Explanation Given: The transition rate formula- r ( N ) = s ( 2 N − 1 ) Equation ( 22 ) is as follows. d N d t = c N − ∫ 1 N r ( u ) d u ( 22 ) The equation for the logistic model d p d t = − A p ( p − p 1 ) p ( 0 ) = p 0 , ( 14 ) where A = k 3 / 2 and p 1 = ( 2 k 1 / k 3 ) + 1 . Initial conditions- t = 0 N ( 0 ) = N 0 Approach: Solve the given differential equation- d N d t = c N − ∫ 1 N r ( u ) d u , Here the transition rate is r ( N ) = s ( 2 N − 1 ) . Use the following exponential and logarithmic properties, wherever required- ln ( a ) = b a = e b ln ( a b ) = ln a − ln b ln ( a b ) = ln a + ln b Calculation: Substitute r ( N ) = s ( 2 N − 1 ) in equation ( 22 ) and simplify- d N d t = c N − ∫ 1 N r ( u ) d u = c N − s ∫ 1 N ( 2 u − 1 ) d u = c N − s [ u 2 − u ] 1 N = c N − s [ N 2 − N ] = c N − s N 2 + s N = N ( c + s ) − s N 2 d N d t = s N [ ( c + s s ) − N ] Substitute W = ( c + s s ) - d N d t = s N [ W − N ] d N d t = − s N [ N − W ] ( 1 ) From equation ( 1 ) - d N d t = s N [ W − N ] d N N [ W − N ] = s d t Integrate the both sides of above equation. ∫ d N N [ W − N ] = s ∫ d t 1 W ∫ [ 1 [ W − N ] + 1 N ] d N = s t + C 1 W [ ln N − ln ( W − N ) ] = s t + C 1 W [ ln N ( W − N ) ] = s t + C ( 2 ) Substitute initial conditions in equation ( 2 ) -

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Chapter 5.5, Problem 9E

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To show:

Differential equation,

dNdt=cN−∫1Nr(u)du takes the form of the equation for the logistic model, when the transition rate formula, is

r(N)=s(2N−1).

and solve dNdt=cN−∫1Nr(u)du for this case.

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Chapter 5.5, Problem 8E

Chapter 5.5, Problem 10E

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Differential equation, d N d t = c N − ∫ 1 N r ( u ) d u takes the form of the equation for the logistic model, when the transition rate formula, is r ( N ) = s ( 2 N − 1 ) . and solve d N d t = c N − ∫ 1 N r ( u ) d u for this case.

Solution Summary: The author explains that the differential equation takes the form of the equation for the logistic model, when the transition rate formula, is r(N)=s

Concept explainers

Videos

To show:

Differential equation,

dNdt=cN−∫1Nr(u)du takes the form of the equation for the logistic model, when the transition rate formula, is

r(N)=s(2N−1).

and solve dNdt=cN−∫1Nr(u)du for this case.

Want to see the full answer?

Chapter 5 Solutions

Differential equation, d N d t = c N − ∫ 1 N r ( u ) d u takes the form of the equation for the logistic model, when the transition rate formula, is r ( N ) = s ( 2 N − 1 ) . and solve d N d t = c N − ∫ 1 N r ( u ) d u for this case.

Solution Summary: The author explains that the differential equation takes the form of the equation for the logistic model, when the transition rate formula, is r(N)=s

Concept explainers

Videos

To show: Differential equation, dNdt=cN−∫1Nr(u)du takes the form of the equation for the logistic model, when the transition rate formula, is r(N)=s(2N−1). and solve dNdt=cN−∫1Nr(u)du for this case.

Want to see the full answer?

Chapter 5 Solutions

To show:

Differential equation,

dNdt=cN−∫1Nr(u)du takes the form of the equation for the logistic model, when the transition rate formula, is

r(N)=s(2N−1).

and solve dNdt=cN−∫1Nr(u)du for this case.