Hopf-Pitchfork Bifurcation in a Phytoplankton-Zooplankton Model with Delays

Abstract

The purpose of this paper is to study the dynamics of a phytoplankton-zooplankton model with toxin delay. By studying the distribution of the eigenvalues of the associated characteristic equation, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, we find that the system can undergo a Hopf bifurcation at the positive equilibrium, and we derive the critical values where Hopf-Pitchfork bifurcation occurs.

1. Introduction

The study of the dynamical interaction of zooplankton and phytoplankton is an important area of research in marine ecology. Phytoplanktons are tiny floating plants that live near the surface of lakes and ocean. They provide food for marine life, oxygen for human being, and also absorb half of the carbon dioxide which may be contributing to the global warming [1]. Zooplanktons are microscopic animals that eat other phytoplankton. Toxins are produced by phytoplankton to avoid predation by zooplankton. The toxin producing phytoplankton not only reduces the grazing pressure on them but also can control the occurrence of bloom; see Chattopadhyay et al. [2] and Sarkar and Chattopadhyay [3]. Phytoplankton-zooplankton models have been studied by many authors [4–9]. In [6], models of nutrient-plankton interaction with a toxic substance that inhibit either the growth rate of phytoplankton, zooplankton, or both trophic levels are proposed and studied. In [7], authors have dealt with a nutrient-plankton model in an aquatic environment in the context of phytoplankton bloom. Roy [8] has constructed a mathematical model for describing the interaction between a nontoxic and a toxic phytoplankton under a single nutrient. Saha and Bandyopadhyay [9] considered a toxin producing phytoplankton-zooplankton model in which the toxin liberation by phytoplankton species follows a discrete time variation. Biological delay systems of one type or another have been considered by a number of authors [10, 11]. These systems governed by integrodifferential equations exhibit much more rich dynamics than ordinary differential systems. For example, Das and Ray [5] investigated the effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in the estuarine system. In this paper we present a phytoplankton-zooplankton model to investigate its dynamic behaviors. The model we considered is based on the following plausible toxic-phytoplankton-zooplankton systems introduced by Chattopadhayay et al. [2] 𝑑𝑃𝑑𝑡=𝑟₁𝑃𝑃1−𝐾−𝑎𝑃𝑍,𝑑𝑍𝑑𝑡=𝑏𝑍^𝑡_−∞𝐺()𝑃(𝑠)𝑃()𝑡−𝑠𝑑𝑠−𝑐𝑍−𝑑𝑡−𝜏𝑒+𝑃(𝑡−𝜏)𝑍, (1) where 𝑃(𝑡) and 𝑍(𝑡) are the densities of phytoplankton and zooplankton, respectively. 𝑟₁, 𝐾, 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are positive constants. 𝜏 is toxin delay, 𝐺(𝑠) is the delay kernel and a non-negative bounded function defined on [0,∞] as follows: ^∞₀𝐺(𝑠)𝑑𝑠=1,𝐺(𝑠)=𝜎𝑒^−𝜎𝑠,𝜎>0. (2)

For a set of different species interacting with each other in ecological community, perhaps the simplest and probably the most important question from a practical point of view is whether all the species in the system survive in the long term. Therefore, the periodic phenomena of biological system are often discussed [12–16].

The primary purpose of this paper is to study the effects of toxin delay on the dynamics of (1). That is to say, we will take the delay 𝜏 passes through a critical value, the positive equilibrium loses its stability and bifurcation occurs. By studying the distribution of the eigenvalues of the associated characteristic equation, the pitchfork bifurcation curve of the system is obtained. Furthermore, we derive the critical values where Hopf-Pitchfork bifurcation occurs.

The paper is structured as follows. In Section 2, we discuss the local stability of the positive solutions and the existence of Pitchfork bifurcation. In Section 3, the conditions for the occurrence of Hopf-Pitchfork bifurcation are determined.

2. Stability and Pitchfork Bifurcation

In this section, we focus on investigating the local stability and the existence of Pitchfork bifurcation of the positive equilibrium of system (1). It is easy to see that system (1) has a unique positive equilibrium 𝐸^∗(𝑃^∗,𝑍^∗), where 𝑃^∗=𝑐+𝑑−𝑏𝑒+(𝑐+𝑑−𝑏𝑒)²+4𝑏𝑐𝑒,𝑍2𝑏_∗=𝑟𝑎𝑃1−^∗𝐾>0, (3) where (𝐻₁)∶𝐾>𝑃^∗.

Let 𝑊(𝑡)=^𝑡_−∞𝜎𝑒^{−𝜎(𝑡−𝑠)}𝑃(𝑠)𝑑𝑠. (4) By the linear chain trick technique, then system (1) can be transformed into the following system: 𝑑𝑃𝑑𝑡=𝑟₁𝑃𝑃1−𝐾−𝑎𝑃𝑍,𝑑𝑍𝑃()𝑑𝑡=𝑏𝑍𝑊−𝑐𝑍−𝑑𝑡−𝜏𝑒+𝑃(𝑡−𝜏)𝑍,𝑑𝑊(𝑡)(𝑡).𝑑𝑡=𝜎𝑃−𝜎𝑊 (5) It is easy to check that system (5) has an unique positive equilibrium 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) with 𝑃^∗=𝑊^∗ provided that the condition (𝐻₁) holds.

Let 𝑃=𝑢+𝑢^∗, 𝑍=𝑣+𝑣^∗, and 𝑊=𝑤+𝑊^∗; then system (5) can be transformed into the following system: 𝑢̇(𝑡)𝑟=−𝐾𝑃^∗𝑢(𝑡)−𝑎𝑃^∗𝑣(𝑡)(𝑡)𝑣(𝑡)−𝑟−𝑎𝑢𝐾𝑢²(𝑡),𝑣(̇𝑡)=−𝑑𝑒(𝑒+𝑃^∗)𝑍^∗𝑢(𝑡−𝜏)+𝑏𝑍^∗𝑤(𝑡)+_{𝑖+𝑗+𝑘≥2}𝑓₂^{(𝑖𝑗𝑘)}𝑢^𝑖(𝑡−𝜏)𝑣^𝑗𝑤^𝑘,𝑤̇(𝑡)=𝜎𝑢(𝑡)−𝜎𝑤(𝑡), (6) where 𝑓₂^{(𝑖𝑗𝑘)}=1𝜕𝑖!𝑗!𝑘!^{𝑖+𝑗+𝑘}𝑓₂𝜕𝑢^𝑖(𝑡−𝜏)𝜕𝑣^𝑗𝜕𝑤^𝑘,𝑓₂𝑢()=𝑏𝑣𝑤−𝑐𝑣−𝑑𝑡−𝜏𝑒+𝑢(𝑡−𝜏)𝑣. (7)

Then linearizing system (6) at 𝐸^∗(𝑃^∗,𝑍^∗,𝑊^∗) is 𝑢̇(𝑡)𝑟=−𝐾𝑃^∗𝑢(𝑡)−𝑎𝑃^∗𝑣(𝑡),𝑣(̇𝑡)=−𝑑𝑒(𝑒+𝑃^∗)𝑍^∗𝑢(𝑡−𝜏)+𝑏𝑍^∗𝑤(𝑡),𝑤̇(𝑡)=𝜎𝑢(𝑡)−𝜎𝑤(𝑡). (8) It is easy to see that the associated characteristic equation of system (11) at the positive equilibrium has the following form and thus the characteristic equation of system (5) is given by 𝐹(𝜆)=𝜆³+𝑝₂𝜆²+𝑝₁𝜆+𝑝₀−𝑞₁𝜆+𝑞₀𝑒^−𝜆𝜏=0, (9) where 𝑝₂=𝜎+𝑟𝑃^∗𝐾,𝑝₁=𝑟𝑃^∗𝜎𝐾,𝑝₀=𝑎𝑏𝑃^∗𝑍^∗𝑞𝜎,₁=𝑎𝑑𝑒𝑃^∗𝑍^∗(𝑒+𝑃^∗)²,𝑞₀=𝑎𝑑𝑒𝑃^∗𝑍^∗𝜎(𝑒+𝑃^∗)². (10) Obviously, 𝑝₂>0, 𝑝₁>0, 𝑝₀>0, 𝑞₁>0, and 𝑞₀>0.

From (9), the following lemma is obvious.

Lemma 1. If the condition 𝐻₂∶𝑝₀=𝑞₀ holds, then 𝜆=0 is always a root of (9) for all 𝜏≥0.

Let 𝑑₀=(𝑏/𝑒)(𝑒+𝑃^∗)², 𝑑₁=𝑝₁(𝑒+𝑃^∗)²/(1−𝜏𝜎)𝑎𝑒𝑃^∗𝑍^∗, and 𝑑₂=(𝑒+𝑃^∗)²[𝜏²𝑎𝑏𝑃^∗𝑍^∗𝜎−2(𝜎+(𝑟𝑃^∗/𝐾))]/2𝜏𝑎𝑒𝑃^∗𝑍^∗; we have the following results.

Lemma 2. Suppose that the condition (𝐻₂) is satisfied. (i)If 𝑑=𝑑₀≠𝑑₁, then (9) has a single zero root. (ii)If 𝑑=𝑑₁≠𝑑₂, then (9) has a double zero root.

Proof. Clearly, 𝜆=0 is a root to (9) if and only if 𝑝₀=𝑞₀, which means 𝑑=𝑑₀. Substituting 𝑑=𝑑₀ into 𝐹(𝜆) and taking the derivative with respect to 𝜆, we obtain 𝐹^′(𝜆)_𝑑=𝑑0=3𝜆²+2𝑝₂𝜆+𝑝₁−𝑞₁(1−𝜏𝜆)−𝜏𝑞₀𝑒^−𝜆𝜏. (11) Then we can get 𝐹^′(0)_𝑑=𝑑0=𝑝₁−𝑞₁−𝜏𝑞₀. (12) For any 𝜏>0, by solving (12), we can obtain 𝑑=𝑑₁. If 𝑑=𝑑₀≠𝑑₁, 𝐹^′(0)≠0 which means that 𝜆=0 is a single zero root to (9), and hence the conclusion of (i) follows.
From (11), it follows that 𝐹^′′(𝜆)_𝑑=𝑑1=6𝜆+2𝑝₂−𝑞₁−2𝜏+𝜏²𝜆+𝜏²𝑞₀𝑒^−𝜆𝜏. (13) Then we get𝐹^′′(0)_𝑑=𝑑1=2𝑝₂−𝜏²𝑞₀−2𝑞₁𝜏. (14) For any 𝜏>0, by solving (14), we can obtain 𝑑=𝑑₂. If 𝑑=𝑑₁≠𝑑₂, 𝐹^′′(0)≠0 which means that 𝜆=0 is a double zero root to (9), and hence the conclusion of (ii) follows. This completes the proof.

From Lemma 2, we have the following result.

Theorem 3. Suppose that (𝐻₂) holds if 𝑑=𝑑₀≠𝑑₁, then, the system (5) undergoes a Pitchfork bifurcation at the positive equilibrium.

3. Hopf-Pitchfork Bifurcation

In the following, we consider the case that (9) not only has a zero root, but also has a pair of purely imaginary roots ±𝑖𝜔 (𝜔>0), when 𝑑=𝑑₀≠𝑑₁ holds.

Substituting 𝜆=𝑖𝜔 (𝜔>0) and 𝑑=𝑑₀ into (9) and separating the real and imaginary parts, one can get −𝜔³+𝑝₁𝜔−𝑞₁𝜔cos(𝜔𝜏)+𝑞₀sin(𝜔𝜏)=0,−𝑝₂𝜔²+𝑝₀−𝑞₁𝜔sin(𝜔𝜏)−𝑞₀cos(𝜔𝜏)=0. (15)

It is easy to see from (15) that 𝜔⁶+𝐷₂𝜔⁴+𝐷₁𝜔²+𝐷₀=0, (16) where 𝐷₂=𝑝²₂−2𝑝₁,𝐷₁=𝑝²₁−2𝑝₀𝑝₂−𝑞²₁,𝐷₀=𝑝²₀−𝑞²₀. (17)

Let 𝑧=𝜔². Then (16) can be written as ℎ(𝑧)=𝑧³+𝐷₂𝑧²+𝐷₁𝑧+𝐷₀. (18) In terms of the coefficient in ℎ(𝑧) define Δ by Δ=𝐷²₂−3𝐷₁. It is easy to know from the characters of cubic algebraic equation that ℎ(𝑧) is a strictly monotonically increasing function if Δ≤0. If Δ>0 and 𝑧^∗=(√Δ−𝐷₂)/3<0 or Δ>0, 𝑧^∗=(√Δ−𝐷₂)/3>0 but ℎ(𝑧^∗)>0, then ℎ(𝑧) has always no positive root. Therefore, under these conditions, (9) has no purely imaginary roots for any 𝜏>0 and this also implies that the positive equilibrium 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) of system (1) is absolutely stable. Thus, we can obtain easily the following result on the stability of positive equilibrium 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) of system (1).

Theorem 4. Assume that (𝐻₁) holds and Δ≤0 or Δ>0 and 𝑧^∗=(√Δ−𝐷₂)/3<0 or Δ>0, 𝑧^∗>0 and ℎ(𝑧^∗)>0. Then the positive equilibrium 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) of system (5) is absolutely stable; namely; 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) is asymptotically stable for any delay 𝜏≥0.

In what follows, we assume that the coefficients in ℎ(𝑧) satisfy the condition (𝐻₃)  Δ=𝐷²₂−3𝐷₁>0, 𝑧^∗=(√Δ−𝐷₂)/3>0, ℎ(𝑧^∗)<0.Then, according to Lemma  2.2 in [17], we know that (16) has at least a positive root 𝜔₀; that is, the characteristic equation (9) has a pair of purely imaginary roots ±𝑖𝜔₀. Eliminating sin(𝜔𝜏) in (15), we can get that the corresponding 𝜏_𝑘>0 such that (9) has a pair of purely imaginary roots ±𝑖𝜔₀, 𝜏_𝑘>0 are given by 𝜏_𝑘=1𝜔₀arccos−𝑞₁𝜔⁴₀+𝑝₁𝑞₁−𝑝₂𝑞₀𝜔²₀+𝑝₀𝑞₀𝑞²₁𝜔²₀+𝑞²₀+2𝑘𝜋𝜔₀,(𝑘=0,1,2,…). (19)

Let 𝜆(𝜏)=𝑣(𝜏)+𝑖𝜔(𝜏) be the roots of (9) such that when 𝜏=𝜏_𝑘 satisfying 𝑣(𝜏_𝑘)=0 and 𝜔(𝜏_𝑘)=𝜔₀. We can claim that 𝑑()sgnRe𝜆𝑑𝜏_{𝜏=𝜏𝑘}=sgnℎ^′𝜔²₀. (20) In fact, differentiating two sides of (9) with respect to 𝜏, we get 𝑑𝜆𝑑𝜏⁻¹=−3𝜆²+2𝑝₂𝜆+𝑝₁−𝑞₁𝑒^−𝜆𝜏+𝑞₁𝜆+𝑞₀𝜏𝑒^−𝜆𝜏𝑞₁𝜆+𝑞₀𝜆𝑒^−𝜆𝜏=−3𝜆²+2𝑝₂𝜆+𝑝₁𝑒^𝜆𝜏𝑞₁𝜆+𝑞₀𝜆+𝑞₁𝑞₁𝜆+𝑞₀𝜆−𝜏𝜆. (21) Then 𝑑()sgnRe𝜆𝑑𝜏_{𝜏=𝜏𝑘}=sgnRe𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔0}=sgnRe−3𝜆²+2𝑝₂𝜆+𝑝₁𝑒^𝜆𝜏𝑞₁𝜆+𝑞₀𝜆+𝑞₁𝑞₁𝜆+𝑞₀𝜆−𝜏𝜆_{𝜆=𝑖𝜔0}−𝑝=sgnRe¹−3𝜔²₀+2𝑝₂𝜔₀𝑖cos𝜔₀𝜏_𝑘+𝑖sin𝜔₀𝜏_𝑘𝑞₁𝜔₀𝑖+𝑞₀𝜔₀𝑖+𝑞₁𝑞₁𝜔₀𝑖+𝑞₀𝜔₀𝑖1=sgnΛ𝑝₁−3𝜔²₀cos𝜔₀𝜏_𝑘−2𝑝₂𝜔₀sin𝜔₀𝜏_𝑘×𝑞₁𝜔²₀−𝑝₁−3𝜔²₀sin𝜔₀𝜏_𝑘+2𝑝₂𝜔₀cos𝜔₀𝜏_𝑘×𝑞₀𝜔₀−𝑞²₁𝜔²₀1=sgnΛ3𝜔²₀−𝑝₁𝜔₀𝑞₁𝜔₀cos𝜔₀𝜏_𝑘−𝑞₀sin𝜔₀𝜏_𝑘−2𝑝₂𝜔²₀𝑞₁𝜔₀sin𝜔₀𝜏_𝑘+𝑞₀cos𝜔₀𝜏_𝑘−𝑞²₁𝜔²₀1=sgnΛ3𝜔⁶₀+2𝑝²₂−𝑝₁𝜔⁴₀+𝑝²₁−2𝑝₀𝑝₂−𝑞²₁𝜔²₀𝜔=sgn²⁰Λ3𝜔⁴₀+2𝐷₂𝜔²₀+𝐷₁𝜔=sgn²⁰Λℎ^′𝜔²₀=sgnℎ^′𝜔²₀, (22) where Λ=𝑞²₁𝜔⁴₀+𝑞²₀𝜔²₀. It follows from the hypothesis (𝐻₃) that ℎ^′(𝜔²₀)≠0 and therefore the transversality condition holds.

Lemma 5. All the roots of (9), except a zero root, have negative real parts when 𝑝₁>𝑞₁; (i) of Lemma 2 and 𝜏∈[0,𝜏₀) hold.

Proof. Consider 𝜆³+𝑝₂𝜆²+𝑝₁−𝑞₁𝜆+𝑝₀−𝑞₀=𝜆𝜆²+𝑝₂𝜆+𝑝₁−𝑞₁. (23) It is easy to get that the roots of (23) are 𝜆₁=0 and 𝜆_2,3=(−𝑝₂±𝑝²₂−4(𝑝₁−𝑞₁))/2. If 𝑝₁−𝑞₁>0, all the roots of (23), except a zero root, have negative real parts. We complete the proof.

Summarizing the previous discussions, we have the following result.

Theorem 6. Suppose that the conditions (𝐻₁), (𝐻₂), and (𝐻₃) are satisfied. (i)If 𝑑=𝑑₀≠𝑑₁ and 𝜏∈[0,𝜏₀), then the system (1) undergoes a Pitchfork bifurcation at positive equilibrium 𝐸^∗. (ii)If 𝑑=𝑑₀≠𝑑₁ and 𝜏=𝜏₀, then system (1) can undergo a Hopf-Pitchfork bifurcation at the positive equilibrium 𝐸^∗.

4. Conclusions

In this section, we present some particular cases of system (1) as follows: 𝑑𝑃𝑑𝑡=𝑟₁𝑃𝑃1−𝐾−𝑎𝑃𝑍,𝑑𝑍𝑃()𝑑𝑡=𝑏𝑍𝑃−𝑐𝑍−𝑑𝑡−𝜏𝑒+𝑃(𝑡−𝜏)𝑍. (24) From [2], we know that the system (24) undergoes a Hopf bifurcation at the positive equilibrium. In this paper, we get the condition that (9) has a zero root and also get the conditions that (9) has double zero roots. Furthermore, we obtain the conditions that (9) has a single zero root and a pair of purely imaginary roots. Under this condition, system (1) undergoes a Hopf-Pitchfork bifurcation at the positive equilibrium. Especially, when 𝜏=0, system (24) reduces to 𝑑𝑃𝑑𝑡=𝑟₁𝑃𝑃1−𝐾−𝑎𝑃𝑍,𝑑𝑍𝑃(𝑡)𝑑𝑡=𝑏𝑍𝑃−𝑐𝑍−𝑑𝑒+𝑃(𝑡)𝑍. (25) We can conclude that the positive equilibrium 𝐸(𝑃^∗,𝑍^∗,𝑊^∗) is locally asymptotically stable in the absence of toxin delay.

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions on the paper. The research of the authors was supported by the Fundamental Research Fund of Henan University (2012YBZR032).