Nutrient-phytoplankton-zooplankton models with a toxin

Abstract

Models of nutrient-plankton interaction with a toxic substance that inhibits either the growth rate of phytoplankton, zooplankton or both populations are proposed and studied. For simplicity, it is assumed that both nutrient and the toxin have the same constant input and washout rate. The effects of toxin upon the existence, magnitude, and stability of the steady states are examined. Numerical simulations demonstrate that the system can have multiple attractors when phytoplankton's nutrient uptake rate is inhibited by the toxin.

Plankton-toxin interaction with a variable input nutrient

“…If nutrient concentration is input constantly instead of periodically in system (1), i.e., if N 0 (t) ≡ N 0 , then σ 0 reduces to αmf (N 0 ) − δ − D, a threshold defined in [19]. The other thresholds σ 1 , ρ 0 , ρ 1 , andρ 1 are the same as those introduced in [19] when N 0 (t) ≡ N 0 .…”

“…The other thresholds σ 1 , ρ 0 , ρ 1 , andρ 1 are the same as those introduced in [19] when N 0 (t) ≡ N 0 . In this case, the results obtained in this study recover those results from the previous study [19].…”

“…The other thresholds σ 1 , ρ 0 , ρ 1 , andρ 1 are the same as those introduced in [19] when N 0 (t) ≡ N 0 . In this case, the results obtained in this study recover those results from the previous study [19]. These thresholds can be viewed as the average net growth rates of phytoplankton and zooplankton under different circumstances when either toxin is absent, or present with effects to different trophic levels.…”

“…[3]. To the best of our knowledge, this research along with a study in [19] are the first investigations of using mathematical models to explore toxic effects upon plankton interaction.…”

“…In natural nutrient-plankton systems, waters flowing into the system bring input of fluxes of nutrients and outflows also carry out nutrients [4,10,32,33]. The rate of waters flowing in and out of the system is a constant and denoted by D. However, unlike the study in [19], we assume that the input nutrient concentration N 0 (t) is varied periodically around N 0 with N 0 (t) = N 0 + ar(t), where N 0 > 0, 0 < a < N 0 , and r(t) is τ -periodic with mean value zero and |r(t)| ≤ 1 for t ≥ 0. The consideration of periodic input of nutrients is motivated by the tidal, day/night, or seasonal cycles occurring in nature.…”