Math Insight

Applet: The dynamics of a population with variable harvesting each time period

The population size $p_t$ in time period $t$ of a species that is being harvested (hunted) is plotted for 100 time periods, and the values $p_t$ are shown in the chart at the right. At time 0, the population size is $p_0$, which you can change by dragging the blue dot or typing a number in the box. In each time period, the population grows by a factor of $r$ but $h(p_t)$ individuals are harvested and removed from the population, so that the dynamics of the population size follow $p_{t+1}-p_t = rp_t-h(p_t)$. You can change $r$ by typing a number in its box. When the linear h checkbox is checked, the function $h$ is of the form $h(p_t)=a+bp_t$, and you can change the coefficients $a$ and $b$ by typing numbers in their boxes. When the linear h checkbox is unchecked, you can type in an arbitrary function for $h(p_t)$, typing p_t for $p_t$. For the linear $h$, the equilibrium value $p_t=E=a/(r-b)$ is plotted by the horizontal cyan line. For the general form of $h$, equilibria may exist, but they are not calculated. You can zoom the vertical axis in and out by clicking the buttons with arrows.

Video introduction.

Applet file: population_with_variable_harvesting.ggb

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