Biological applications of bifurcation theory

For a more rigorous example, consider the dynamical system shown in Figure 2, described by the following equation:

${\displaystyle {\dot {x}}=-x^{2}+r}$

where r is once again the control parameter (labeled ε in Figure 2). The system's fixed points are represented by where the phase portrait curve crosses the x-axis. The stability of a given fixed point can be determined by the direction of flow on the x-axis; for instance, in Figure 2, the green point is unstable (divergent flow), and the red one is stable (convergent flow). At first, when r is greater than 0, the system has one stable fixed point and one unstable fixed point. As r decreases the fixed points move together, briefly collide into a semi-stable fixed point at r = 0, and then cease to exist when r < 0.

In this case, because the behavior of the system changes significantly when the control parameter r is 0, 0 is a bifurcation point. By tracing the position of the fixed points in Figure 2 as r varies, one is able to generate the bifurcation diagram shown in Figure 3.

Other types of bifurcations are also important in dynamical systems, but the saddle-node bifurcation tends to be most important in biology. Real biological systems are subject to small stochastic variations that introduce error terms into the dynamical equations, and this usually leads to more complex bifurcations simplifying into separate saddle nodes and fixed points. Two such examples of "imperfect" bifurcations that can appear in biology are discussed below. Note that the saddle node itself in the presence of error simply translates in the x-r plane, with no change in qualitative behavior; this can be proven using the same analysis as presented below.

Figure 4. Unperturbed (black) and imperfect (red) transcritical bifurcations, overlaid. u and p are referred to as x and r, respectively, in the rest of the article. As before, solid lines are stable, and dotted unstable.

A common simple bifurcation is the transcritical bifurcation, given by

${\displaystyle {dx \over dt}=rx-x^{2}}$ and the bifurcation diagram in Figure 4 (black curves). The phase diagrams are shown in Figure 5. Tracking the x-intercepts in the phase diagram as r changes, there are two fixed point trajectories which intersect at the origin; this is the bifurcation point (intuitively, when the number of x-intercepts in the phase portrait changes). The left fixed point is always unstable, and the right one stable.

Figure 5. Ideal transcritical bifurcation phase portraits. Fixed points are marked on the x-axis, each trajectory in a different color. The direction of the arrows indicates which direction they move as r increases. The red point is unstable, and the blue point is unstable. The black dot at the origin is stable for r < 0, and unstable for r > 0.

Now consider the addition of an error term h, where 0 < h << 1. That is,

${\displaystyle {dx \over dt}=rx-x^{2}-h}$

The error term translates all the phase portraits vertically, downward if h is positive. In the left half of Figure 6 (x < 0), the black, red, and green fixed points are semistable, unstable, and stable, respectively. This is mirrored by the magenta, black, and blue points on the right half (x > 0). Each of these halves thus behaves like a saddle-node bifurcation; in other words, the imperfect transcritical bifurcation can be approximated by two saddle-node bifurcations when close to the critical points, as evident in the red curves of Figure 4.

Figure 6. Imperfect transcritical bifurcation phase portraits. Five values of r are shown, given relative to the two critical points. Note that the y-intercept value is the same as h, or the magnitude of the imperfection. The green and blue points are stable, while the green red and magenta are unstable. The black dots indicate semistable fixed points.

Besides observing the flow in the phase diagrams, it is also possible to demonstrate the stability of various fixed points using linear stability analysis. First, find the fixed points in the phase portrait by setting the bifurcation equation to 0:

${\displaystyle {\begin{aligned}{dx \over dt}=f(x)&=rx-(x)^{2}-h\\0&=rx^{*}-(x^{*})^{2}-h\end{aligned}}}$

Using the quadratic formula to find the fixed points x*:

${\displaystyle {\begin{aligned}x^{*}&={-r\pm {\sqrt {r^{2}-4(-1)h}} \over 2(-1)}\\&={r\pm {\sqrt {r^{2}+4h}} \over 2}\\&\approx {r\pm {\sqrt {r^{2}}} \over 2}\end{aligned}}}$

where in the last step the approximation 4h << r ² has been used, which is reasonable for studying fixed points well past the bifurcation point, such as the light blue and green curves in Figure 6. Simplifying further,

${\displaystyle {\begin{aligned}x^{*}&\approx {r\pm r \over 2}\\&={\begin{cases}0,&{\text{fixed point closer to origin}}\\r,&{\text{fixed point further from origin}}\end{cases}}\end{aligned}}}$

Next, determine whether the phase portrait curve is increasing or decreasing at the fixed points, which can be assessed by plugging x* into the first derivative of the bifurcation equation.

${\displaystyle {\begin{aligned}f'(x)&=r-2x\\f'(0)&=r={\begin{cases}>0,&{\text{if }}r>0\rightarrow {\text{unstable (magenta)}}\\<0,&{\text{if }}r<0\rightarrow {\text{stable (green)}}\end{cases}}\\f'(r)&=-r={\begin{cases}<0,&{\text{if }}r>0\rightarrow {\text{stable (blue)}}\\>0,&{\text{if }}r<0\rightarrow {\text{unstable (red)}}\end{cases}}\end{aligned}}}$

The results are complicated by the fact that r can be both positive and negative; nonetheless, the conclusions are the same as before regarding the stability of each fixed point. This comes as no surprise, since the first derivative contains the same information as the phase diagram flow analysis. The colors in the above solution correspond to the arrows in Figure 6.

The buckling beam example from earlier is an example of a pitchfork bifurcation (perhaps more appropriately dubbed a "trifurcation"). The "ideal" pitchfork is shown on the left of Figure 7, given by

${\displaystyle {dx \over dt}=rx-x^{3}}$

and r = 0 is where the bifurcation occurs, represented by the black dot at the origin of Figure 8. As r increases past 0, the black dot splits into three trajectories: the blue stable fixed point that moves right, the red stable point that moves left, and a third unstable point that stays at the origin. The blue and red are solid lines in Figure 7 (left), while the black unstable trajectory is the dotted portion along the positive x-axis.

As before, consider an error term h, where 0 < h << 1, i.e.

Figure 8. Ideal pitchfork bifurcation phase portraits. Fixed points are marked on the x-axis, each trajectory in a different color. The direction of the arrows indicates which direction they move as r increases. The red and blue points are stable, and there is a third unstable fixed point at the origin, indicated by the black dot. For r = r_crit = 0, the black dot also indicates a semi-stable point that appears and splits into the other three trajectories as r increases.

${\displaystyle {dx \over dt}=rx-x^{3}+h}$

Once again, the phase portraits are translated upward an infinitesimal amount, as shown in Figure 9.Tracking the x-intercepts in the phase diagram as r changes yields the fixed points, which recapitulate the qualitative result from Figure 7 (right). More specifically, the blue fixed point from Figure 9 corresponds to the upper trajectory in Figure 7 (right); the green fixed point is the dotted trajectory; and the red fixed point is the bottommost trajectory. Thus, in the imperfect case (h ≠ 0), the pitchfork bifurcation simplifies into a single stable fixed point coupled with a saddle-node bifurcation.

A linear stability analysis can also be performed here, except using the generalized solution for a cubic equation instead of quadratic. The process is the same: 1) set the differential equation to zero and find the analytical form of the fixed points x*, 2) plug each x* into the first derivative ${\displaystyle f'(x)={d \over dx}{dx \over dt}}$, then 3) evaluate stability based on whether ${\displaystyle f'(x^{*})}$ is positive or negative.

Figure 9. Imperfect pitchfork bifurcation phase portraits. Four different values of r relative to r_crit are shown. Note that the y-intercept value is the same as h, or the magnitude of the imperfection. The red and blue points are stable, while the green one (previously hidden at the origin) is unstable. As in Figure 5, the black dot indicates a semi-stable point that appears and splits into the red and green ones as r increases.

Combined saddle-node bifurcations in a system can generate multistability. Bistability (a special case of multistability) is an important property in many biological systems, often the result of network architecture containing a mix of positive feedback interactions and ultra-sensitive elements. Bistable systems are hysteretic, i.e. the state of the system depends on the history of inputs, which can be crucial for switch-like control of cellular processes.[2] For instance, this is important in contexts where a cell decides whether to commit to a particular pathway; a non-hysteretic response might switch the system on-and-off rapidly when subject to random thermal fluctuations close to the activation threshold, which can be resource-inefficient.