# Math Insight

### Applet: Evolution of a bistable population

The evolution of the size $x(t)$ of a population is illustrated in three different ways. In the right panel, each green diamond represents one individual in the population, so the the population size $x(t)$ is given by the number of green diamonds. You can change time $t$ by dragging the red point on the red slider in the left panel or by clicking the triangle in the lower-left corner of one of the panels to start the animation. The state space, or space of possible values of $x(t)$, is represented by the blue line in the left panel. The initial condition $x_0$, or population size at time $t=0$, is shown by the large blue point in the state space. The population size $x(t)$ at time $t$ is shown by the small red point on the state space (visible only if $t > 0$). If you check the “plot versus time” box, then the right panel displays a green curve indicating how the population size changes as a function of time. The height of the blue dot on the curve indicates the initial population size $x_0$ at time $t=0$, and the height of the red dot on the curve indicates $x(t)$ for the time $t$ shown in the left panel.

The evolution of the population size shows bistable behavior so that the population size either crashes to zero (i.e., the population dies out) or the population size approaches a larger value (i.e., the population survives). The fate of the population depends on the initial condition $x_0$ (which you can change by typing a value in the box or dragging one of the blue points). You can change the value of the larger population size (and other parameters affecting how the population evolves) by checking the “advanced options” and changing the values of the parameters. You can also change the name of the state variable $x$ to another variable by entering a different letter in the state variable box.

The population size evolves according to the differential equation $$\diff{x}{t} = rx(x-\alpha)(x-\beta).$$ This equation has three steady states, $x=0$, $x=\alpha$, and $x=\beta$. As long as you leave the growth rate $r$ positive, the smallest and largest steady state are stable and the middle steady state is unstable, leading to bistable behavior of the population. (Note: if you make the growth rate $r$ too large, the applet will not accurately show this behavior unless you make the time step $dt$ sufficiently small.)

Applet file: evolution_bistable_population.ggb