Original articles
Bifurcations and multistability in a food chain model with nanoparticles

https://doi.org/10.1016/j.matcom.2021.06.017Get rights and content

Highlights

  • We investigate the effect of nanoparticles in a three-species aquatic model.

  • Coexisting attractors of different kinds are observed in the system.

  • Shrimp-shaped periodic structures are found in the chaotic regime.

  • Period-doubling and period-bubbling transition to chaos are detected in the system.

  • Depletion rate can be used as a control strategy to prevent population extinction.

Abstract

In the present paper, we report the effect of nanoparticles in a three-species aquatic food chain model. It is assumed that due to the unavoidable interactions with nanoparticles, the growth rate of basal prey reduces. Fascinating dynamical scenarios in the model are explored in different bi-parametric spaces. We show that the model exhibits coexisting attractors of different kinds (periodic–periodic/periodic–chaotic) in some regions of the bi-parametric space. We also discover well-organized shrimp-shaped periodic structures in the chaotic regime, indicating the coexistence of order (regular pattern) and chaos. Further, a period-bubbling transition to chaos is found near the inner boundaries of the shrimp-shaped structure. It is also observed that managing the depletion rate of the nanoparticles can be used as a control strategy to prevent population extinction in an aquatic food chain system.

Introduction

Nanoparticles (NPs) are particles existing on a nanometer scale ranging between 1 and 100 nanometers in all three dimensions with a surrounding interfacial layer. Scientific research on NPs is gaining more and more attention due to their charming properties and potential applications in different fields such as Biology, Physics, Chemistry, and Nanotechnology [23]. NPs are everywhere in our day-to-day products like cosmetics, garments, paints, batteries, sunscreens, fertilizers, processed foods, and many others. Consequently, a large amount of NPs are being discharged into the water bodies, and mostly the marine system through direct and indirect disposal during the time of production and usage. Although NPs are very useful, and in some cases, there are no viable alternatives to replace them, they pose possible dangers in medical applications as well as to the surrounding environment. It occurs mostly due to their high surface-to-volume ratio. Experimental studies have shown that silver, nickel, cobalt, and titanium dioxide (TiO2) NPs are harmful to aquatic creatures and create toxic effects on fish, phytoplankton, etc. [5], [9], [19]. Moreover, TiO2 NPs have a considerable effect on the growth of various phytoplankton species [33]. A recent study [6] based on the impact of TiO2 and cerium dioxide (CeO2) NPs on a marine diatom, revealed that these NPs can change the growth, photosynthetic activity, and damage cellular components of phaeodactylum tricornutum. Another interesting study [14] based on the influence of engineered NPs on harmful algal blooms (HABs) species by taking into account the effects of TiO2, ZnO and Al2O3 NPs showed that HAB species cell growth is mostly affected through ZnO NPs due to its toxic effect. Experimental observations have also revealed that phytoplankton populations are highly vulnerable to the exposure of TiO2 nanoparticles mostly due to their highly photoactive nature causing considerable damage even under natural levels of ultraviolet radiation [20]. Miler et al. [21] experimented with ZnO, CeO, CuO, and AgO NPs on marine phytoplankton and found that all these ENPs (engineered nanoparticles) have a detrimental effect on the population growth. Sendra et al. [37] studied the effect of CeO2 NPs on phytoplankton and they elucidated the fact that positively-charged CeO2 NPs alter cell complexity more and increase cytotoxicity. It is to be noted that although NPs can be formed through various natural phenomena such as volcanic eruption, woodland fire, clay minerals, and desert dust storms [38], but the transmission of NPs due to increasing human activities may be much more severe for the environment.

In a single species population model, Verhulst [43] modified Malthus’s model [17] in a more natural setup. Lotka–Volterra (LV) predator–prey model was the first interacting species model [16], [44], and after the development of this model, many researchers have modified and extended it by considering various types of functional responses. Among the significant components of predator–prey modeling, functional response is a key element representing the predator consumption rate of the prey species (usually modeled as proportional to the rate of feeding). Functional responses are formulated by considering many ecological factors such as the density of prey species, the efficiency with which predators can search out and kill the prey, the time required for handling the captured prey, and competition among predators. Commonly used functional responses are (i) Lotka–Volterra type, which is linearly increasing, (ii) Holling type-II, which is concave increasing, (iii) Holling type-III, which is sigmoid increasing, and (iv) Holling type-IV, which is nonmonotonic. Among them, Holling type-II functional response has received the lion’s share of attention. Many authors experimentally inspected and provided a conceptual understanding of the applicability of Holling type-II response in predator–prey interactions [8], [22]. On the other hand, several researchers theoretically explored the dynamics of various predator–prey systems with Holling type-II functional response [3], [10], [11].

In population ecology, many studies have been carried out to explore chaos through interactive models in different contexts [10], [24], [25]. Hastings and Powell [10] studied a three-species food chain model in a continuous-time setup. They considered logistic growth in the prey population and Holling type-II functional response for the predator–prey interaction. This work is renowned for being the first food chain model exhibiting chaotic dynamics. After this pioneering work of Hastings and Powell [10], exploring chaos in ecological systems has received substantial attention. However, in the real world population chaos is rare. A possible explanation for such would be the presence of several ecological factors, which make the system dynamics regular and thus unable to exhibit chaos. To date, several ecological phenomena have been documented to have the ability to control chaos in three-species food chain models. Some of those ecological phenomena are: refuge [7], toxic inhibition [2], disease in prey [4], additional food [34], predator switching [26], migration [27], [28], and fear effect [29].

Many researchers have employed the Hastings–Powell (HP) model on populations living in aquatic systems. For example, Samanta et al. [36] considered phytoplankton, zooplankton, and fish as prey, middle predator, and top predator, respectively, and studied the impact of cascading migration on the three species food chain system. As the phytoplankton–zooplankton–fish interaction forms the fundamental basis of marine ecosystems, many considered this three-species food chain system [35], [41] as their base model. Phytoplankton, the primary and most important producer of the marine food web system, plays a major role in the global carbon cycle, and nanoparticles have a great impact on their growth. As a result, their impact on the aquatic food chain system is an interesting domain of research. Here, we first consider the Hastings–Powell (HP) model in the aquatic environment with phytoplankton, zooplankton, and fish as basal prey, middle predator, and top predator, respectively. There is enough existing literature on the fact that the phytoplankton is affected by the presence of NPs [18], [39]. In the present paper, we introduce NPs to the three-species food chain system and assume that NPs cause a reduction in the growth rate of phytoplankton. The rest of the paper is organized as follows. In Section 2, we formulate the mathematical model based on the HP model with NPs. Preliminary mathematical analyses such as local stability, Hopf bifurcation of the system around the interior equilibrium point are investigated in Section 3. We explore the rich dynamics of our modeled system with the help of extensive numerical simulations in Section 4 and finally, end the paper with a brief conclusion.

Section snippets

Mathematical model

In 1991, Hastings and Powell [10] proposed and analyzed a three-species continuous-time food chain model with Holling type-II functional response. This model is famous for exhibiting chaotic dynamics. The Hastings–Powell (HP) model is governed by the following equations: dxdt=x1xa1xy1+b1x,dydt=a1xy1+b1xa2yz1+b2yd1y,dzdt=a2yz1+b2yd2z,where x(t), y(t) and z(t) are the densities of prey, middle predator and top predator populations at time t, respectively. ai, bi and di (i=1,2) are the

Mathematical preliminaries

In this section, we carry out some preliminary mathematical analysis of the system (2).

Numerical simulations

To explore possible rich dynamics of our system (2), we took the help of MATLAB to perform extensive numerical simulations. For the correctness of our results, we tested our findings with several standard ODE solvers. Our very basic assumption is that, due to the interactions with NPs, basal prey’s growth rate drops by some noticeable margin. We choose the HP model [10] as the base with an ambition of taking the advantage of it being well-known as well as well-studied by ecologists. To fully

Conclusion

In the present paper, we have investigated the effects of nanoparticles on a tri-trophic food chain model in an aquatic system (phytoplankton–zooplankton–fish). We assumed that the basal prey’s interactions with nanoparticles reduce its growth rate. We observed that an adequate increase in basal prey’s exposure to nanoparticles can stabilize the once chaotic system through multiple time switching between higher-periodic and chaotic attractors. We also observed that a further increase in

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.

Acknowledgments

We are thankful to the anonymous reviewers and the handling editor for their valuable comments and suggestions which helped us to improve the quality of the manuscript.

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    All authors contributed equally to this work.

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