Math Insight

Applet: Equilibria when harvesting natural populations

A plot showing the equilibria for logistic growth with harvesting. The updating function $f(x)=x+r(1-x)-hx$ is shown by the red curve along with the diagonal line $x=y$ in blue. The two equilibria, which are the intersections between the curves, are shown by the blue and green points. The green point corresponds to the equilibrium $E_1=0$. As you can observe by the slope of $f$ at the equilibria, as long as the harvesting rate $h$ is less than the low density growth rate $r$, the equilibrium $E_2$ corresponding to the blue point is positive and stable, and the equilibrium $E_1=0$ is unstable. If the harvest rate is larger than the low density growth rate, then the second equilibrium $E_2$ becomes negative an unphysical. The only physically relevant equilibrium is $E_1=0$, which is stable. You can change $r$ and $h$ by typing values in their boxes. To display the sustainable harvest, which is the stable equilibrium multiplied by the harvest rate $h$,check the “show harvest” box.

Applet file: harvest_natural_populations_equilibria.ggb

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