Elsevier

Ecological Complexity

Volume 46, March 2021, 100927
Ecological Complexity

A nonautonomous model for the effects of refuge and additional food on the dynamics of phytoplankton-zooplankton system

https://doi.org/10.1016/j.ecocom.2021.100927Get rights and content

Highlights

  • We consider interacting dynamics of phytoplankton-zooplankton with refuge and toxins released by phytoplankton.

  • We incorporate seasonal variations in the level of nutrients, refuge and toxin liberation by phytoplankton.

  • Our analysis shows high toxicity and refuge cause extinction of zooplankton.

  • Additional food supports the survival of zooplankton population and controls the phytoplankton bloom.

  • We observe that seasonality in nutrients and toxins released by phytoplankton generate higher periodic solutions.

Abstract

In this paper, a mathematical model for the interacting dynamics of phytoplankton-zooplankton is proposed. The phytoplankton have the ability to take refuge and release toxins to avoid over predation by zooplankton. The zooplankton are provided some additional food to persist in the system. The phytoplankton are assumed to be affected directly by external toxic substances whereas zooplankton are affected indirectly by feeding on the affected phytoplankton. We incorporate seasonal variations in the model, assuming the level of nutrients, refuge and the rate of toxins released by phytoplankton as functions of time. Our results show that when high toxicity and refuge cause extinction of zooplankton, providing additional food supports the survival of zooplankton population and controls the phytoplankton population. Prey refuge and additional food have stabilizing effects on the system; higher values of the former results in extinction of zooplankton whereas phytoplankton disappear for larger values of the latter. Seasonality in nutrients level and toxins released by phytoplankton generate higher periodic solutions while time-dependent refuge of phytoplankton causes the occurrence of a period-three solution. The possibility of finding additional food for zooplankton may push back the ecosystem to a simple stable state from a complex dynamics.

Introduction

Planktonic species experience large changes in population size over a short period of time, which enables ecologists relatively easily to observe population-level responses to deviation in several biotic and abiotic factors (Yoshida, 2005). For the pelagic ecosystem, phytoplankton variability plays a critical role to the survival and recruitment of fish and other invertebrate populations (Platt et al., 2003). Some species of phytoplankton are known to produce toxic substances in the aquatic environment impacting economic, ecological and human health (Hallam et al., 1983). For example, Noctiluca scintillans (Macartney) Kofoid & Swezy, 1921, Phaeocystis globosa Scherffel, 1900, Alexandrium catenella (Whedon & Kofoid) Balech, 1985, Dinophysis acuminata Claparède & Lachmann, 1859 etc., are harmful to planktonic organisms (Hallegraeff, 1993, Nielsen, Kiørboe, Bjørnsen, 1990). The ability of phytoplankton to release toxins has the capacity to mitigate the planktonic blooms by reducing the predation pressure of zooplankton (Chattopadhyay et al., 2002). The spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton has been explored by Chakraborty et al. (2015). Toxicity causes patterns, stripes and spots, in the distribution of nutrient and phytoplankton. These lead to spatiotemporal oscillations if a threshold in toxicity is attained. Several environmental and physical factors influence the phytoplankton release of toxic chemicals, whose influence on other aquatic species may vary significantly (Chakraborty, Roy, Chattopadhyay, 2008, Chatterjee, Pal, Chatterjee, 2011, Johansson, Graneli, 1999, Shilo, 1971). Chakraborty et al. (2007) considered periodic toxin release by phytoplankton to study seasonally recurring blooms. Indeed, phenomena such as tides, alternation of day and night, seasonal cycles all induce temporal inhomogeneity in the aquatic nutrient concentration.

Biodiversity and ecosystem functions have been altered by intensive anthropogenic activities (Das et al., 2009). Industrial revolution, agricultural and urban sprawl etc., have severely impacted on ecological systems. Toxins alter diatoms cell structures, metabolism and community composition and hinder phytoplankton growth (Labille, Brant, 2010, Miao, Schwehr, Xu, et al., 2009, Miao, Zhang, Luo, et al., 2010, Miller, Bennett, Keller, Pease, Lenihan, 2012). Algae accumulate heavy metals and other potential contaminants by absorption or by adsorption to the cell wall. The increase in the (heavy) metal content of acidifying environment can affect the phytoplankton biomass. Johnson et al. (1970) concluded that high acidity lowers the level of available inorganic carbon, resulting in a reduction of primary production of phytoplankton. The Hill reaction of photosynthesis is affected by chemical pollutants, which results in growth inhibition of phytoplankton (Walsh, 1972). Slawyk et al. (1976) documented that minimized light penetration due to the presence of suspended industrial effluents caused a reduction in the rate of uptake of nitrate and ammonia by marine phytoplankton. Anthropogenic toxicants can, therefore, affect the process of photosynthesis and other related aspects of energy utilization and assimilation, and thus influence species abundance and induce mortality in the plankton community. Phytoplankton may be a vital link in the transfer of anthropogenic contaminants from water to zooplankton and other pelagic consumers. Pollutants can have direct or indirect toxic effects on zooplankton, including lethal or sublethal effects (Walsh, 1978). Varying sensitivities to anthropogenic toxicants among zooplankton species could cause changes in community structure by affecting variables such as rate of increase, mortality, and population density. Sprules (1975) documented that major changes in composition of crustacean zooplankton communities of industrially acidified lakes were related to pH level. Industrial acidification caused significant decline in the number of species and changes in species dominance, both of which were affected as pH level dropped from 7.0 to 3.8. Almeda et al. (2013) conducted ship-, shore- and laboratory-based crude oil exposure experiments to investigate the impacts of crude oil on mesozooplankton survival and showed that zooplankton mortality ranged from 12% to 96% depending on crude oil concentrations and station. Many other heavy metals act as chelating agents, which also affect various enzymes, cofactors etc., which hinder the biochemical process, ultimately impairing important life processes or causing substantial mortality in zooplankton. Recent mathematical studies explored the growth suppression mechanism of environmental toxins (Mandal, Tiwari, Samanta, Venturino, Pal, 2020, Panja, Mondal, Jana, 2017, Rana, Samanta, Bhattacharya, et al., 2015), showing that environmental toxins destabilize the system leading to persistent oscillations. A higher contact rate between environmental toxins and phytoplankton dampens the phytoplankton and zooplankton equilibria, while their removal from water restores the stable coexistence of planktons. The indirect effect of toxins on zooplankton by feeding on contaminated phytoplankton has been investigated by Chakraborty and Das (2015).

In a lake ecosystem, the refuge for phytoplankton is commonly obtained by benthic sediments, where eggs can be produced and temporary escape from the predation of zooplankton is found (Schindler and Scheuerell, 2002), but also water stratification may constitute a phytoplankton’s temporary refuge (Wiles et al., 2006). The empirical study by Mullin et al. (1975) suggested that refuge can prevent prey extinction by stabilizing the community equilibrium reducing the oscillatory tendency of predator-prey interactions. In the toxin-producing phytoplankton-zooplankton model by Li et al. (2017), a constant refuge capacity and toxin may act as biological controls for the onset or termination of algal blooms. However, the effectiveness of refuges for phytoplankton depends on various factors such as temperature, radiation and winds (Talling, 1966), implying that a time dependent refuge modeling is more realistic. Moreover, providing additional food to predators is widely recognized as one of the best techniques for biological conservation programs (Srinivasu and Prasad, 2010). Suspended organic particles, detritus, bacteria, etc., act as alternative food sources for zooplankton population (Chakraborty and Chattopadhyay, 2008). In a very recent study, Mandal et al. (2020) assumed zooplankton’s logistic growth, i.e., treating them as generalist predators, feeding not just on phytoplankton, but also on other food sources. This results in killing out the oscillations and restoring the system to a stable state.

The aims of this study are two folds: first we see how the presence of additional food supports the survival of zooplankton in the system where phytoplankton took refuge and release toxins to avoid predation. The effects of environmental toxins are also studied. Secondly, we investigate how the seasonal variations of nutrients concentration, refuge, and toxin released by phytoplankton regulate the dynamics of the system. These ecological questions will be answered by model analysis and numerical simulations for different parameter setups.

The rest of the paper is organized as follows: in the next section, we propose our nonautonomous model; for the corresponding autonomous model, equilibria and their local stability behaviors are summarized. The nonautonomous system is analyzed in Section 3; boundedness, permanence, non-permanence, existence of periodic solution, and global attractivity of boundary as well as positive periodic solutions are discussed. In Section 4, numerical simulations are conducted, which will help us to have an intuitionistic view of the effects of key parameters on the occurrence and termination of phytoplankton bloom. The paper ends with conclusions and discussions.

Section snippets

The seasonally-forced model

Several mathematical and ecological researches have documented complex dynamical behaviors of multi-species food chain models with variable lengths (Kot, 2001, Pastor, 2011). Our model for the study of possible effects of anthropogenic toxicants on planktonic ecosystem is composed of phytoplankton and their grazer zooplankton, all homogeneously distributed over space. The growth of phytoplankton fully depends on the level of nutrients whereas the growth of zooplankton depends on phytoplankton

Mathematical analysis of system (2.1)

Let g(t) be a generic continuous periodic function with period ω. Denote bygu=suptRg(t),gl=inftRg(t),g¯=1ω0ωg(t)dt.

Numerical results

Here, we report the simulations performed to investigate the behaviors of systems (2.1) and (2.2) using the Matlab variable step Runge-Kutta solver ode45. Unless otherwise mentioned, the set of parameter values will be the same as in Table 2.

Discussion

In this paper, we proposed a mathematical model for the dynamical interactions of phytoplankton-zooplankton. The nutrients level in the system is assumed to be constant. The phytoplankton population is assumed to take refuge to avoid predation of zooplankton; also release toxin products to reduce over predation of zooplankton. Toxicants released into the aquatic systems from industries affect both phytoplankton as well as zooplankton. The zooplankton population may find some additional food. We

CRediT authorship contribution statement

Arindam Mandal: Formal analysis, Software. Pankaj Kumar Tiwari: Writing - original draft, Software. Samares Pal: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank the associate editor and anonymous reviewers for valuable comments, which contributed to the improvement in the presentation of the paper. The research work of Arindam Mandal is supported by University Grants Commission, Government of India, New Delhi, in the form of Senior Research Fellowship (Ref.No:19/06/2016(i)EU-V). The research of Samares Pal is partially supported by Science and Engineering Research Board, Government of India (Grant No. CRG/2019/003248).

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