Global dynamics of a di ff usive phytoplankton-zooplankton model with toxic substances e ff ect and delay

: This paper examines a di ff usive toxic-producing plankton system with delay. We first show the global attractivity of the positive equilibrium of the system without time-delay. We further consider the e ff ect of delay on asymptotic behavior of the positive equilibrium: when the system undergoes Hopf bifurcation at some points of delay by the normal form and center manifold theory for partial functional di ff erential equations. Global existence of periodic solutions is established by applying the global Hopf bifurcation theory.


Introduction
In aquatic systems such as rivers, lakes and oceans, harmful algal blooms (HABs) have attracted considerable scientific attention in recent years. The effect of toxin-producing phytoplankton (TPP) on zooplankton is one reason for such attention. Researchers have made great progress in mathematical modeling in this field [1][2][3][4][5][6]. To fully understand the mechanism of planktonic blooms and to formulate reasonable control measures, Chattopadhyay et al. [3] build a ODE phytoplankton-zooplankton model with toxin effect, using data gathered from the coastal region of West Bengal and part of Orissa, India. These researchers investigated plankton models with different predational response functions and toxin liberation processes to obtain rich dynamics. They also considered the diffusivity of plankton influenced by ocean currents and tides, and the time delay effect of the phytoplankton toxin on zooplankton. Wang et al. [7] consider a ODE plankton system with Holling II response function and linear toxin processes. These authors concluded that the system undergoes Hopf bifurcation at positive equilibrium and Hopf-transcritical bifurcation when the parameters satisfy a particular condition.
In spatial plankton models, for example, Chaudhuri et al. [4] consider that the system includes both non-toxic and toxic phytoplankton and the system shows toxic phytoplankton-induced spatiotemporal patterns when one phytoplankton releases toxin. Further, to describe the reduction of zooplankton due to toxin-producing phytoplankton, plankton systems with discrete delay are presented in References [2,8] to show the effect of delay. There is o large body of literatures describing the dynamics of aquatic models [6,[9][10][11].
In fact, the problem we analyze is based on Chen et al. [1], which considers a model as follows: (1.1) They conclude that the plankton model (1.1) occurs as a bistable phenomenon.
In real-world conditions, plankton affected by tides and turbulence may move and diffuse across lakes and seas. Thus, such diffusion should be considered in studying the dynamics of plankton models. Meanwhile, the effect of phytoplankton toxin on zooplankton is in the form of a time-delay. We assume that no plankton species enter or leave at the boundaries of their environments. Based on these considerations, we present the plankton model as follows: for t > 0, x ∈ Ω = [0, lπ]. u represents the TPP population, and v represents the zooplankton population. Here, parameters β, r, θ and c are positive. β denotes the conversion ratio, r denotes the mortality rate of zooplankton due to natural death and higher predation, θ denotes the rate of toxin liberation by TPP population, and c denotes the half-saturation constant, here, we consider the case c > 1. d 1 and d 2 represent the diffusion coefficients of phytoplankton and zooplankton, respectively. Here, the initial conditions of system (1.2) are considered as where φ = (φ 1 , φ 2 ) is uniformly continuous, and the homogeneous Neumann boundary conditions are imposed as where ∂ ∂ν denotes the outward normal derivative on ∂Ω. This paper proceeds as follows. Section 2 gives the uniform boundedness of solutions. Section 3, by constructing the upper and lower solutions, establishes the global attractivity of positive equilibrium of the system without time-delay. Further, taking time delay as a branch parameter, we give the sufficient condition for which the system undergoes Hopf bifurcation at the interior equilibrium. In Section 4, under the assumption condition, we analyze the existence of Hopf bifurcation. Section 5 considers the global existence of these bifurcating periodic solutions. Finally, in Section 6 presents several numerical simulations.

Basic properties of solutions
This section shows the boundedness of the solutions of system (1.2).
Then, in view of the Hopf boundary lemma and the boundary condition, it follows that This implies u(x 1 , t 1 ) < 1.
By adding the two equations in system (1.2) with the form βu + v, it follows that By the maximum principle in Reference [12], this implies that This completes the proof. □ Clearly, system (1.2) has two boundary equilibria, E 0 = (0, 0) and E 1 = (1, 0).

Stability of the positive equilibrium
Let us now analyze the stability of plankton system (1.2) at the interior equilibrium. By referring to [1, Lemma 2.3], we know system (1.2) has a unique positive equilibrium E * (u − , v − ) when the parameters satisfy the following condition (H1) c > 1, r cθ > 1, and β > (c + 1)(r + θ).
When τ = 0, system (1.2) becomes (3.1) Applying the upper and lower solutions method, we show that the interior equilibrium E * (u − , v − ) for system (3.1) is globally attractive under assumption (H1). We denote In this section, note that ( Theorem 3.1. Under assumption (H1), the interior equilibrium E * for the plankton system (3.1) is globally attractive.
Proof. Firstly, we can know from system (3.1) by the convergence of the logistic equation, the comparison principle of parabolic equation and Lemma 2.1, then for a sufficiently small positive number ε, there must be a t 1 > 0 such that, for t ≥ t 1 , Moreover, for any initial value and it is evident that there exists a K > 0 such that In applying the upper and lower method in References [13,14], we define the two sequences (c m 1 ,c m 2 ) and (ĉ m 1 ,ĉ m 2 ) as follows:c m 1 =c m−1 Further, for the dynamics of system (1.2), we fix the parameters d 1 , d 2 , r, θ and c, and pick delay τ as the bifurcation parameter.
Firstly, in the phase space C = C([−τ, 0], X), we linearize system (1.2) to analyze the stability of the positive equilibrium (δ, v δ ), and haveŻ and the corresponding L 1 and L 2 are Then by a simple derivation, the characteristic equation and then Eq (3.5) has two roots given by Here, we establish the definition: Therefore, if (H1) holds, then βA 1 B 1 − θδ(1 − δ) > 0, and adding the condition (H2), and we obtain that D n − θδ(1 − δ) > 0. Thus, we obtain the following results.  In the following, with the help from Ruan and Wei [15], we analyze the existence of purely imaginary eigenvalues λ = ±iω(ω > 0) to study the stability of the positive equilibrium E * . Plugging λ = iω into Eq (3.3), we obtain −ω 2 − T n iω + D n − e −iωτ θδ(1 − δ) = 0, n = 0, 1, 2, · · · , then it follows from separating the real and imaginary parts that we rewrite the above Eq (3.6) by z = ω 2 , and get and Eq (3.7) has a pair of roots given by When assumptions (H1) and (H2) hold, we find that has two positive roots with n = n 0 }.
By the above calculation, we have Reλ ′ (τ + n,k ) > 0, Reλ ′ (τ − n,k ) < 0 for n ∈ N, k ∈ N 0 . Proof. Taking the differential of both sides of Eq (3.3) with respect to τ, we obtain dλ dτ then we simply calculate to obtain Completing the proof. □ Thus, it is straightforward that τ + n,0 ≤ τ − n,0 for all n ∈ N, and we know thatτ = τ + n,0 is the smallest point changing the stability of the linearized system (1.2) when the other parameters are fixed.
(i) If 2D n − T 2 n < 0 for all n ∈ N 0 , then the positive equilibrium E * of system (1.2) is locally asymptotically stable.
Thus, by a series of derivations and calculations, we can determine the value of g 21 . Then, we analyze the bifurcation property according to the following expression: (ii) If β 2 < 0(> 0), then the bifurcating periodic solutions on the center manifolds are orbitally stable(unstable); (iii) If T 2 > 0(< 0), then the bifurcating periodic solutions are period increases(decreases).
Further, by Theorem 3.6, we obtain the following result.

Analysis of global Hopf bifurcation
In the above section, the sufficient condition for the occurence of Hopf bifurcation at E * is given. In this section, we analyze the global dynamics of system (1.2) near the equilibrium E * . First, we cite the global Hopf bifurcation result in Reference [17].
Proof. By contradiction, assume that system (1.2) has τ-periodic solution, in other words, system (3.1) then has a periodic solution. We know that system (3.1) also has the same equilibria as system (1.2), that is, and an interior equilibrium E * . For system (3.1), it is straightforward that the u-axis and v-axis are the invariable manifold, and the orbits of the system do not intersect each other, thus, the solutions can not cross the coordinate axes. Hence, it follows that it must be the interior equilibrium E * if there exists any periodic solution within the first quadrant. From the above discussion in Lemma 2.1, the positive equilibrium E * is globally attractive, then system (3.1) has no nontrivial positive periodic solution, this means that system (1.2) has no nontrivial τ-periodic solution. This leads to contradiction. Completing the proof. □ Theorem 5.3. Assume condition (H1) holds and 2D n − T 2 n > 0, T 2 n − 4D n + 4θ 2 δ 2 (1 − δ) 2 > 0. When λ = iω + n 0 , then for each τ > τ + n 0 ,k (k = 0, 1, . . .) system (1.2) has at least k + 1 periodic solutions.

Simulations
In the following, let Ω = (0, π), and we choose the set of parameters as

Summary and discussion
This paper's main contribution is that it provides analytic results for the reaction-diffusion TPPzooplankton model with Holling II response function. Here, for system (3.1), when time delay is considered, it is found that time delay is a factor that causes the dynamic behavior of the system to destabilize, that is, the positive equilibrium of system (1.2) changes from globally asymptotically stable into unstable. Furthermore, from a biological point of view, plankton populations fluctuates periodically over time. Finally, numerical simulation shows that the plankton system (1.2) with time delay discussed in this paper better describes real-world problems.