Chapter_5_Solutions

Chapter_5_Solutions - CHAPTER 5 PDEs in The Life Sciences...

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Unformatted text preview: CHAPTER 5 PDEs in The Life Sciences 1. Age-Structured Models Exercise 1: Write 1 = integraldisplay 8 3 4 e ( r +0 . 03) a da =- 4 r + 0 . 03 parenleftBig e 8( r +0 . 03)- e 3( r +0 . 03) parenrightBig , and use a software package or calculator to solve for r . Exercise 2: First note that u ( a,t ) = 0 for a > t + , since f ( a ) = 0 for a > . For (a) observe that the renewal equation (5.9) is B ( t ) = integraldisplay t B ( t- a ) e a da + integraldisplay f ( a- t ) e t da. The first integral becomes, upon changing variables to s = t- a , integraldisplay t B ( t- a ) e a da = integraldisplay t B ( s ) e ( t s ) ds. The second integral is integraldisplay f ( a- t ) e t da = integraldisplay t + t u e t da = u e t . For (b), differentiate (using Leibniz rule) B ( t ) = integraldisplay t B ( s ) e ( t s ) ds + u e t to get B ( t ) = integraldisplay t B ( s ) e ( t s ) ds (- ) + B ( t )- u e t = ( - ) B ( t ) . For (c) note that the last equation is the differential equation for growth- decay and has solution B ( t ) = B (0) e ( ) t . Therefore the solution from (5.7)(5.8) is given by u ( a,t ) = , a > t + u e t , t < a < t + B (0) e ( ) t e a , < a < t. 1 2 5. PDES IN THE LIFE SCIENCES Finally, for part (d) we have, using part (c), N ( t ) = integraldisplay t u ( a,t ) da + integraldisplay t + t u ( a,t ) da = integraldisplay t B (0) e ( ) t e a da + integraldisplay t + t u e t da = B (0) e ( ) t (1- e t ) + u e t ....
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Chapter_5_Solutions - CHAPTER 5 PDEs in The Life Sciences...

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