# Math Insight

### Applet Introduction Video: The dynamics of a population with harvesting of a fixed number each time period

This video is an introduction to the applet: The dynamics of a population with harvesting of a fixed number each time period.

This applet illustrates how the population size of some animal species evolves over time when a fixed number of the species are regularly harvested, which is a euphemism for killed, usually for the physical or economic benefit of the harvester. The horizontal axis represents time $t$ in terms of time bins of some arbitrary length. You can imagine each unit of time represents a year, a month or a day. Here we are plotting 100 of these time bins. The vertical axis represents some measurement of the size of the population, presumably the number of individuals in the population, but it doesn't really matter what measurement is used. We use the variable $p$ for the population size, so that $p_t$ is the population size in time bin $t$. Each green dot represents the population size during a particular time period. Hmm, something strange seems to be happening to the population size here.... Oh well, I'll let you worry about that.
The evolution of the population size is given by this equation. The change in the population size from time $t$ to time $t+1$ is this expression on the right. Right now it is set to 0.2 times $p_t$ minus 800, but we'll change that in a minute. The equation determines how the population at a time bin is calculated from the previous population size. Since the initial time is time bin zero, the initial population size is $p_0$. Right now, it is set to 3995, but we can change it either by typing a new number in the box or dragging the blue point with the mouse. The general form of the right side of the equation is some number $r$ times $p_t$ minus another number $h$. The number $r$ is the population growth rate and $h$ is the number of individuals harvested in each time bin. You can change these values by typing numbers in the boxes. You can see that the equation changes to reflect the new values.
The applet also calculates one special value for you, which is the equilibrium value $E$. It is displayed at the top and represented by the horizontal cyan line. The population change is zero if $p_t$ is $h$ divided by $r$, which is 800 with the current values. If you start with a population size that is exactly at the equilibrium, the population doesn't change at all in future times.