Deterministic vs stochastic models
Biological systems can be represented either without explicit randomness or with randomness included as part of the model. This gives the distinction between deterministic and stochastic models.
Deterministic models
In a deterministic model, the initial conditions and parameter values determine a single model trajectory. Repeating the calculation with exactly the same inputs gives the same result.
For example,
\[\frac{dN}{dt}=rN,\qquad N(0)=N_0\]has a uniquely determined solution
\[N(t)=N_0e^{rt}.\]Stochastic models
A stochastic model includes random quantities or random events. Starting from the same initial state can therefore produce different possible trajectories.
Instead of asking only for one trajectory, stochastic modelling may investigate quantities such as
\[P(\text{extinction before time }T),\qquad E[N(t)],\qquad \operatorname{Var}(N(t)).\]Comparison
| Deterministic | Stochastic |
|---|---|
| Same inputs give the same trajectory | Same inputs can give different trajectories |
| Describes a specified systematic evolution | Represents variability explicitly |
| Often studied with difference or differential equations | Often studied with Markov models, stochastic simulation or SDEs |
Why both are useful
Stochastic is not automatically better than deterministic. The appropriate choice depends on the biological question, system size, available information and importance of variability. Deterministic models can describe dominant or average behaviour efficiently, while stochastic models are especially useful when the distribution of possible outcomes matters.