Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Kejun Zhuang,Dezhi Liu,Gao Jia

doi:10.1155/2019/4590915

Abstract

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

Highlights

Oceans have a major role in the global carbon cycling and so directly impact the pace and extent of climate change [1]
The results indicate that spatial diffusion has a vital role in the spatiotemporal dynamics of the planktonic model and spatial pattern may occur
By the Hopf bifurcation theory [25], we know that σ2 determines the bifurcation direction: if σ2 > 0 (σ2 < 0), the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for τ > τ0 (τ < τ0); β2 determines the stability of bifurcating periodic solutions: if β2 > 0 (β2 < 0), the periodic solutions are stable; T2 determines the monotonicity of the period of periodic solutions: if T2 > 0 (T2 < 0), the period increases

Summary

Introduction

Oceans have a major role in the global carbon cycling and so directly impact the pace and extent of climate change [1]. In model (2), all the coefficients are positive constants, r and s are the growth rates of two phytoplankton populations, respectively, K is the carrying capacity, α and β are the maximum zooplankton ingestion rate and maximum zooplankton conversion rate, respectively, μ is the death rate of zooplankton, θ is the rate of toxin liberation by toxic phytoplankton, and θ1 is the specific predation rate of zooplankton population on toxic phytoplankton. (x, t) ∈ (0, L) × [−τ, 0] , where P(x, t), T(x, t), and Z(x, t) denote the densities of nontoxic phytoplankton, toxin-producing phytoplankton, and zooplankton at location x and time t, respectively, Δ is the usual Laplace operator, L denotes the depth of the water column, and the homogeneous Neumann boundary condition means that no plankton species is entering or leaving the column at the top or the bottom.

Long-Time Behavior

Steady State

Hopf Bifurcation Induced by Time Delay

Stability and Direction of the Bifurcation

Numerical Simulation

Time t