Fig 1.
Probability of extinction of small stochastic populations.
Probability of extinction in terms of the initial population size around L for deterministic (σ = 0) and demographic stochastic (σ = 1) uncontrolled populations and populations controlled with different intensities by ALC (A) and ATH (B). Calculations are based on Models (1) and (2) with r = 4.5, and s = 0.002 (L ≈ 6.015). For a given initial population size, the probability of extinction has been obtained for the first 100 generations and over 1000 replicates.
Fig 2.
ALC can slow down the convergence to extinction of small deterministic populations.
The black curve corresponds to the production function of the uncontrolled Model (1) with r = 4.5, and s = 0.002, and the red curve to the population controlled by ALC with intensity c = 0.5.
Fig 3.
Stochastic Allee and collapse thresholds.
Stochastic Allee and collapse thresholds as functions of control intensity for different levels of noise and for different strengths of the Allee effect in the range of bistable dynamics (A to D) and in the range of essential extinction (E to H). Calculations are based on Models (1) and (2) with r = 4.5 and . For a given initial population size, the probability of extinction has been obtained for the first 100 generations and over 5000 replicates. The right-hand side panels show stochastic collapse threshold only for ATH since they exist under ALC only for extremely small intensities.
Fig 4.
Large populations can be driven to extinction if there is a collapse threshold U and population size exceeds that threshold. A: The collapse threshold exists if limx→+∞f(x) < L. Then there is a U such that f(x) > L for all x ∈ (L, U) and f(x) < L for all x ∈ (U, +∞). B: There is no collapse threshold if limx→+∞f(x) ≥ L, because then f(x) > L for all x > L.
Fig 5.
Persistence and extinction depend on the initial population size for ALC.
Deterministic population sizes over the first 100 generations as a function of the initial value for ALC. Parameter values K = 400, r = 4.5, s = 0.002, σ = 0 (deterministic), and c = 0.007 (the value of L/U is 0.007614).
Fig 6.
Probability of outbreak in terms of the initial population size.
The population is controlled by (A) ALC and (B) ATH. Population dynamics are deterministic (σ = 0) or with demographic stochasticity (σ = 1). Calculations are based on Models (1) and (2) with r = 4.5, and s = 0.002 (L ≈ 6.015). Population outbreaks are considered to occur when the number of individuals exceeds (K + f(d))/2. For a given initial population size, the outbreak probability has been obtained for the first 100 generations and over 1000 replicates.
Fig 7.
Numerical simulations for the gypsy moth model.
(A) Comparison of model time series for populations of the defoliator gypsy moth. The black curve corresponds to the uncontrolled system, the blue one to the system controlled by ALC with c = 0.9 and the red one to ATH with h = 0.9. (B) Box plots of the maximum population density of defoliators for the uncontrolled system and systems controlled by ATH and ALC with different intensities. Calculations are based on Model (3) with λ = 74.6, ϕ = 20, A = 0.967, B = 0.14, k = 1.06 and σ = 0.5. Initial densities in (A) are x0 = 10 for the defoliator and z0 = 7 for the pathogen. Values in (B) have been obtained from 100 time series with initial population densities uniformly distributed in [0.01, 100] and a time horizon of 50 generations.