Explanation Given information: Spherical colonies: Suppose the volume of a spherical colony is proportional to the number of individuals in it and growth occurs only at the surface-resource interface of the colony. Calculation: Given that the growth occurs only at the surface-resource interface of the spherical colony, therefore, the population changes according to the equation N t + 1 = N t + I Where I = K S is the input of new individuals and is proportional to the surface area S of the colony, with proportionality constant R . so that, if r t denotes the radius of the spherical colony at time t , then I can be written as I = 4 π K r t 2 , where we know that S = 4 π r 2 for a sphere. Using the given fact that the volume of the spherical colony is proportional to the number of individuals in it, then we have V t = A N t , where V t and N t correspond to the time step t , and A is the constant of proportionality. Since we know that the volume of sphere is V = 4 3 π r 3 , then the expression for V t can be written in the form 4 3 π r t 3 = A N t , solving the equation for r t we have r t 3 = 3 4

15th Edition

ISBN: 9781305177291

Author: James Stewart, Troy Day

Publisher: Cengage Learning

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Differential Equations

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Chapter 1.6, Problem 36E

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Chapter 1.6, Problem 35E

Chapter 1.6, Problem 37E

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Chapter 1 Solutions

Biocalculus: Calculus For Life Sciences

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difference equation that models the population.

Solution Summary: The author explains the difference equation that models the population based on the volume of a spherical colony and its surface-resource interface.

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Chapter 1 Solutions