Biology · Systems & Quantitative Biology

Feedback, stability, oscillations and switching

Two rules both have a fixed state at four. One moves nearby values toward four; the other moves them away. A zero rate identifies a fixed state, but its response to a small displacement determines a different property: stability.

Research Design, Data & Scientific Practice introduces observations, comparisons and uncertainty. Gene Regulation & Cell Signalling develops biological feedback, while Population Ecology & Life-History Patterns develops rates and population boundaries. Their English chapters are available in this working manuscript; publication remains pending. Start with arithmetic, averages and ordinary graph reading. A derivative is introduced as an instantaneous rate of change. Exponentials and the circular sine–cosine example are explained where used; a first reading can follow their values and arrows, then return to their formulas with further mathematical preparation.

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By the end, you can

Distinguish a fixed state, its local stability, multiple stable states and a periodic orbit without inferring them from feedback signs alone.

A fixed state may return or repel

A fixed state, also called an equilibrium here, has zero rate under the model rule. In dx/dt = 4 - x, the rate is zero at x = 4. Below four it is positive, and above four it is negative. These signs move a nearby state toward four. In dx/dt = x - 4, the same fixed state has the opposite signs and moves nearby states away.

For these two linear rules the trajectories can be written exactly. Starting at x₀, the returning trajectory is 4 + (x₀ - 4)e⁻ᵗ, and the repelling trajectory is 4 + (x₀ - 4)eᵗ. The exponential factors shrink or grow the displacement from four. The displayed table samples times zero, one quarter and one half; the formula describes the continuous trajectory between them.

A returning fixed state in this scalar example is asymptotically stable: nearby displacements approach zero as time increases. Stability is conditional on the specified dynamics. A biological system may also have disturbances, changing parameters, spatial structure or additional states. The labels in this mathematical comparison do not establish an observed organism's regulatory stability.

The same fixed value can return or repel

The curves are exact solutions for the assigned rules. Extending a repelling amount interpretation beyond its nonnegative domain would require a different justified model.

Read the diagram in words

Both rules have zero rate at four. Starting at two or six, the first rule shrinks the displacement and the second enlarges it. Four series use the same time and state scales; the repelling lower branch remains positive throughout the displayed interval.

Return from below
Nondimensional time	Assigned scalar state
0	2
0.25	2.44239843385719
0.5	2.786938680574733

x=4+(2-4)exp(-t)

Return from above
Nondimensional time	Assigned scalar state
0	6
0.25	5.55760156614281
0.5	5.213061319425266

x=4+(6-4)exp(-t)

Departure from below
Nondimensional time	Assigned scalar state
0	2
0.25	1.4319491666245172
0.5	0.7025574585997436

x=4+(2-4)exp(t)

Departure from above
Nondimensional time	Assigned scalar state
0	6
0.25	6.568050833375483
0.5	7.297442541400256

x=4+(6-4)exp(t)

State change per time unit
Assigned scalar state	Rate points toward four	Rate points away from four
2	2	-2
6	-2	2

Fixed state: four: 4

Original nondimensional scalar rules with constant coefficients and fixed state four. Time 0..0.5 is displayed so every sampled state remains positive, including the lower repelling branch. No biological recovery time or valid indefinite amount model is asserted.

A nonlinear rule can have more than one destination

The original rule dx/dt = -(x - 1)(x - 2)(x - 3) has zero rate at one, two and three. Between zero and one its rate is positive. Between one and two it is negative. Between two and three it is positive, and above three it is negative. Those signs lead toward one or three and away from two.

A starting state one and a half moves toward one; a starting state two and a half moves toward three. With the same parameter values, different starting states can have different long-term destinations. This is bistability in the assigned scalar model: two attracting fixed states separated by an unstable one. A steep single response curve alone would not establish that property.

Positive feedback combined with appropriate nonlinearity can support bistability in some biological models, but a sign label is insufficient. Nor does this fixed-parameter polynomial demonstrate a hysteresis experiment. Hysteresis would require a specified changing-input comparison that reveals history-dependent responses. The example establishes the destinations of this one mathematical rule and leaves that separate input comparison open.

Initial state selects one of two stable destinations

The rate curve and sign arrows show bistability for this scalar model. They do not constitute a measured response curve or a demonstrated hysteresis comparison.

Read the diagram in words

The polynomial rate is zero at one, two and three. Its signs point toward one and three and away from two. Starting at one and a half leads toward one, whereas starting at two and a half leads toward three under the same fixed rule.

Instantaneous state rate
Nondimensional state	Instantaneous state rate
0	6
0.5	1.875
1	0
1.5	-0.375
2	0
2.5	0.375
3	0
3.5	-1.875

Zero-rate state
Zero-rate state	Nondimensional state
Lower attracting fixed state	1
Middle repelling fixed state	2
Higher attracting fixed state	3

State increases / State decreases
Nondimensional state	Nondimensional state	State increases / State decreases	Attracting destination under this fixed rule
0	1	1	1
1	2	-1	1
2	3	1	3
3	3.5	-1	3

Attracting destination under this fixed rule
Nondimensional state	Nondimensional state	Instantaneous state rate	Attracting destination under this fixed rule
Start in the lower basin	1.5	-0.375	1
Start in the higher basin	2.5	0.375	3

dx/dt=-(x-1)(x-2)(x-3)

Original nondimensional scalar polynomial on the displayed domain 0..3.5. It supplies two attracting fixed states and one repelling fixed state, but no actual signalling circuit, changing control input, hysteresis path or biological switching procedure.

An orbit can repeat without attracting nearby orbits

An oscillation repeats a changing state over time. In the original circular example, x = 2 + cos t and y = 2 + sin t. Cosine and sine describe the horizontal and vertical coordinates of a unit circle as its angle increases. Adding two centres the circle at (2, 2), so both state coordinates stay between one and three.

The associated rule is dx/dt = -(y - 2), dy/dt = x - 2. The selected trajectory returns after a period of 2π time units. Nearby initial points follow circles with different radii; they do not converge onto the displayed circle. It is a neutral periodic orbit, not an attracting limit cycle. Repetition and attraction are distinct dynamical properties.

Biological oscillations may involve feedback, delays, nonlinear responses and multiple timescales. Their explanation requires suitable change rules and time-resolved evidence. A negative loop or a periodic mathematical drawing alone supplies neither. Check which behaviour the model actually demonstrates: fixed-state return, multiple destinations, periodicity or recovery of a particular cycle after perturbation.

A repeating circle need not attract another circle

The two state coordinates have equal scales. Time orders the points but is not a spatial coordinate, and periodicity does not imply attraction.

Read the diagram in words

The five records move counterclockwise from (3, 2) through (2, 3), (1, 2), (2, 1) and back to (3, 2) over 2π time units. The same rule keeps the radius fixed for nearby starts, so another circle does not converge onto the displayed one.

Selected unit-radius orbit
Selected unit-radius orbit	Nondimensional time	First state coordinate	Second state coordinate
Start	0	3	2
Quarter period	1.5707963267948966	2	3
Half period	3.141592653589793	1	2
Three quarters	4.71238898038469	2	1
One period	6.283185307179586	3	2

Periodic, without attraction to this orbit
First state coordinate	Second state coordinate	r	T
2	2	1	6.283185307179586

x=2+cos(t)

y=2+sin(t)

dx/dt=-(y-2)

dy/dt=x-2

Original nondimensional two-state rule centred at (2, 2), with selected initial state (3, 2). Its unit-radius orbit is periodic but neutral: neighbouring initial radii remain different. No attracting limit cycle, actual biological clock or oscillator design is supplied.

Worked example

Read directions before interpreting a trajectory

At x = 2, the returning rule has rate +2 and the repelling rule rate -2. At x = 6, their rates are -2 and +2. Both rules have a fixed state at four, but the direction on either side differs.

For the polynomial, the rate at 1.5 is -0.375, and at 2.5 it is +0.375. These signs lead away from two toward different attracting fixed states. A state exactly at two would remain there in the exact unperturbed rule, even though that fixed state is unstable.

For the circular trajectory, the five quarter-period records are (3, 2), (2, 3), (1, 2), (2, 1) and (3, 2). The final record repeats the initial one. A circle of another radius would also repeat, so this table does not demonstrate attraction toward the unit-radius orbit.

Check your understanding

Two models both have zero rate at four. Does this prove the same stability?
Show reasoning
No. Stability depends on nearby dynamics. The returning rule drives a displacement toward four; the repelling rule drives it away, despite the shared fixed state.
Why does the polynomial example have two stable destinations?
Show reasoning
Its rate changes sign toward one on both sides of one and toward three on both sides of three. The intermediate fixed state two has signs pointing away. These directions distinguish the two basins of attraction.
What does the circular example establish, and what does it leave open?
Show reasoning
It establishes a periodic trajectory with period 2π under its assigned rule. Nearby circles keep different radii, so it does not establish an attracting limit cycle or a robust biological clock.

What to carry forward

Examine the actual change rule and nearby trajectories; fixed states, bistability, periodicity and attraction answer different questions.

References

Resources supporting this lesson’s explanations and examples.