Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

The local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.


Introduction
The rapid population and equally rapid decline are features of phytoplankton. Phytoplankton are temperature and nutrients sensitive. Their growth rate and reduction depend upon several features like temperature, nutrients, season, and the place of occurrence. Phytoplankton is commonly found in two types: spring blooms and red tides. As the name shows, spring blooms outbreak is seasonal and depends upon availability of nutrients. They are more sensitive as compared to other phytoplankton. Their outbreaks and survival are shorter than those of others. The name red tides is derived from their appearance.
The red tides are more usual and formal in coastal areas strong enough to absorb heat and toxic nutrients to survive. Red tides survive for several months, and their blooming is for some weeks or months. The evaluation in red tide species helps them to survive in the environment. The phytoplankton is divided into two categories: the first one shows rapid growth, while the second shows low rate of growth. The growth rate of these species depends upon environmental conditions and availability of nutrients. Some species of phytoplankton secrete a large amount of toxins that is the cause of fishes. When a bloom of a specific destructive phytoplankton occurs, the aggregate impact of all the poison released may influence other life forms, causing mass mortality. Such extraordinary concentrations or blooms are responsible for the enormous localized mortality watched in fishes and invertebrates in different places [1]. There has been a worldwide increment in harmful plankton blossoms in the final three decades, see the cited bibliography therein for more consultation on this topic [2][3][4][5], and considerable logical consideration towards harmful algal sprouts has been paid in recent years [6,7].
Hereafter before giving the mathematical modeling of the phytoplankton-zooplankton model, first we give some characteristics of red tides. It is pointed out in [1] that a phytoplankton-zooplankton model describes the detail of any particular species such as location, reproduction, and the following other features of different species of red tides.
• On the basis of existence, red tides are classified into two classes. The population of the first one remains constant throughout seasons of several months and that of the second does not remain constant. • The evolutionary process in that species is very slow, which helps to control the environmental conditions preventing from the predators and the effects of toxins. • The growth rate of red tides depends upon different environmental factors such as temperature, nutrients, trace elements, and pollution. This mechanism also describes the survival of that species in a changing environment. The population of that species increases or decreases due to environmental factors. • The occurrence of that species in a season is cyclic in nature: after getting maturity it returns to the original forum. This model also describes the mechanism of their rapid growth and recycling in the environment. It is also pointed out in [1] that a phytoplankton-zooplankton model explains the red tide environment as a constant system with population emerging with time, exemplified by ordinary differential equations. There are many expedient nominees for the role of refractory variable, but these are arranged into two classes: intrinsic and extrinsic. Within the mathematical model each complement would be denoted by a pair of coupled ordinary differential equations. Many accessible refractory variables are neither entirely intrinsic nor extrinsic. A similar situation would apparently hold for iron concentrations. The attribute of the elicit mechanism ascertained in the model lies in the interaction of growth rate of phytoplankton with the grazing rate of zooplankton. The mechanism is then modeled in the phytoplankton evolution as follows: where P and Z respectively denote the populations of phytoplankton and zooplankton. Moreover, the parameter r denotes the gross rate of production of phytoplankton, R m is the maximum specific predation rate, K is carrying capacity, and α governs how quickly that maximum is attained as prey densities increase. It is specified in [1] that Holling type-III utilized in several biological models and most commonly Holling type-III shape are as the Michaelis-Menten grazing function R m Z P α+P and the Ivlev grazing function R m Z(1e -λP ). Advance headstrong calculated within the system is the populace of zooplankton. In this manner, the rate of generation of zooplankton is controlled by the population thickness of phytoplankton, whereas their misfortune from the framework is through passing and natural predation by higher individuals of the nourishment web. So the full phytoplanktonzooplankton system takes the following form [1]: It is noted that in (2), parameter μ is the specific rate of elimination of zooplankton by passing out, and predacity is modeled as being relative to zooplankton population and γ is the ratio of biomass consumed to biomass of new herbivores produced. Now model (2) takes the following form: by using the following transformations: Moreover, (3) reduces to the following form: by using transformation Finally, on dropping tildes, the continuous-time phytoplankton-zooplankton model (5) becomes of the following form: It is well known that discrete-time models described by difference equations are more sensible than the continuous time models when populations have non-overlapping eras. Besides, discrete-time models also give more proficient computational results for numerical simulations and give a rich dynamics as compared to the continuous ones. Our extensive numerical simulations clearly show that the discrete system shows much more complex behavior as compared to the continuous one. Moreover, in spite of the fact that there are so many living circumstances in which it is characteristic to discover an occasion in discrete-time intervals, e.g., the propagation plan of phytoplankton-zooplankton. Subsequently, in order to show more reasonable and related to the living circumstance, we consider here a discrete form. So, by the forward Euler scheme, the discrete analogue of phytoplankton-zooplankton model (7) takes the following form: where h denotes the step-size. Our main contributions in this article include: • Topological classifications around fixed points of phytoplankton-zooplankton model (8). • Exploration of periodic points of phytoplankton-zooplankton model (8).
• Comprehensive bifurcation analysis around fixed points by bifurcation theory. • Investigation of chaos by the feedback control method for phytoplankton-zooplankton model (8). • Validation of the obtained results numerically. The next section is about the study of fixed points along with a linearized form of phytoplankton-zooplankton model (8), whereas topological classifications around fixed points are briefly studied in Sect. 3. Section 4 is purely dedicated to the explanation of periodic points of phytoplankton-zooplankton model (8). The comprehensive bifurcation analysis around fixed points is given in Sect. 5. Section 6 is about the investigation of chaos by the feedback control method for phytoplankton-zooplankton model (8). Theoretical results are numerically verified in Sect. 7, whereas conclusion of the paper is given in Sect. 8.

Fixed points along with linearized form of phytoplankton-zooplankton model (8)
In the present section, the existence of fixed points along with a linearized form of phytoplankton-zooplankton model (8) are given. In the following lemma, we first summarize the result regarding the existence of fixed points of phytoplankton-zooplankton model (8) in the region R 2 + = {(P, Z) : P, Z ≥ 0} as follows.

Topological classifications around
), (12) becomes Further characteristic equation of (25) is where Finally, the roots of (26) are Hereafter the following two lemmas give the complete topological classifications around ) of phytoplankton-zooplankton model (8) if < 0 and ≥ 0, respectively.
) of phytoplankton-zooplankton model (8), the following topological classifications hold: ) is a stable focus if ) is an unstable focus if (31) holds and ) of phytoplankton-zooplankton model (8), the following topological classifications hold: ) is a stable node if ) is an unstable node if (35) holds and
where f 1 and f 2 are depicted in (11). After straightforward computation, from (38) one gets the following desired statement: In a similar way, one can prove that the rest of fixed points F P0 (1, 0) and F + PZ ( ων 2 1-ω , ) of phytoplankton-zooplankton model (8) are periodic points of prime period-1.
) of phytoplankton-zooplankton model (8) is a periodic point of prime period-1.
Proof (i) In view of (38) we have Therefore from (40) one has the required result.
(ii) In view of proof (i) of Theorem 4.3, one gets the required statement: (iii) The following computation shows the required statement:

Bifurcation analysis around F 00 (0, 0)
In the following, we prove that around F 00 (0, 0), the occurrence of possible bifurcations is flip bifurcation and transcritical bifurcation.
Proof Under the same manipulation as we have done in Sect. 5.1.1, it is recalled that F 00 (0, 0) is non-hyperbolic if (17) holds. So, by choosing γ as a bifurcation parameter, the central manifold of the following map (17) holds. Thus it manifests that phytoplankton-zooplankton model (8) undergoes a transcritical bifurcation at F 00 (0, 0).
Proof It is noted here that phytoplankton-zooplankton model (8) is invariant with respect to Z = 0, and in order to explore the said bifurcation, we restrict the model on the line Z = 0, where (8) takes the form From (47) one has the following one-dimensional map with h as a bifurcation parameter: Finally, if h = h * = 2 β and P = P * = 1, then from (48) one gets: and ∂f ∂h h * = 2 β ,P * =1 The computed condition, which is depicted in (51), violates the nondegenerate condition, and hence this implies the fact that if (γ , h, β, ω, ν) ∈ F| F P0 (1,0) , then there exists no flip bifurcation around F P0 (1, 0) of phytoplankton-zooplankton model (8).

Bifurcation analysis around F
In the following, we prove that around F + PZ ( ων 2 1-ω , βν( ) the occurrence of possible bifurcations are Neimark-Sacker and flip bifurcations.

Numerical simulations
The obtained results are numerically verified in this section for fixing suitable values of involved parameters. Here the following two cases are considered for the completeness of this section. Case 1: Let h = 0.95, β = 0.55, ν = 0.6, ω = 0.23, then from (33) one gets γ = 3.0548345625166413, which is the value of a bifurcation parameter. So, the interior fixed ) of phytoplankton-zooplankton model (8) is a stable (respectively unstable) focus if 0 < γ < 3.0548345625166413 (respectively γ > 3.0548345625166413). To show this fact deeply, if one chooses the bifurcation value γ = 1.7 < 3.0548345625166413, then Fig. 1(a) shows that F + PZ (0.3279214350000127, 0.5270170961180427) of the discrete phytoplankton-zooplankton model (8) is a stable focus; moreover, Fig. 1(b)-1(f ) shows the same qualitative behavior if bifurcation values respectively are γ = 1.97, 2.1, 2.35, 3.01, 3.01345 < 3.0548345625166413. On the other hand, if γ = 3.3 > 3.0548345625166413, then Fig. 2(a) shows that the positive fixed point ) of phytoplankton-zooplankton model (8) changes behavior, and as a consequence an attracting invariant closed curve appears. Now numerically we have to show that if γ = 3.3 > 3.0548345625166413, then phytoplankton-zooplankton model (8) undergoes the supercritical N-S bifurcation, that is, the discriminatory quantity < 0. So, if γ = 3.3, then from (57) one gets Using (93) and (94) Fig. 2(b)-2(j) that attracting invariant closed curves appear, and therefore the discrete phytoplankton-zooplankton model (8) undergoes a supercritical N-S bifurcation, i.e., for the said bifurcation values, < 0 (see Table 1). Finally, bifurcation diagrams along with maximum Lyapunov exponent are drawn and presented in Fig. 3.

Figure 8
Graphs of n vs P n and Z n for controlled system (85)