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### Course: Differential equations>Unit 1

Lesson 6: Logistic models

# The logistic growth model

The logistic differential equation dN/dt=rN(1-N/K) describes the situation where a population grows proportionally to its size, but stops growing when it reaches the size of K.

## Want to join the conversation?

• So I get the addition of a cap on population growth in order to account for carrying capacity. However, isn't there also a necessity to include some form of threshold. For example, if you only have one individual, population physically cannot grow (unless it's about plants). And even if you have two individuals, there's a huge change they won't come in contact, and thus breeding wouldn't occur. So how would a minimum threshold factor in?
• One way to put that into the equation is with what is known as the Allee effect. For this, you introduce another constant, which I will call A, and expand the equation to dN/dt=rN(1-N/K)((N/A)-1). With this, A is then the minimum population necessary to achieve positive growth, since if N<A, the (N/A)-1 term will be negative (making dN/dt negative), if N=A, the (N/A)-1 term is 0 (making dN/dt zero), and if N>A, the (N/A)-1 term will be positive (making dN/dt positive as long as N is still smaller than K).
• At he said that, if the population starts at zero, it will stay zero forever. If that's the case how are humans here today because more than 5 billion years ago the population of everything was zero? XD
• If you think about it, everything on Earth currently had a single common ancestor 5 billion years ago. There are many theories for the origin of this common ancestor, but we know that it must exist because if it didn't exist, there would be no organisms to reproduce and gradually mutate into every organism we see on Earth today. Hope this helps! Note that the single ancestor could be created by external factors but the creation of later life is purely through biological processes.
• Won't K itself decay over time? I mean the resources of this planet are really finite.
• Yes, but this is just a simple model. It doesn't cover everything.
• Can a similar differential equation be used to demonstrate the opposite effect?
For example the progressive decrease in an initial population to asymptote at a specified low level?
Thanks
• yes, because when N > K , the factor ( 1 - N/K ) would be negative , so the rate will be negative and it becomes less and less negative when N approaches K and the function would be asintotic to N = K
• How are you supposed to know to use (1-N/K)? For any modeling with differential equations, how do you just know how to set it up? (i.e. - From Sal's video on Newton's law of cooling, how were you supposed to know that dT/dt = -k(T-T(a))?)
• 1-N/K was just an assumption that was made. You make these assumptions based on your intuition of what is happening. Then you check whether it actually fits reality. As for Newton's law of cooling, it became easier, because dT/dt = -k(T-T(a)) is an actual scientific law.
(1 vote)
• Wouldnt (K-N)/K also fulfill the needs of what want to have in the parenthesis?
(1 vote)
• `1 - N/K = (K-N)/K` Both expressions are equal, so you can put any of those there.
• why the Nature of K is defferent than the nature of N ie: N is the poulation in function of time and K is the limitation of the enviorement. they should be the same if they are part of a fraction?

and what if K is dependant of N ie: in real life resources are produced at a certain level in proportion of the population?
• K is the limitation of the population within the environment. So, like N, its units are 'number of population'. N changes with time but at any specific time is just the population count. K is a constant population count, representing the highest number N could reach.

If K is dependent on N, as in I'm interpreting you as meaning that for example as N increases then K increases, then it isn't a logistic function to start with. You could develop a function with a further dampening factor that expressed K as a function of N I guess.
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• I am wondering in this type of differential equations, we only see the change of population (N) with respect to only one factor (time --> dt), but how to include other factors like amount of food , water , temperature, ...etc which also affects (N) ??
• You take all these factors into account when you assign a k value: the more you know about resources et cetera, the more sensible the estimate for k is going to be. Of course each "microvariable" will change over time but keep in mind these two things

1) sometimes these changes are negligible with respect to the span of time you are considering for your population study

2) you can't model exaclty a complex (and often chaotic) system
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• How would we find the equation of the model where there is a dampening oscillation of the population around the cap as shown in the previous video?