Complex dynamic of plankton–fish interaction with quadratic harvesting and time delay

Model. Earth Syst. Environ. (2016)2:204 DOI 10.1007/s40808-016-0248-x ORIGINAL ARTICLE Complex dynamic of plankton–fish interaction with quadratic harvesting and time delay Amit Sharma1 • Anuj Kumar Sharma 2 • Kulbhushan Agnihotri3 Received: 29 September 2016 / Accepted: 31 October 2016 Ó Springer International Publishing Switzerland 2016 Abstract The present paper deals with a food chain model consisting of three species phytoplankton, zooplankton and fish. We have divided the present paper into two parts. In the first part, we have assumed that the fish population is harvested using a non-linear harvesting function. Considering this rate of harvesting ‘E’ as a control parameter, we have estimated different ranges of harvesting parameter for maintaining the sustainability in the plankton ecosystem. Moreover, the bifurcation analysis of the system is carried out using normal form theorem by taking ‘E’ as bifurcation parameter. In the second part, a digestion delay corresponding to zooPlankton–fish interaction is introduced for more realistic consideration of the real world problem. Taking harvesting parameter in the stability range, the effect of time delay on the given system is investigated. This research demonstrate that for a certain range of delay, system enters into the excited state with the existence of stability switches which seems new findings for the Plankton–fish system. Explicit results are derived for stability and direction of the bifurcating periodic solution by using normal form theory and center manifold arguments. To validate our analytical findings numerical simulations are also executed. Keywords Plankton  Fish  Quadratic harvesting  Time delay  Stability switches  Normal form  Center manifold theorem & Amit Sharma 1 DAV Institute of Engineering and Technology, Jalandhar, Punjab, India 2 LRDAV College, Jagraon, Punjab, India 3 SBSSTC, Ferozpur, Punjab, India Introduction A major concern in population ecology is to understand how a population of a given species influences the dynamics of populations of other species. The dynamic relationship between predator and their prey has long been and continue to be one of the dominant themes in mathematical ecology due to its universal existence and importance (Berryman and Millstein 1989). The pioneering work of May (1976) has established some mathematical models based on certain ecological principles to explore the complexity of the ecological system. The research of the last two decades have demonstrated that very complex dynamics can arise in three or more species food chain models (Hastings and Powell 1991; Klebanoff and Hastings 1993; Rai and Upadhyay 2004; Gakkhar and Naji 2005; Upadhyay and Chattopadhyay 2005) including quasi-periodic or chaos. A great number of theoretical studies indicates that, indeed, plankton systems are, in principle, capable to generate their own chaos. Preypredator models (e.g., phytoplankton-zooplankton interaction) in seasonally varying environments have been proved to be chaotic (Inoue and Kamifukumoto 1984; Klebanoff and Hastings 1994; Kuznetsov and Rinaldi 1996). Chaotic behavior in tritrophic food chain interactions have been shown in Gilpin (1979), Hogeweg and Hesper (1978), Scheffer (1991). It is well established that toxin has great impact on phytoplankton-zooplankton interaction and can be used as a mechanism of controlling complexity (bloom) of the plankton ecosystem (Pal et al. 2007; Chakarborty et al. 2008; Sarkar and Chattopadhyay 2003; Mukhopadhyay and Bhattacharyya 2006; Hansen 1995; Ives 1987; Buskey and Hyatt 1995). Recently, Upadhyay and Chattopadhyay (2005), Upadhyay et al. (2007), Upadhyay and Rao (2009) have studied a series of mathematical models 123 204 Page 2 of 17 representing the prey(TPP)-specialist predator(Zooplankton)-generalist predator(Fish) interaction in the context of deterministic chaos in the plankton system. They have studied the role of toxin as a control parameter in stabilizing the plankton system by using different response functions for fish grazing. Further, the introduction of strictly planktivorous fish to lakes can alter plankton communities via cascading interactions in food webs, where strong fish predation on zooplankton leads to reduction of their grazing pressure on algae. Many limnological studies have focused on this dramatic impact of fish on plankton communities (Carpenter et al. 1987; Leavitt and Findlay 1994; Pace et al. 1999). Furthermore, enhancement of piscivorous fish by stocking and catch restrictions has been attempted in several eutrophic or hypertrophic waters, and has resulted in the elimination of excess algae (Carpenter et al. 1987; Rinaldi and Solidoro 1998; Benndorf et al. 1984; McQueen et al. 1989; Lazzaro 1987). Recently, research carried by authors in Strock et al. (2013) reveals that the introduction of white perch in eutrophic lake reduces the algal pigments concentration and results high label of stability in trophic interaction due to the cascading effects of white perch. Attayde et al. (2010) in their model, also supports the idea that omnivory decreases the amplitude of limit cycles and increases the persistence in plankton dynamics. Keeping in view the above findings, here, we have considered a tri-trophic food chain model of Plankton–fish interaction. We have assumed that the generalist predator (fish) is harvested using a nonlinear harvesting function (Feng 2014). Our major concern in this research to determine the impact of non-linear harvesting on the Plankton– fish interaction by predicting different ranges of harvesting parameter. So far many researcher (Upadhyay et al. 2007; Upadhyay and Rao 2009, references there in) have studied that toxicated phytoplankton may be used as controller and can stabilize the Plankton–fish dynamic. In this paper, we have introduced quadratic harvesting of the fish population and proposed different ranges of harvesting for regulating agencies so that they can utilize it for future harvesting along with the stabilization of the ecosystem. The organization of our paper is as follows: In Sect. 2, we have developed a mathematical model followed by its properties in Sect. 2.1. The stability and bifurcation analysis of the given model system with rate of harvesting as a bifurcation parameter is presented in Sect. 3. In Sect. 4, we have assumed that there is a gestation delay in fish population and considering this time delay as a bifurcation parameter, we have discussed its possible impact on the Plankton–fish dynamic. The stability and bifurcation 123 Model. Earth Syst. Environ. (2016)2:204 properties are provided in Sect. 5. The justification of our analytical findings by numerical simulation and concluding remarks are given in Sects. 6 and 6.2. The mathematical model Let p(t) be population density of the toxin producing phytoplankton (prey) which is predated by individuals of specialist predator zooplankton of population density z(t) a (...truncated)