01 · Lesson 11

Dynamical systems

A dynamical system combines a state, a rule for change and an initial condition. Its central question is not only “what is the value now?” but “how can the system behave through time?”

Scenario: will a population settle, grow or collapse?

A population grows when rare but experiences competition as it approaches environmental capacity. We want to understand long-term behaviour for different starting populations without solving a new problem separately for every initial value.

The logistic model is

\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),\qquad r,K>0.\]

State and trajectory

The state contains the variables needed to determine future change under the model. Here the state is the single value \(N\). In an SIR model it is \((S,I,R)\).

A trajectory is the sequence or curve of states generated from an initial condition. Different initial states can produce different trajectories under the same rule.

Equilibria

An equilibrium has zero rate of change. For \(dN/dt=f(N)\), solve \(f(N^*)=0\). The logistic equilibria are

\[N^*=0\qquad\text{and}\qquad N^*=K.\]

An equilibrium represents a state that remains unchanged in the exact model. It is not automatically stable or biologically attainable.

Direction of change and phase line

For \(0<N<K\), \(dN/dt>0\), so population increases.
For \(N>K\), \(dN/dt<0\), so population decreases.
At \(N=0\) or \(N=K\), the rate is zero.

A phase line records these directions on the state axis. It reveals qualitative behaviour without requiring an explicit solution formula.

Stability

An equilibrium is locally stable if sufficiently nearby trajectories remain near it and asymptotically stable if they also approach it. For a one-dimensional ODE, a useful test is

\[f'(N^*)<0\quad\Rightarrow\quad\text{locally asymptotically stable},\]

while \(f'(N^*)>0\) indicates instability. For the logistic model, \(0\) is unstable and \(K\) is stable for positive populations.

Multivariable systems

For \(d\mathbf x/dt=\mathbf f(\mathbf x)\), trajectories move through a state space or phase plane. Nullclines show where one component’s derivative is zero. Their intersections are equilibria. The Jacobian matrix describes the local linearisation, and its eigenvalues help determine local stability.

Oscillations, multiple stable states, thresholds and bifurcations can occur in nonlinear systems. These are developed carefully in Dynamical Systems in Biology rather than repeated here.

Parameters and bifurcations

A parameter change can alter not only numerical values but the number or stability of equilibria. Such a qualitative change is a bifurcation. In biology this may represent a threshold between disease elimination and persistence, population survival and extinction, or steady and oscillatory behaviour.

Conditions and limitations

Define the complete state and valid state space.
Distinguish local conclusions from global behaviour.
Check whether trajectories preserve non-negativity and conservation.
Do not infer stability from a single plotted trajectory.
Remember that deterministic long-term behaviour may omit stochastic extinction or rare transitions.

What this lesson adds

You can now identify states and trajectories, find equilibria, use direction and local stability, and recognise how parameters can change qualitative biological behaviour.