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• W2078589814 abstract "We study the following equations for a two-level ecosystem consisting of primary resources and grazing consumers: (1) An augmented logistic equation ẋ(t) = rx(t) [ 1 - x(t)k[1 - D(t)] ] - u(t)x(t), where x > 0 is grazer biomass (e.g. fish or reindeer), u ⩾ 0 is a per-unit-x harvest rate, and 0 < r < ∞ and 0 < K < ∞ are constants. D(t) represents the fractional depletion at time t of the renewable resource (e.g., algae or grass). Thus 0 ⩽ D < 1, and the usual carrying capacity K is reduced to K[1 - D(t)]. (2) A depletion equation D(t) = ∫-∞t F[x(τ)]1 + bD(τ) P(t - τ)dτ where F[x(τ)] is the rate at which x(τ) is increasing D(τ) at time τ, and P(t - τ), 0 ⩽ P ⩽ 1, is the fraction of the depletion incurred at τ that is still unrenewed att. (3) A discounted harvesting function J = ∫0∞ e-δt[ s - kx(t) ] u(t)x(t)dt to be maximized subject to (I) and (II). s is revenue per-unit harvested biomass, and kx(t) is the per-unit cost of harvesting biomass at the stock level x(t). [s - kx(t)] > 0 for all x(t) of interest. We prove that (I), with u=0, and (II) are unstable about their unique nontrivial equilibrium point under perturbations of the integral factor in the Ḋ(t) equation, for all 0 < L < ∞. Computer solutions virtually assure, however, that the system is globally stable about a small neighborhood containing its equilibrium underperturbations in D(t) and x(ϴ), t - L ⩽ ϴ ⩽ t. Our interpretation of the instability is that under certain conditions, events such as grass fires, algal dieoffs, and resource stocking could destabilize an otherwise stably equilibrated system. We solve two general problems associated with (III): in the first, the asymptotic x, D, and u are found for (III) when L = 0; and in the second, 0 < L < ∞. An example is given in which a nearly 50% increase in asymptotic u results from including D's delay dynamics." @default.
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• W2078589814 date "1979-11-01" @default.
• W2078589814 modified "2023-09-28" @default.
• W2078589814 title "Depletion models for ecosystems with continuously delayed resource renewals" @default.
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• W2078589814 doi "https://doi.org/10.1016/0025-5564(79)90004-x" @default.
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