A population grows according to the logistic equation dP/dt = rP(1-P/K). If P(0) = P₀, find P(t) and analyze its behavior.
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Step-by-Step Solution
Step 1: Separate variables
The given logistic differential equation is dtdP=rP(1−KP). To solve this, we first separate the variables P and t to prepare for integration. We move all terms involving P to one side and all terms involving t to the other side.
Step 2: Integrate both sides using partial fractions
Before integrating, we can simplify the left side by multiplying the numerator and denominator by K. Then, we use partial fraction decomposition for the integral on the left side. The integral of the right side is straightforward.
Step 3: Perform partial fraction decomposition
We decompose the fraction P(K−P)K into partial fractions PA+K−PB. By equating the numerators, K=A(K−P)+BP. Setting P=0 gives K=AK, so A=1. Setting P=K gives K=BK, so B=1.
Step 4: Integrate and solve for P
Now we integrate the decomposed fractions. The integral of P1 is ln∣P∣ and the integral of K−P1 is −ln∣K−P∣. Combining the logarithms and exponentiating both sides, we get K−PP=Aert, where A=eC.
Step 5: Apply initial condition and solve for P(t)
We use the initial condition P(0)=P0 to find the constant A. Substituting t=0 and P=P0 into the equation K−PP=Aert, we find A=K−P0P0. Substituting A back and solving for P(t) gives the logistic growth function.
Step 6: Analyze the behavior of P(t)
As t→∞, e−rt→0, so P(t)→P0KP0=K. This means the population approaches the carrying capacity K. As t→−∞, e−rt→∞, so P(t)→0. The population starts from P0 and grows towards K if 0<P0<K. If P0>K, the population decreases towards K. If P0=K, the population remains constant at K. If P0=0, the population remains 0.