# صالح احمد محمد مشرى

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 Cardioid

The curve given by the polar equation

 (1)

sometimes also written

 (2)

where .

The cardioid has Cartesian equation

 (3)

and the parametric equations

 (4) (5)

The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with ) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.

The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is .

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.