How Do You Solve Complex Differential Equations in Population Growth Models?

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SUMMARY

The discussion focuses on solving complex differential equations related to population growth models, specifically the Gompertz equation and another model represented by dP/dt = P(10^-1 - (10^-5)P). Participants seek assistance in finding solutions that satisfy initial conditions and analyzing the behavior of P(t) as time approaches infinity. Key techniques include transforming the equations and applying limits to determine population trends over time.

PREREQUISITES
  • Understanding of differential equations, particularly first-order equations.
  • Familiarity with the Gompertz equation and its application in population modeling.
  • Knowledge of logarithmic functions and their properties.
  • Basic calculus skills, including limits and concavity analysis.
NEXT STEPS
  • Study the solution methods for first-order differential equations, focusing on separation of variables.
  • Learn about the implications of the Gompertz equation in biological contexts.
  • Explore techniques for analyzing limits in population models.
  • Investigate concavity and inflection points in the context of population growth graphs.
USEFUL FOR

Students studying calculus, particularly those tackling differential equations in biological applications, as well as educators seeking to enhance their teaching of population growth models.

AngusYoung93
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Hey, everyone. I've been doing my Calculus II homework, and I've been trying these two problems for the past few hours, but I can't even seem to get started on them. A friend of mine recommended I try here because you guys are awesome.

Homework Statement


The Gompertz equation
dP/dt = P(a - b ln(P))
where a and b are positive constants, is another model of population growth.
a) Find the solution of this differential equation that satisfies the initial condition: P(0) = p(sub(0))
b) What happens to P(t) as t -> infinity?
c) Determine the concavity of the graph of P


2. The attempt at a solution

dP/dt = P(a - bQ)
where Q = ln(x)​



Homework Statement



the differential equation
dP/dt = P(10^-1 - (10^-5)P)
models the population of a certain community. Assume P(0) = 2000 and time t is measured in months.
a) Find P(t) and show that lim t -> infinity exists
b) Find the limit


2. The attempt at a solution

I do not even know where to start on this. Any help at all would be nice.
 
Physics news on Phys.org
Write it as
\frac{d\,P}{P\,(a-b\,\ln P)}=d\,t
and set \ln P=u
 

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