Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity

Iran J Sci Technol Trans Sci https://doi.org/10.1007/s40995-020-01042-8 (0123456789().,-volV)(0123456789(). ,- volV) RESEARCH PAPER Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity Nilesh Kumar Thakur1 • Smriti Chandra Srivastava1 • Archana Ojha1 Received: 7 September 2020 / Accepted: 23 November 2020 Ó Shiraz University 2021 Abstract In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment. Keywords Plankton  Toxicity  Local stability  Time delay  Hopf-bifurcation  Chaos 1 Introduction Viral infection in planktonic species affects the bloom dynamics and causes behavioral as well as other changes in the aquatic and marine systems. The capability of regulating the plankton dynamics is still far from understanding. Algal viruses perform a remarkable component in the evolutionary driving force of the aquatic system and responsible for biogeochemical cycles across all the microbial communities. It effectively accounts for the & Nilesh Kumar Thakur Smriti Chandra Srivastava Archana Ojha 1 Department of Mathematics, National Institute of Technology Raipur, Raipur, CG 492010, India idealization of mathematical biology that handles the new ecological and epidemiological challenges. Viruses have notable prospects as mortality agents for phytoplankton and play a dominant role in extinction and survival behavior among all the planktonic species. Several new developments concerned with the dynamical and behavioral complexity of the prey–predator system have been addressed in the area of ecology and epidemiology (Anderson and May 1986; Das and Chattopadhyay 2015; Thakur and Ojha 2020a). Mathematical models are facilitated to demonstrate the qualitative functioning of the prey–predator system and help to examine the long-term relationship among interacting species of the ecosystem. Many authors assumed only one infected population in their model system, i.e., either prey or predator population is infected, whereas others assumed as both the populations are infected (Gao et al. 2020c; Goyal et al. 2020; Singh et al. 2018; Atangana 2017, 2020, 2018). A number of new observations by using fractional derivative operators have 123 Iran J Sci Technol Trans Sci been addressed by Atangana (2017), Atangana (2018), Atangana (2020), Cattani (2018), Gao et al. (2020d), Gao et al. (2020b), İlhan and Kıymaz (2020) and Cattani and Pierro (2013). Eco-epidemiological models are also considered to describe the coronavirus pandemic that describes the real phenomena (Gao et al. 2020a, e). The pioneering work of Kermack and McKendrick (1927) has established a classical SIR model (susceptible, infectious, recovered model) with the idea of plankton disease to study how the population is influenced by infection. Beltrami and Carroll (1994) developed an eco-epidemiological model based on prey–predator in which the prey population seems to be infected by viral contamination and forms an infected group. They found that the system has been destabilized by a minute amount of infection agents otherwise stable tropic configuration noticed. Gakkhar and Negi (2006) studied the role of viral infection and toxic substances on the plankton system and concluded that higher infection rates controlling the plankton blooms. Dhar and Sharma (2010) presented a phytoplankton dynamics along with viral infection and incubation class and found that the absence of incubation class makes the phytoplankton system unstable, whereas the presence of incubation class in the form of delay makes the phytoplankton system stable. A good number of reviews are available on prey– predator dynamics with disease and infection and also their possible ecological and biological impact specified in Biswas et al. (2010), Saifuddin et al. (2016) and Zhao and Jiang (2014). Upadhyay et al. (2008) proposed an ecoepidemiological model based on the Salton sea which contains an infected fish population and tried to explain all the possible ways to the existence of chaos in a detrimental wetland ecosystem. Das et al. (2016) focused on a phytoplankton–zooplankton model system with virally infected species and studied the essential features of plankton dynamics by taking two important parameters, i.e., mortality of phytoplankton and viral infection of zooplankton. Auger et al. (2009) modeled a predator–prey system by using simple Lotka–Volterra equations with disease-affected predator population is considered. Tannoia et al. (2012) discussed a system for transmissible diseases which is disseminating among predators and found that the persistence of oscillation behavior for the system. Bairagi et al. (2007) investigated a comparison-based study of a model with an infected prey–predator population where the predator response function is governed by three different responses function. They observed that when prey is affected with a disease, then species coexistence is not possible whereas some diverse outcomes yield. Later, many investigations have been made by numerous authors to study the eco-epidemiological model in different ecological scenarios where the population is influenced by external toxicity, external disease transmission, prey 123 refuge, Allee effect, etc. (Biswas et al. 2016; Hethcote et al. 2004; Kumar et al. 2019; Venturino 2002). Among them, the toxin-producing phytoplankton–zooplankton system has played a prominent role in marine as well as freshwater ecosystems. (...truncated)