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We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation−u′1+u′2′=λu1+up,−L&lt;x&lt;L,u(−L)=u(L)=0,\left\{\begin{array}{l}-{\left(\frac{{u}^{^{\prime} }}{\sqrt{1+{u}^{^{\prime} 2}}}\right)}^{^{\prime} }=\lambda {\left(\frac{u}{1+u}\right)}^{p},\hspace{1.0em}-L\lt x\lt L,\\ u\left(-L)=u\left(L)=0,\end{array}\right.whereλ\lambdais a bifurcation parameter, andL,p&gt;0L,p\gt 0are two evolution parameters. We prove that on the(λ,‖u‖∞)\left(\lambda ,\Vert u{\Vert }_{\infty })-plane, for0&lt;p≤240\lt p\le \frac{\sqrt{2}}{4}, the bifurcation curve is⊃\supset-shaped bifurcation starting from(0,0)\left(0,0). And forp=1,f(u)=u1+up=1,f\left(u)=\frac{u}{1+u}is a logistic function, then the bifurcation curve is also⊃\supset-shaped bifurcation starting fromπ24L2,0\left(\frac{{\pi }^{2}}{4{L}^{2}},0\right). While forp&gt;1p\gt 1, the bifurcation curve is reversedε\varepsilon-like shaped bifurcation ifL&gt;L∗L\gt {L}^{\ast }, and is exactly decreasing forλ&gt;λ∗\lambda \gt {\lambda }^{\ast }if0&lt;L&lt;L∗0\lt L\lt {L}_{\ast }.

On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type