Hopf Bifurcation Analysis of a Diffusive Nutrient–Phytoplankton Model with Time Delay

: In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.


Introduction
One of the most complex and difficult problems in water pollution treatment is the prevention and control of algal bloom. Due to the complexity of the pollution source and the difficulty factor of material removal, it takes a lot of energy, but it is not very effective. Therefore, scientists search for better methods to prevent and cure algal bloom, especially using mathematical models, in order to find reasonable prevention and cure measures [1][2][3][4][5][6][7]. In addition, many scholars further study the dynamics of the N-P model by considering factors such as time delay and diffusion [8][9][10][11][12]. M. Rehim et al. studied a nutrientplankton-zooplankton system with toxic phytoplankton and three delays, and showed the phenomenon of stability switches [8]. Y. Wang and W. Jiang considered a differential algebraic phytoplankton-zooplankton system with delay and harvesting, and indicated that the toxic liberation delay of phytoplankton may affect the stability of the coexisting equilibrium [10]. In particular, Huppert et al. [13] considered the following N-P model where N is the nutrient level and P is the density of phytoplankton. a denotes the constant external nutrient inflow. b represents the maximal nutrient uptake rate. c represents the maximal conversion rate of nutrients into phytoplankton. d stands for the per capita mortality rate of phytoplankton. e denotes the per capita loss rate of nutrients. Relevant research work has analyzed the reasonable, deterministic, and empirical relationship between the abundance of toxin-producing phytoplankton and the diversity of plankton communities with large amounts of plankton but no toxins (called nontoxic plankton plants, NTP) [14]. In the case of toxic substances released by toxic phytoplankton (TPP), a simple model of vegetative phytoplankton was proposed and analyzed to understand the dynamic changes of the phenomenon of the seasonal mass reproductive cycle. The presence of chemical and toxic substances helps explain this phenomenon [15][16][17]. In [18], Chakraborty et al. considered the effect of toxins produced by toxic phytoplankton on the death of nontoxic phytoplankton, and produced the following equation where θ is the release rate of toxic chemicals by the TPP population, and µ denotes the half-saturation constant.
Since the spatial distribution of nutrients and phytoplankton is inhomogeneous, there is diffusion. In addition, there is a time delay in the conversion from nutrients to phytoplankton. So, we incorporate reaction diffusion and time delay into the model (2), that is where d 1 and d 2 are diffusion coefficients for N and P, respectively. is the Laplace operator. This is based on the assumption that the prey and predator are not stationary and will spread randomly . τ is the time delay that occurs for nutrients to be converted to phytoplankton. For analysis convenience, we have denoted The corresponding problem has the following form The content of the paper is arranged as follows. In Section 2, we study the stability and the existence of the Hopf bifurcation. In Section 3, we analyze the property of Hopf bifurcation. In Section 4, we provide a numerical simulation to verify the previous conclusions. Finally, we conclude this paper.

Stability Analysis
In [18], Chakraborty et al. studied the existence of equilibria. We cite the following result. The equilibrium points satisfy the following equation It can be calculated that trivial equilibrium 1 s , 0 and interior equilibrium (N * , P * ), where N * = 1 hP * +s , and P * is a root of the equation We provide the result from [18] as follows.
Lemma 1. The existence of a positive equilibrium for the model (4) can be divided into the following cases.
Proof. Suppose d 1 = d 2 = 0, τ = 0, and hypothesis (11) hold, we can obtain T 0 < 0, D 0 > 0, so the real part of the roots of the characteristic equation is negative, then the equilibrium (N * , P * ) is locally asymptotically stable.
, and It is easy to verify that a − < d 1 d 2 a 0 < a + under the hypothesis (11). (11) hold. For the system (4), we have the following conclusion.
a 0 , then the equilibrium (N * , P * ) is locally asymptotically stable.
a 0 , then the equilibrium (N * , P * ) is locally asymptotically stable.
Proof. We can obtain T n < 0 and D n > 0 for a ≥ d 1 d 2 a 0 . It can be concluded that all the characteristic roots have a negative real part. Then, the equilibrium (N * , P * ) is locally asymptotically stable (so, statement (1) is true). In the same way, statements (1)-(3) are also correct. Suppose the conditions in statement (4) are true, then at least there is a positive real part of eigenvalue root. Then, the equilibrium (N * , P * ) is Turing unstable.
which leads to Let z = ω 2 , Equation (15) is By direct computation, we have Proof. The roots of Equation (16) are It is easy to verify that z + n > 0 if and only if n ∈ M, and z − n is always negative or a non real number.
Suppose one of the conditions (1)-(3) in Theorem 2 and hypothesis (11) hold, from Equation (14), we can obtain For n ∈ M, then Equation (8) has a pair of purely imaginary roots ± iω n at τ j n , j ∈ N 0 , Proof. From (8), we can obtain Theorem 3. For system (4), assume one of the conditions (1)-(3) in Theorem 2 and hypothesis (11) hold, then we have the following conclusion.
Then, we can obtaiṅ That is, Using the definition of Aτ and (46), we have that for −1 ≤ θ < 0 That is, Similarly, we have That is, Then,

Conclusions
Diffusion and time delay was incorporated into a nutrient-phytoplankton model. The instability and Hopf bifurcation induced by the time delay was studied. Through the central manifold theory and normal form method, some parameters were given to determine the property of bifurcating periodic solutions. The results indicate diffusion may induce Turing unstable. The release rate β of toxic chemicals by the TPP population has a stabilizing and destabilizing effect on the stability of the positive equilibrium. In addition, the time delay can also affect the stability of the positive equilibrium, and it can induce periodic oscillation of prey and predator population density.