Math Insight

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

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

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

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This applet illustrates how the population size of some animal species evolves over time when a variable number of the species are regularly harvested, i.e., killed. The horizontal axis plots time $t$ in terms of some arbitrary time scale. The vertical axis plots population size $p$, and green dots show the population size $p_t$ at different time bins $t$.This applet also displays a list of the population sizes at different time periods. Here, we can see, for example, that the population size at time 4, or $p_4$ is 1464.1. (draw arrow. Add text later.) If $p_4$ is the number of individuals in the population, then it appears we have a tenth of an individual along with 1464 whole ones. We aren't going to worry about the unrealistic nature of having fractional animals running around, but will allow for fractional animals as a mathematical idealization. You can just round the numbers in your head if the fractions bother you.

The initial population size $p_0$, now 1000, is represented in three places. You can change it by typing a new value in the box or by dragging the blue point with the mouse. The population evolves according to this equation, which can be written in general form as this. The population change from time bin $t$ to time bin $t+1$ is due to two factors. The population increases due to reproduction at rate $r$, which is now set at 0.2 but can be changed by typing in a new value in the box. The population decreases due to harvesting at rate $h$. The harvesting rate can depend on the population size, so it is written as a function of population size $p_t$.

When the “linear h” box is checked, harvesting rate is a linear function of population size. We can write this linear function as a constant $a$ plus $b$ times the current population size. The values of $a$ and $b$ are determined by the numbers in these boxes. If we make $b$ zero and $a$ nonzero, then a fixed number of individuals are harvested in each time period.

The equilibrium value, i.e., the population size where the population stays constant is shown at the top and by the horizontal cyan line. When the harvesting is a linear function of population size, there is only one equilibrium.

The harvesting rate can made more complicated by unchecking the “linear h” box. Then you can type in an arbitrary function for the harvesting rate $h(p_t)$. In the expression you type in the box, you write $p_t$ as $p$ underscore $t$. You can type exponents by using the caret, so you can enter $p_t$ to the power of three like this.

When $h$ is an arbitrary function of $p_t$, we don't have a nice way to calculate the equilibria and there might be more than one. The applet attempts to calculate a list of the equilibria and display them as horizontal cyan lines. It's an imperfect algorithm, so it may not get them all. For this harvesting function, there are three equilibria, and we can zoom in to see that indeed there are three horizontal lines. I'm not sure what negative population sizes are supposed to mean. But, the applet dutifully calculates values based on the formula and doesn't filter out crazy values.

If you want to get back to the default situation, you can always click the reset button.