# Math Insight

### Applet: The dynamics of a population with harvesting of a fixed number 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. 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 $a$ individuals are harvested and removed from the population, so that the dynamics of the population size follow $p_{t+1}-p_t = rp_t-a$. (You can change $r$ and $a$ by typing numbers in their boxes.) The equilibrium $p_t=E=a/r$ is plotted by the horizontal cyan line. You can explore how the evolution of the population size has very different behavior depending on the relationship between $p_0$ and $E$. To set the initial conditions to be exactly the equilibrium, you can enter $E$ in the box for $p_0$. You may notice something that doesn't make sense with this model. You can zoom the vertical axis in and out by clicking the buttons with arrows.

Applet file: population_with_harvesting_fixed_number.ggb