Discrete-Time Model of Seasonal Plankton Bloom

Abstract

The most interesting results in modeling phytoplankton bloom were obtained based on a modification of the classical system of phytoplankton and zooplankton interaction. The modifications using delayed equations, as well as piecewise continuous functions with a delayed response to intoxication processes, made it possible to obtain adequate phytoplankton dynamics like in nature. This work develops a dynamic model of phytoplankton-zooplankton community consisting of two equations with discrete time. We use recurrent equations, which allows to describe delay in response naturally. The proposed model takes into account the phytoplankton toxicity and zooplankton response associated with phytoplankton toxicity. We use a discrete analogue of the Verhulst model to describe the dynamics of each of the species in the community under autoregulation processes. We use Holling-II type response function taking into account predator saturation to describe decrease in phytoplankton density due to its consumption by zooplankton. Growth and survival rates of zooplankton also depend on its feeding. Zooplankton mortality, caused by an increase in the toxic substances concentration with high density of zooplankton, is included in the limiting processes. An analytical and numerical study of the model proposed is made. The analysis shows that the stability loss of nontrivial fixed point corresponding to the coexistence of phytoplankton and zooplankton can occur through a cascade of period doubling bifurcations and according to the Neimark-Saker scenario leading to the appearance of quasiperiodic fluctuations as well. The proposed dynamic model of the phytoplankton and zooplankton community allows observing long-period oscillations, which is consistent with the results of field experiments. As well, the model have multistability areas, where a variation in initial conditions with the unchanged values of all model parameters can result in a shift of the current dynamic mode.

Zooplankton effect on the dynamics of competing phytoplankton species

“…The community consists of zooplankton and two phytoplankton species competing for resources: toxic and non-toxic. This work continues our previous study [3], where a simple model of a phytoplankton-zooplankton community consisting of two equations with discrete-time has been proposed.…”

Modeling dynamics of a plankton community using a discrete-time model

“…Such behavior is in good agreement with the hypothesis that blooming species are those able to escape control by microzooplankton through a combination of predation avoidance mechanisms at the beginning of the bloom [2]. Note that such dynamics occurs without artificially complicating the coefficients and the model [3]. Conclusion: Thus, despite its simplicity, the proposed discrete-time model of plankton community dynamics reveals adequate dynamics of interacting species.…”