Stability and bifurcation analysis of a nutrient‐phytoplankton model with time delay

Abstract

We proposed a nutrient‐phytoplankton interaction model with a discrete and distributed time delay to provide a better understanding of phytoplankton growth dynamics and nutrient‐phytoplankton oscillations induced by delay. Standard linear analysis indicated that delay can induce instability of a positive equilibrium via Hopf bifurcation. We derived the conditions guaranteeing the existence of Hopf bifurcation and tracked its direction and the stability of the bifurcating periodic solutions. We also obtained the sufficient conditions for the global asymptotic stability of the unique positive steady state. Numerical analysis in the fully nonlinear regime showed that the stability of the positive equilibrium is sensitive to changes in delay values under select conditions. Numerical results were consistent with results predicted by linear analysis.

Dynamics of a stochastic phytoplankton–toxic phytoplankton–zooplankton system under regime switching

“…Due to the complexity and openness of real aquatic ecosystems, establishing mathematical models is now a classical way to study the planktonic blooms, 15 which can provide quantitative insights into the dynamic mechanisms of changes in plankton populations. In recent years, many deterministic mathematical models for plankton dynamics, such as delayed nutrient-phytoplankton models, 16,17 a diffusive nutrient-toxic phytoplankton model, 18 viral infection nutrient-phytoplankton models, 19,20 a phytoplankton-toxin-producing phytoplankton-zooplankton model, 13 and so on, have been developed and studied extensively, and many interesting results have been shown. However, plankton populations in real aquatic environments often fluctuate unpredictably because of the unpredictability of environmental stochasticity, and these deterministic models do not capture the random environmental fluctuations which is an important feature of aquatic ecosystems.…”

Bifurcation and control of disease spreading networks model with two delays

“…Therefore, the study of differential equations with delays are more general than ordinary differential equations [17][18][19][20][21][22][23][24]. For example, a nutrient-phytoplankton interaction model with a discrete and distributed time delay was considered in Guo et al [18]. Zhang et al [20] studied the dynamic characteristics of fractional-order three-dimensional nonlinear financial system with delay.…”

“…Compared with general differential equations, equation with delays are more complex and more accurate to describe the state of dynamic systems. Therefore, the study of differential equations with delays are more general than ordinary differential equations [17][18][19][20][21][22][23][24]. For example, a nutrient-phytoplankton interaction model with a discrete and distributed time delay was considered in Guo et al [18].…”

Nonlinear dynamics of plankton ecosystem with impulsive control and environmental fluctuations

“…Actually, many mathematical models have been constructed to study the dynamical behaviors of plankton since the pioneering work of Riley et al (1949), and many physical and biological processes underlying the mechanisms of plankton dynamics in the aquatic environments have been investigated (Huppert et al 2002;Dai et al, 2016;Caperon, 1969;Guo et al, 2019;Lin et al, 2005;Zhao et al, 2020). For example, in order to study how the nutrient affects the dynamics of phytoplankton blooms, Huppert et al (2002) presented a simple nutrient-phytoplankton model and identified an important threshold effect that a bloom will only be triggered when nutrients exceed a certain defined level using mathematical model analysis.…”

“…Due to the complexity and openness of real aquatic ecosystems, establishing mathematical models is now a classical way to study the planktonic blooms (Truscott and Brindley, 1994), which can provide quantitative insights into the dynamic mechanisms of changes in plankton populations. In recent years, many deterministic mathematical models for plankton dynamics, such as delayed nutrient-phytoplankton models (Dai et al, 2016;Guo et al, 2019), a diffusive nutrient-toxic phytoplankton model (Chakraborty et al, 2015), viral infection nutrient-phytoplankton models (Li and Gao, 2016;Chattopadhyay et al, 2003), a phytoplankton-toxin producing phytoplankton-zooplankton model (Chattopadhyay et al, 2004), and so on, have been developed and studied extensively, and many interesting results have been shown.…”