S Population of Italy 2 Population Models Population of Study 1 Two Models of

S population of italy 2 population models population

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Population of U. S. Population of Italy 2 Population Models Population of Study 1 Two Models of Population: A population of animals is studied over a period of days Model 1: Linear time-varying model: dP dt = (0 . 05 - 0 . 00088 t ) P, P (0) = 64 Model 2: Nonlinear time-varying model: dP dt = (0 . 3 e - 0 . 01 t - 0 . 1) P 2 / 3 , P (0) = 64 Give the modeling differences and interpretations Solve each differential equation Find when the models predict a maximum and find the maximum population Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (35/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Study 2 Model 1: Linear time-varying model: dP dt = (0 . 05 - 0 . 00088 t ) P, P (0) = 64 Model 2: Nonlinear time-varying model: dP dt = (0 . 3 e - 0 . 01 t - 0 . 1) P 2 / 3 , P (0) = 64 The first model uses a basic Malthusian growth Model 1 has the growth rate dropping linearly in time The second model assumes the population is depending on absorption through surface area Model 2 has the growth rate dropping exponentially in time to a basic linear decay term Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (36/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Study 3 Model 1: Consider: dP dt = (0 . 05 - 0 . 00088 t ) P, P (0) = 64 and rewrite as dP dt + (0 . 00088 t - 0 . 05) P = 0 Integrating factor is μ ( t ) = e R (0 . 00088 t - 0 . 05) dt = e (0 . 00044 t 2 - 0 . 05 t ) , so d dt e (0 . 00044 t 2 - 0 . 05 t ) P ( t ) = 0 Solution is e (0 . 00044 t 2 - 0 . 05 t ) P ( t ) = C or P ( t ) = C e (0 . 05 t - 0 . 00044 t 2 ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (37/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Study 4 Model 1 solution: From before P ( t ) = C e (0 . 05 t - 0 . 00044 t 2 ) = 64 e (0 . 05 t - 0 . 00044 t 2 ) with initial condition. Maximum population occurs when dP dt = 0, so 0 . 05 - 0 . 00088 t m = 0 or t m = 56 . 82 The maximum population is P ( t m ) = 64 e (0 . 05 t m - 0 . 00044 t 2 m ) = 264 . 9 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (38/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Study 5 Model 2: Consider Separable DE : dP dt = (0 . 3 e - 0 . 01 t - 0 . 1) P 2 / 3 , P (0) = 64 and rewrite as Z P - 2 / 3 dP = Z (0 . 3 e - 0 . 01 t - 0 . 1) dt Integrating gives 3 P 1 / 3 ( t ) = - 30 e - 0 . 01 t - 0 . 1 t + C so P ( t ) = C 3 - 10 e - 0 . 01 t - t 30 3 Since P (0) = 64 = ( C 3 - 10 ) or C 3 = 14, the solution is, P ( t ) = 14 - 10 e - 0 . 01 t - t 30 3 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (39/41)
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Introduction Separation of Variables Modified Malthusian Growth Model Population of U. S. Population of Italy 2 Population Models Population of Study 6 Model 2 solution: From before P ( t ) = 14 - 10 e - 0 . 01 t - t 30 3 with initial condition.
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