A question regarding Logistic population model

In summary, the conversation discusses the logistic population model and the concept of differentiation of a constant function. The speaker also asks for clarification on the logic behind the derivative of a constant function and its relation to the initial value of the population.
  • #1
issacnewton
998
29
Hi
I am going through Edx course on Introduction to Differential Equations by Paul Blanchard (BUx: Math226.1x). At one point, he is explaining the Logistic population model.$$\frac{dp}{dt} = kp\left(1- \frac{p}{N}\right) $$ After this, he says that since the right hand side does not involve [itex]t[/itex], if [itex]p(0) = 0[/itex] then [itex]\frac{dp}{dt} = 0 [/itex] for all [itex]t[/itex]. I don't quite get his logic here. Can anyone explain this please ?

Thanks
 
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  • #2
If you can answer a question like the one below you get the gist of what is going on.
Example: How many people does it take to start reproducing more people? Or mammals or bacteria?
 
  • #3
Jim, it will take 2 persons or in case of bacterias, it will just take one.
 
  • #4
Can anybody give more hints ?
 
  • #5
If there are none to start with, how can you reproduce?
 
  • #6
Ok that makes sense. But if we have [itex]\frac{dp}{dt}[/itex] function of [itex]t[/itex], then does [itex]p(0) = 0[/itex] still lead to [itex]\frac{dp}{dt} = 0[/itex] for all [itex]t[/itex] ?
 
  • #7
IssacNewton said:
After this, he says that since the right hand side does not involve [itex]t[/itex], if [itex]p(0) = 0[/itex] then [itex]\frac{dp}{dt} = 0 [/itex] for all [itex]t[/itex]. I don't quite get his logic here.

The derivative of a constant function is zero -meaning the derivative of a constant function is the "zero function". For example, if ##f(x) = 3x + 15## when you compute ##\frac{df}{dx} ## as the derivative of ##3x## plus the derivative of ##15## what do you get for the answer when you differentiate the ##15##?

Note that ##15## must be taken to denote the constant function ##g(x) = 15## in order to differentiate it because we differentiate functions, not single numbers.
 

1. What is the Logistic population model?

The Logistic population model is a mathematical model used to describe the growth of a population over time. It takes into account factors such as carrying capacity and population growth rate to predict the population size at any given time.

2. How is the Logistic population model different from other population models?

The Logistic model differs from other models, such as the exponential growth model, by taking into account the carrying capacity of a population. This means that as the population grows, the growth rate decreases until it reaches a stable equilibrium.

3. What is carrying capacity and how does it affect the Logistic population model?

Carrying capacity is the maximum number of individuals that an environment can support without causing harm or depletion of resources. In the Logistic population model, the carrying capacity is a key factor in determining the rate of population growth and the stability of the population over time.

4. Can the Logistic population model accurately predict population growth in real-world scenarios?

The Logistic population model is a simplified mathematical representation of population growth and may not accurately reflect the complexities of real-world scenarios. However, it can provide valuable insights and predictions when used with accurate data and assumptions.

5. How can the Logistic population model be applied in practical situations?

The Logistic population model has various applications, including predicting the growth of animal and plant populations, managing natural resources, and studying human population dynamics. It can also be used in industries such as agriculture and healthcare to make informed decisions about resource allocation and sustainability.

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