1. **Problem Statement:** We want to explore a logistic differential equation and understand its bifurcation diagram. 2. **Logistic Differential Equation:** The logistic equation models population growth with a carrying capacity and is given by: $$\frac{dx}{dt} = rx\left(1 - \frac{x}{K}\right)$$ where $r$ is the growth rate and $K$ is the carrying capacity. 3. **Bifurcation Parameter:** To study bifurcations, we vary the parameter $r$ and observe how the equilibrium points change. 4. **Equilibrium Points:** Set $\frac{dx}{dt} = 0$: $$rx\left(1 - \frac{x}{K}\right) = 0$$ This gives equilibria at: $$x = 0 \quad \text{and} \quad x = K$$ 5. **Stability Analysis:** - For $x=0$, the stability depends on $r$; if $r>0$, $x=0$ is unstable. - For $x=K$, it is stable if $r>0$. 6. **Bifurcation Diagram:** Plotting equilibria $x$ versus parameter $r$ shows a transcritical bifurcation at $r=0$. 7. **Summary:** The logistic equation exhibits a bifurcation at $r=0$ where stability of equilibria exchanges. Final answer: The logistic differential equation is $$\frac{dx}{dt} = rx\left(1 - \frac{x}{K}\right)$$ with bifurcation at $r=0$.