Explanation Given: The Gompertz law N ( t ) = e c ( 1 − e − b t ) / b has been observed experimentally for the growth of some tumors. A transition rate r ( N ) of the form b ( 1 + ln N ) predicts Gompertzian growth. Initial conditions. t = 0 P ( 0 ) = 1 Approach: The transition rate is as follows. r ( N ( t ) ) = b ( 1 + ln N ( t ) ) ( 1 ) Differential equation for P ( t ) is given by d P d t = c P − r ( N ) P ( 2 ) Q ( t ) is given by- P ( t ) + Q ( t ) = P ( t ) e b t ( 3 ) Use the following logarithmic properties, wherever required ln ( x n ) = n ln ( x ) ln e = 1 ln 1 = 0 Calculation: Substitute e c ( 1 − e − b t ) / b for N ( t ) in equation ( 1 ) and simplify. r ( N ( t ) ) = b ( 1 + ln N ( t ) ) = b ( 1 + ln e c ( 1 − e − b t ) b ) = b ( 1 + c ( 1 − e − b t ) b ) = b + c ( 1 − e − b t ) From equation ( 2 ) d P d t = c P − r ( N ) P = ( c − r ( N ) ) P Substitute r ( N ( t ) ) in equation ( 2 ) d P d t = ( c − r ( N ) ) P = ( c − ( b + c ( 1 − e − b t ) ) ) P = ( − b + c e − b t ) P d P P = ( − b + c e − b t ) d t Integrate- ∫ d P P = ∫ ( − b + c e − b t ) d t ln P = − b t − c b e − b t + C ( 4 ) Substitute initial conditions in equation ( 4 )

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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ISBN: 9780137394524

Author: R. Nagle, Edward Saff

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Chapter 5.5, Problem 7E

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P(t) and Q(t) for the tumor growth model.

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