Theory

The package evaluates perturbations acting on matrix population models of the form

\[\mathbf{n}_{t+1} = \mathbf{A}_t \mathbf{n}_t,\]

where \(\mathbf{n}_t\) is the population vector at time \(t\) and \(\mathbf{A}_t\) is the projection matrix.

Perturbation regimes

Each perturbation regime is defined by three quantities:

magnitude: proportional reduction applied to the selected matrix entries;
duration: number of consecutive time steps during which a perturbation is active;
period: number of time steps between the onset of consecutive perturbation events.

A duration of one time step corresponds to a pulse perturbation, whereas longer durations represent sustained perturbations over multiple projection intervals.

Population reduction

For a given perturbation regime, population reduction is defined as

\[\rho = 100 \left(1 - \frac{N_{\mathrm{pert}}(T)} {N_{\mathrm{base}}(T)}\right),\]

where \(N_{\mathrm{pert}}(T)\) is the final population size under the perturbed dynamics and \(N_{\mathrm{base}}(T)\) is the final population size under the unperturbed baseline dynamics.

Integrated vulnerability

Integrated vulnerability is computed as the mean population reduction across the simulated perturbation space:

\[\Phi = \langle \rho(m,d,p) \rangle_{\Omega},\]

where \(m\) is perturbation magnitude, \(d\) is perturbation duration, \(p\) is perturbation period, and \(\Omega\) is the set of simulated perturbation regimes.