Question

How does logistic growth occur?

Answer 1

Logistic growth of population occurs when the rate of its growth is proportional to the product of the population and the difference between the population and its carrying capacity `M`, i.e.,

`{dP}/{dt}=kP(M-P)`, where `k` is a constant,

with initial population `P(0)=P_0`.

As you can see above, the population grows faster as the population gets larger; however, as the population gets closer to its carrying capacity `M`, the growth slows down.

by separating variables and integrating,

`Rightarrow int 1/{P(M-P)}dP=int kdt`

by Partial Fraction Decomposition,

`Rightarrow 1/M int(1/P+1/{M-P})dP=int kdt`

by multiplying by `M`,

`Rightarrow int(1/P+1/{M-P})dP=int kMdt`

`Rightarrow ln|P|-ln|M-P|=kMt+C_1`

`Rightarrow ln|P/{M-P}|=kMt+C_1`

`Rightarrow |P/{M-P}|=e^{kMt+C_1}=e^{kMt}cdot e^{C_1}`

`Rightarrow P/{M-P}=pm e^{C_1}e^{kMt}=Ce^{kMt}`

Since `P(0)=P_0`,

`P_0/{M-P_0}=Ce^{kM(0)}=C`

So, the equation becomes

`P/{M-P}=P_0/{M-P_0}e^{kMt}`

by solving for `P`, we have the logistic equation

`Rightarrow P(t)=M/{1+(M/P_0-1)e^{-kMt}}`.

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I hope that this was helpful.