Fig 1.
Steady-state diagram and corresponding eigenvalues of the fisheries model as a function of predator death rate μP.
(a-c) steady states showing a S-bend shape for J and P variables with color-coded lower (blue), middle (green) and upper (red) branches. Population A shows a linear trend with μp. (d, e) Real and imaginary parts of linearized model. Saddle-node bifurcation points are marked B1 and B2 and indicated by vertical dashed grey lines. Model parameters are c = 1, b = 1, μJ = 0.05, μA = 0.1. Eigenvalues are not relevant for since predator population goes negative.
Fig 2.
Numerically obtained fluctuation variance of model variables prior to saddle-node bifurcation.
(a–d) For each value of μP, starting with μP = 0.4, the model is simulated for 60,000 time units with resolution of 0.01 time units, from which the population variances are computed. μP is incremented geometrically towards the catastrophic collapse at μP = μB2. Death rates are perturbed every time unit using white noise with standard deviation of σnoise = 2 × 10−6. (a) Noise added to the juvenile population (b) Noise added to the adult population (c). Independent noise added to all three populations. (d) Identical, fully correlated, noise added to all three populations. (e) An extra experiment where noise is added to P only population.
Fig 3.
Fluctuation variance prior to saddle-node bifurcation with independent noises added to all three populations.
Experimental and theoretical variances are plotted for all three populations while approaching catastrophe. A fixed step Euler method with Δt = 0.01 is used for numerical simulations each for 6000 time units. Starting with μP = 0.4, it is incremented towards the saddle-node point at μB2. The distance from bifurcation point is geometrically reduced enabling more experiments close to bifurcation. Three independent white noise sources are added to each population as described in Eqs (14)–(16) with standard deviation of σnoise = 2 × 10−6.
Fig 4.
Tracking of dominant eigenvalue and its decomposed eigenvector towards saddle-node bifurcation point.
(a) The real part of dominant eigenvalue and (b) the corresponding decomposed eigenvector as a function of predator mortality rate μP.
Fig 5.
Observability coefficients of fisheries model.
Observability coefficient of system variables , and
as a function of μP calculated using the definition in Eq (24). Lie derivatives are used to construct the observability matrix from which the observability coefficients are calculated. See text for details. A geometrically spaced vector of predator mortality rate is used to cover the range 0.4 ≤ μP ≤ μB2.