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3. We go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: dP P k(t)P dt M(t) In this case the equation is not autonomous, so we can't use phase line analysis. We will instead find explicit analytical solutions. (a) Show that the substitution z = 1/P transforms the equation into the linear equation Pz=) KIt).! (1- dz k(t) + k(t)z = M(t) dt ZMA) (b) Using your résult in (a), show that if k is constant but M varies, the general solution is KH)ZMH)-) ekt (P)- 4)= =") mo P(t) = C + S Modt kekt %3D ZMA) (c) Similarly, show that if M is constant but k varies, the general solution is M P(t) 1+ CMe-Sk(t) dt " (d) Consider the special case where M is constant but k decreases in time as k = e-t. Suppose that the initial population is less than M. What happens to the population in the long run? Does it make sense?

3. We go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: dP P k(t)P dt M(t) In this case the equation is not autonomous, so we can't use phase line analysis. We will instead find explicit analytical solutions. (a) Show that the substitution z = 1/P transforms the equation into the linear equation Pz=) KIt).! (1- dz k(t) + k(t)z = M(t) dt ZMA) (b) Using your résult in (a), show that if k is constant but M varies, the general solution is KH)ZMH)-) ekt (P)- 4)= =") mo P(t) = C + S Modt kekt %3D ZMA) (c) Similarly, show that if M is constant but k varies, the general solution is M P(t) 1+ CMe-Sk(t) dt " (d) Consider the special case where M is constant but k decreases in time as k = e-t. Suppose that the initial population is less than M. What happens to the population in the long run? Does it make sense?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. We go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: dP P = k(t)P(1- dt M(t) In this case the equation is not autonomous, so we can't use phase line analysis. We will instead find explicit analytical solutions. (a) Show that the substitution z = 1/P transforms the equation into the linear equation 2= ] KIH) ! (1- dz + k(t)z = dt k(t) M(t) ZMP) (b) Using your résult in (a), show that if k is constant but M varies, the general solution is KH) ZMA)-) (P)- {4)=) ekt P(t) C + S Modt kekt ZMA) (c) Similarly, show that if M is constant but k varies, the general solution is M P(t) = 1+ CMe-Sk(t)dt " (d) Consider the special case where M is constant but k decreases in time as k = e=t. Suppose that the initial population is less than M. What happens to the population in the long run? Does it make sense?
Transcribed Image Text:3. We go back to the logistic model for population dynamics (without harvesting), but we now allow the growth rate and carrying capacity to vary in time: dP P = k(t)P(1- dt M(t) In this case the equation is not autonomous, so we can't use phase line analysis. We will instead find explicit analytical solutions. (a) Show that the substitution z = 1/P transforms the equation into the linear equation 2= ] KIH) ! (1- dz + k(t)z = dt k(t) M(t) ZMP) (b) Using your résult in (a), show that if k is constant but M varies, the general solution is KH) ZMA)-) (P)- {4)=) ekt P(t) C + S Modt kekt ZMA) (c) Similarly, show that if M is constant but k varies, the general solution is M P(t) = 1+ CMe-Sk(t)dt " (d) Consider the special case where M is constant but k decreases in time as k = e=t. Suppose that the initial population is less than M. What happens to the population in the long run? Does it make sense?
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