The integration of the Allee effect equation, ds/dt = s(r-a(s-b)^2), can be found by using the method of separation of variables. This involves separating the dependent variable, s, from the independent variable, t, and then integrating both sides of the equation.
To begin, we can rewrite the equation as:
ds/(r-a(s-b)^2) = dt
Next, we can use the substitution u = s-b, which will simplify the equation to:
ds/(r-au^2) = dt
Now, we can integrate both sides of the equation with respect to their respective variables:
∫ ds/(r-au^2) = ∫ dt
We can use the substitution v = √(r/a)u, which will result in:
(1/√(r/a)) ∫ du/u^2 = ∫ dt
Applying the power rule of integration, we get:
(1/√(r/a)) (-1/u) = t + C
Substituting back for u and rearranging the equation, we get:
s(t) = b + (√(a/r))cot(√(ar)t + C)
Where C is the constant of integration.
Therefore, the integration of the Allee effect equation is given by:
s(t) = b + (√(a/r))cot(√(ar)t + C)
In summary, the integration of the Allee effect equation can be found by using the method of separation of variables and applying the power rule of integration. This solution can help us understand the behavior of the population over time and how it is affected by the Allee effect.
