User:
Case without interaction
Storyboard 
To facilitate the analysis, one can first consider a species within the ecosystem as if there were no others and solve the model to understand the behavior.
Under this concept we realize that every species is restricted to the existence of resources that it requires to survive. In this sense, every species is limited in its development, even in the case in which it is argued that it has no natural enemies.
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Asymptotic solution
Equation
The equation
| $ \displaystyle\frac{d n }{d t } = r n + \alpha n ^2$ |
tends to an asymptotic solution equal to
| $ n_{\infty} = - \displaystyle\frac{ r }{ \alpha }$ |
\n\nwhich only makes sense if that value is positive. On the other hand, the equation for small populations reduces to\n\n
$\displaystyle\frac{dn}{dt}\sim r n$
which only makes sense if the
Therefore, the model only makes sense if
the factor $r_i$ is always positive
and
the diagonal factor (self-interaction) $\alpha_{ii}$ is negative
The latter can be understood in the context that an excessive increase will be slowed down by resources not associated with a species (for example, space, light, chemicals, etc.).
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Model simplification
Equation
If only one species is assumed to exist, the equation
| $\displaystyle\frac{d n_i }{dt}= r_i n_i + \displaystyle\sum_j \alpha_{ji} n_i n_j$ |
reduces to the equation
| $ \displaystyle\frac{d n }{d t } = r n + \alpha n ^2$ |
since the remaining populations, including the mixed terms in
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Simplified model solution
Equation
The equation
| $ \displaystyle\frac{d n }{d t } = r n + \alpha n ^2$ |
with the condition
| $ n_{\infty} = - \displaystyle\frac{ r }{ \alpha }$ |
can be solved by giving us the solution
| $ n(t) = \displaystyle\frac{n_{\infty}}{1 + (n_{\infty}/n_0 - 1)e^{-rt}}$ |
where
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