot@pr@tD, 8t, 0, 15<, PlotRange 820, <D Out[205]= FunctionA8t<, t I t ME Out[206]= In[207]:= Remove@PD; α = 0.; DSolve@8P'@tD α HDe@P@tDD Su@P@tDDL, P@0D P0<, P, td; pr =Pê. %@@1DD Plot@pr@tD, 8t, 0, 15<, PlotRange 820, <D Out[210]= FunctionA8t<, t I t ME Out[211]=
7 2 Exercise10(unit1).nb In[212]:= α = 0.41; DSolve@8P'@tD α HDe@P@tDD Su@P@tDDL, P@0D P0<, P, td; pr =Pê. %@@1DD Plot@pr@tD, 8t, 0, 15<, PlotRange 820, <D Out[215]= FunctionA8t<, t I t ME Out[216]= Solve@De@pD Su@pD, pd 88p 24<< The price tends to the price of equilibrium. H We don't need to solve the differential equation every time we change the value of α. We can do it this way: L Remove@"Global` "D In[219]:= De = Function@8p<, 100 3pD; Su = Function@8p<, 20+2pD; Remove@PD; P0 = 45; DSolve@8P'@tD α HDe@P@tDD Su@P@tDDL, P@0D P0<, P, td; pre = P ê. %@@1DD α = 0.1; Plot@pre@tD, 8t, 0, 15<, PlotRange 820, <D Out[224]= FunctionA8t<, t I t ME Out[226]=
8 Exercise10(unit1).nb 3 LimitA24 0.5t I t M, t InfinityE 24. α = 0.; Plot@pre@tD, 8t, 0, 15<, PlotRange 820, <D In[227]:= α = 0.41; Plot@pre@tD, 8t, 0, 15<, PlotRange 820, <D Out[228]=
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