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Mathematical part

Assumption

1. Let P(t) denote the size of a population at time t.
2. The model for exponentiel growth begins with

for  some  k>0   .

3. In this model the relative or specific, growth rate defined by
,Assume as a constant.

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Calculation

In many cases,

​

​

​

is not constant but a function of P,Let

Suppose an environment is capable of substaining no more than a fixed
number K of individuals in its population .The quality is called the
carrying capacity of the environment

Special cases:

so now, without human intervation,we assume that the population
would behaves logistically ,then the logistic eqaution is,

 

 

 


where r > 0 is the intrinsic growth rate and K is the environment  carring capacity.
 

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