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So I have an equation to describe population growth

population growth equation

where a and b are constants. Typically b is small so that initially, since the population p(t) is small, the squared term can be neglected and the population growth is exponential.

how do I solve the differential equation for p(t) with p(t = 0) = 1 where a = 2 and b = 0.05 and how do I find the value of p(t) as t → ∞

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    $\begingroup$ Is this a math question or a question about how to solve this using Wolfram Mathematica? $\endgroup$ – Chris K Oct 23 '20 at 23:51
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    $\begingroup$ This DE is separable so you have $$\frac{dp}{p(a+b p)} = dt $$ $\endgroup$ – Cesareo Oct 24 '20 at 10:29
  • $\begingroup$ If $p$ has an asymptote, necessarily $ap-bp^2=0$ so either $p=0$ or $p=b/a$. $\endgroup$ – anderstood Oct 24 '20 at 12:23
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Clear[p]
p[t_] = p[t] /. 
  DSolve[{p'[t] == 2 p[t] - 0.05 p[t]^2, p[0] == 1}, p, t][[1]]
Plot[p[t], {t, 0, 5}]

enter image description here

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