Phase Line

The vertical line given by *H* is the phase line associated with the differential equation

Recall that the phase line carries information on the nature of the constant solutions (or equilibria) with respect to their classification as sources, sinks, or nodes. This classification is given by the sign of the function . The graph of for different values of *H* is given below

Putting everything together we get the following diagram (which is called the **bifurcation diagram**)

Instead of just drawing some phase lines, we will usually color the regions. The next picture illustrates this very nicely:

Let us use this diagram to discuss the fate of the fish population as the parameter *H* increases. When *H*=0 (no fishing), the fish population tends to the carrying capacity *P*=1 which is a sink. If *H* increases but stays smaller than 0.25, then the fish population still tends to a new and smaller number

which is a also sink. When *H* is increased more and exceeds 0.25, then the differential equation has no equilibrium points (constant solutions). The fish population is decreasing and crosses the t-axis at finite time. This means that the fish population will vanish completely in finite time. Hence, in order to avoid such a catastrophic outcome, *H* needs to be slightly lower than 0.25, which is called the optimal harvesting rate. You should also keep in mind that a slightly smaller number will be a better choice than *H*=0.25 itself, since for *H*=0.25 the only equilibrium point *P* = 0.5 is not a sink (in fact, it is a node) and as soon as the population *P* falls below 0.5, we will again witness extinction in finite time.

The next animation illustrates the behavior of the solutions as *H* changes. The animation is based on the differential equation

*P*'=*P*(1-*P*/5)-*h*.In this case, the bifurcation occurs, when

*h*=1.25.