We consider a continuous-time model describing the interaction between phytoplankton and zooplankton using a Holling type-II response. We then transform this continuous-time model into a discrete-time counterpart using a fractional-order discretization method. The paper explores the local stability of this obtained system concerning all equilibrium points and establishes the global asymptotic stability of its positive fixed point. The study also demonstrates that, under specific mathematical conditions, the system undergoes a Neimark–Sacker bifurcation around its positive equilibrium point. To effectively manage this bifurcation, two modified hybrid control techniques are introduced. The paper concludes by presenting illustrative numerical examples that validate the theoretical findings and assess the effectiveness and feasibility of the newly proposed control strategies. In addition, a comparative analysis is conducted between the modified hybrid techniques and an existing hybrid approach.
1. Introduction
Planktons are a diverse group of organisms found in the air (or water) that lack the ability to propel themselves against currents (or breeze). The organisms that constitute plankton are known as plankters. They play a crucial role as a food source for various small and large marine creatures such as fish, bivalves, and whales [1]. Oceanic plankton comprises archaea, bacteria, protozoa, algae, and migratory or drifting animals found in the saltwater of the sea and tidal waters. Freshwater or river plankton is similar to sea plankton but is found in natural lakes and streams. Plankton is commonly associated with occupying water. However, there are also floating varieties known as aeroplankton, which spend part of their lives adrift in the atmosphere. These can include plant pollen, wind-dispersed seeds, spores, airborne microorganisms from global dust storms, and marine plankton propelled into the air by ocean spray [1].
Although many planktonic species are small in size, plankton encompasses a wide range of creatures with varying masses, including larger animals like jellyfish. Planktons are defined primarily by their level of reactivity and ecological niche rather than by phylogenetic classification [2–7]. There are two main functional (or trophic level) groups within plankton:
(1)
Phytoplankton (from the Greek phyton, meaning plant) are autotrophic algae that thrive in shallow waters where there is sufficient light for photosynthesis.
(2)
Zooplankton (from the Greek zoon, meaning animal) are tiny protozoans (e.g., shellfish and other animals) that obtain their food from other planktons. This category also includes some larvae and eggs of larger nektonic creatures, such as crustaceans and fish [8].
Multiple phytoplankton classes produce toxins that are harmful to other organisms, leading to their categorization as poisonous algae. However, the exact nature of these toxic substances, their production mechanisms, and their ecological significance are not fully understood. These toxins could be released into the environment, possessing allelochemical properties that aid in fending off competitors or grazers, such as zooplankton. Alternatively, they might primarily remain within the phytoplankton cells, exerting toxic and deterrent effects on zooplankton or mixotrophic organisms. Some of these toxins may even play an aggressive role, functioning as venom to immobilize their prey [9].
In the plankton community, the species on the first trophic level are called phytoplankton, and their predators are called zooplankton [10]. Phytoplankton is microscopic and has a single-celled structure. Moreover, some of them are producers of a high amount of oxygen and reducers of carbon dioxide. These planktons are called “Beneficial algal blooms” and live in oceans, rivers, lakes, and other water sources favorable for their life [11]. Mathematically modeling the relationships among plankton classes offers a potential approach to enhancing understanding for anyone involved in studying the ecological dynamics of plankton populations. In investigations into models of phytoplankton-zooplankton interactions, the notable impact of toxin-producing plankton on zooplankton populations becomes apparent [11]. Recently, Belova et al. [12] have discussed the spreading of nutrients and the dynamics of phytoplankton inhabitants in the Azov ocean by considering salinity and high temperature. Mukhopadhyay and Bhattacharyya [13] discussed a plankton-nutrient model linked to the marine atmosphere, focusing on planktonic blooms. Gao et al. [14] investigated chaos in a phytoplankton-zooplankton system subjected to periodic and seasonal forcing. Rhodes et al. [15] proposed two scientific models concerning the plankton environment, demonstrating viral infection in phytoplankton and viruses. The effect of predation on competitive elimination and the coexistence of competitive predators were contemplated by Huppert et al. [16]. Furthermore, a one-phytoplankton two-zooplankton model with harvesting was presented and explored by them. Huppert et al. [16] scrutinized the impact of predation on the appropriate elimination and coexistence of predators. Furthermore, they deliberated a framework encompassing one phytoplankton and two zooplankton species, taking into account harvesting. Pei et al. [17] delved into a nutrient-phytoplankton model to analyze the dynamic behavior of phytoplankton blooms. Chattopadhyay et al. [18] introduced a mathematical model with a temporal delay in poison release by phytoplankton.
The mathematical modeling of phytoplankton-zooplankton interactions was done by authors of [19]. Observations made by them during the modeling of phytoplankton-zooplankton interactions are as follows.
It is assumed that the toxin-producing plankton has a logistic growth [19].
The intrinsic rate of growth and environmental carrying capacity for phytoplankton species are, respectively, represented by r and k.
f(p) and g(p) are response functions for phytoplankton and zooplankton populations, respectively.
They have considered that p(t) represents the density of phytoplankton population, and density of zooplankton population is represented by population variable z(t).
Rate of predation of phytoplankton by zooplankton is symbolized by β.
b > 0 is the biomass of zooplankton.
Rate of mortality of zooplankton and rate of production of toxin by toxin-producing plankton are, respectively, denoted by δ and ρ.
Given these assumptions, the phytoplankton-zooplankton model presented is as follows [19]:
(1)
Holling [20] outlined three distinct response functions for predator-type categories: Holling type-I, Holling type-II, and Holling type-III responses. Kuang and Freedman [21] explored the dynamics of a prey-predator model utilizing the Holling type-II response. Furthermore, the research indicated that examining the dynamic properties of prey-predator models incorporating Holling responses leads to more effective outcomes than models without them.
The Holling type-II response is often illustrated and defined using a rectangular hyperbola, with the mathematical expression as follows:
(2)
where λ is a half-saturation constant. By employing the Holling type-II response, we obtain the following expression for system (1):
(3)
To explore the intriguing dynamics of phytoplankton-zooplankton models, readers can refer to the works cited in references [22, 23]. When dealing with nonoverlapping populations, it becomes more convenient to study the dynamics of natural or biological systems using difference equations as an alternative to differential equations. To delve into the extensive investigation of mathematical modeling encompassing both discrete- and continuous-time models in the fields of biology and ecology, readers are encouraged to consult the referenced works [24–28].
In addition, difference equations offer improved reflection and examination of chaos in natural systems [29]. Thus, studying population models in discrete form proves to be fascinating.
Various mathematical methods exist to discretize continuous mathematical models into discrete form, including the following:
Euler method
Method of piecewise arguments
Nonstandard finite difference schemes
Runge–Kutta method
In recent years, numerous studies have focused on the dynamical analysis of the model (3) in both discrete and continuous forms (see [13–17]). However, the model, in its original form, cannot be examined using the fractional-order discretization method. Applying the fractional-order discretization method to any continuous system can lead to unexpected changes in its dynamical behavior when converted into its discrete counterpart. These changes may include the existence of chaos in the discretized system or the emergence of different types of bifurcation not experienced by the original system.
Fractional-order differential equations (FOD) have proven to be a valuable tool for modeling genetic effects and various natural processes [30]. Many scientific phenomena, such as nonlinear oscillations of earthquakes, wave propagation models, hydrologic models, and diffusion of wave models, are effectively modeled using fractional derivatives [31–33]. Compared to integer-order differential equations, using FOD to study natural interactions often yields more accurate results [30, 34, 35]. As a result, many researchers have employed fractional-order differential equations for dynamical studies of population models in mathematical biology [36, 37].
There are many well-known definitions for fractional-order derivatives. Among them, the most precise and useful definition is given as follows [38]:
(4)
where n ∈ Z+, and α ∈ (n − 1, n) is the order of fractional derivative of g(t). ℷθ represents the θ ordered Riemann–Liouville operator (see Caputo [38]), which has the following mathematical form:
(5)
Here, Γ(θ) is Euler’s gamma function. Moreover, the fractional-order counterpart of system (3) is given as follows [39, 40]:
(6)
where p0 > 0 and z0 > 0 are initial conditions. The Caputo operator is motivated by its ability to integrate initial conditions, providing a more physically meaningful explanation. Its advantages consist of smoother solutions and a direct physical explanation compared to other fractional derivatives. The memory effect in the Caputo operator is useful for capturing long-range dependencies in diverse scientific applications. As in the case of our model, the global stability of the system is due to this effect as it shows a long-term dependence on initial conditions. To provide additional information on various methods of fractional-order derivatives, we direct the reader to the works of authors [41–43]. Now, proceeding the discretization by using piecewise arguments (see [38]), we get
(7)
Suppose that t ∈ 0, s[0, s), then t/s ∈ [0, 1), and we have
Repeating this process of discretization n times, we get the following system (see [44]):
(12)
In addition, when t/s⟶n + 1, then system (10) takes the following form:
(13)
2. Interpreting External Parameters
Here, we have the ecological interpretation of the noninherent parameters in system (11).
(i)
α: The parameter α, which represents the order of the fractional operator in the system, does not possess a direct biological interpretation. Instead, it serves as a mathematical choice to quantify memory or nonlocality within the ecological context being studied. A higher α value implies a longer memory effect, which becomes relevant when past interactions between zooplankton and phytoplankton significantly impact their current dynamics. In essence, the choice of the fractional Riemann–Liouville operator’s order allows for a more advanced analysis of how memory and nonlocal effects influence the behavior of these populations in response to changing environmental conditions.
(ii)
s: In this situation, the choice of s plays a critical role in balancing accuracy and computational efficiency. A smaller s provides a more precise representation of the continuous-time system, as it closely captures the dynamics at each time step. However, this precision comes at the cost of increased computational workload since it requires more frequent model evaluations. On the other hand, a larger s reduces computational demands but may sacrifice accuracy by taking larger steps between evaluations. In the context of numerical simulations, like modeling biological population dynamics, s serves as a parameter that determines the size of time steps used to update the zooplankton and phytoplankton populations (pn and zn), allowing researchers to tailor their approach based on the trade-off between accuracy and computational resources.
(iii)
μ, μ1: In our model and in many ecological systems, the control parameter μ is an external factor or variable that can be adjusted or changed deliberately to influence how the system behaves. This parameter is not an inherent part of the system’s natural dynamics but is introduced to either study the effects of external interventions or to manage the system’s behavior for specific purposes. For instance, μ could represent human interventions in a phytoplankton-zooplankton system, such as introducing a new predator species, implementing pest control measures, altering habitat structures, or adding nutrients to an aquatic environment. These interventions can exert significant impacts on how the system’s dynamics unfold and are crucial for understanding and managing ecological systems effectively.
3. Novelty of Work
This research brings a fresh perspective to the existing body of knowledge by innovatively discretizing a continuous-time model of phytoplankton and zooplankton dynamics. Unlike previous studies that focused on equilibrium points and stability analysis, our work extends this exploration to global stability, honing in on the positive fixed point. Delving into Neimark–Sacker bifurcation and establishing the absence of period-doubling bifurcation, our findings provide new angles to current understanding. Moreover, we introduce and compare practical chaos control methods, specifically through modified hybrid approaches, addressing areas that have not been extensively covered in existing literature. In responding to the demand for a deeper grasp of ecological models, our research contributes valuable insights into the dynamics and control strategies within this field. The subsequent sections of this article aim to investigate fixed points and assess the local stability of the system described by (13), study the Neimark–Sacker bifurcation around the positive fixed point in system (13), employ a modified control strategy to manage the Neimark–Sacker bifurcation in system (13), and provide numerical simulations to validate our theoretical conclusions.
4. Analysis of Fixed Points and Local Stability
To acquire fixed points for system (13), we examine the ensuing set of two-dimensional equations:
(14)
When system (14) is solved, three fixed points can be identified: (p0, z0) = (0, 0), (p1, z1) = (k, 0), and
(15)
Furthermore, if
(16)
then p∗ > 0 and z∗ > 0. The following lemma explains the condition that aligns with the Jury criterion to ensure the local stability of equilibrium points (refer to [45]).
Lemma 1 (see [45].)Consider the characteristic equation , derived from a 2 × 2 Jacobian matrix J(p, z) associated with system (11) at its equilibrium points. Let T and D denote the trace and determinant of J(p, z), respectively, and suppose . Then, the following assertions can be made:
(i)
The conditions |ξ1| < 1 and |ξ2| < 1 hold if and only if and D < 1
(ii)
The conditions |ξ1| > 1 and |ξ2| > 1 hold if and only if and D > 1
(iii)
The conditions |ξ1| < 1 and |ξ2| > 1 or |ξ1| > 1 and |ξ2| < 1 hold if and only if
(iv)
The occurrence of complex roots characterized by ξ1 and ξ2 with |ξ1| = 1 = |ξ2| occurs if and only if T2 − 4D < 0 and D = 1
Given that ξ1 and ξ2 are solutions to the equation , the point (p, z) acts as a sink when |ξ1| < 1 and |ξ2| < 1, rendering it locally asymptotically stable. It is identified as a repeller or source when |ξ1| > 1 and |ξ2| > 1. The point becomes a saddle point under the conditions |ξ1| < 1 and |ξ2| > 1, or |ξ1| > 1 and |ξ2| < 1. Finally, (p, z) is classified as nonhyperbolic if condition (iv) is satisfied.
Initially, we examine the stability conditions for (13) with respect to the stationary point (p0, z0). The matrix J(p, z) computed at (p0, z0) is expressed as
(17)
Moreover, the characteristic equation of is as follows:
(18)
has two characteristic values, namely, ξ1 = rsα + Γ(1 + α)/Γ(1 + α) and ξ2 = Γ(1 + α) − sαδ/Γ(1 + α). Furthermore, it is evident that |ξ1| exceeds 1 for all parameter values. Thus, by treating the condition |ξ1| > 1 as true, we can now delineate stability criteria for (13) centered on (p0, z0).
Proposition 2. If (p0, z0) = (0, 0) is an equilibrium point of system (11), it acts as a source if the condition Γ(1 + α) > sαδ is satisfied and as a saddle if the condition 2Γ(1 + α) < sαδ is satisfied.
Our subsequent aim is to examine the local stability of system (13) focusing on the point (p1, z1). Denote as the Jacobian matrix of system (13) at the equilibrium (p1, z1). This matrix is represented as
(19)
Furthermore, by analyzing , we can derive the corresponding characteristic polynomial:
(20)
Therefore, employing Lemma 1 enables the formulation of the subsequent statement regarding the local stability of (13) around the point (p1, z1) = (k, 0).
Proposition 3. If (p1, z1) = (k, 0) is a fixed point of system (11), then it is stable if and only if
(21)
(p1, z1) represents an unstable fixed point exclusively when
(22)
(p1, z1) represents a saddle point exclusively when
(23)
The point (p1, z1) can never be nonhyperbolic fixed point.
At the end, we have certain findings concerning the local stability of system (11) around the point (p∗, z∗) = (γδ/b − δ − ρ, rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(δ + ρ − b)2). Furthermore, both components of (p∗, z∗) are positive if and only if (13) remains true.
Local stability in ecology refers to the behavior of a system in the neighborhood of an equilibrium point. If equilibrium is locally stable, it means that small perturbations in the populations of phytoplankton and zooplankton will eventually drop, and the system will return to the equilibrium. A locally stable equilibrium suggests that the ecological system has a tendency to resist small disturbances, maintaining a relatively constant balance between phytoplankton and zooplankton populations. The Jacobian matrix for system (11) evaluated at the point (p∗, z∗) can be computed as follows:
(24)
Let
(25)
denote the characteristic polynomial derived from the matrix , where
(26)
and
(27)
Considering that (16) holds true, and conducting mathematical calculations, it is deduced that
(28)
Therefore, one can analyze the local stability of system (13) around (p∗, z∗) = (γδ/b − δ − ρ, rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(δ + ρ − b)2) using the Proposition 4 (see Lemma 1). To see the topological classification of positive fixed point (p∗, z∗), see Figure 1.
Topological classification of positive fixed point for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, and s, δ ∈ (0, 1), with initial conditions p0 = 0.380975609 and z0 = 1.6789467026.
Proposition 4. Assuming that equation (13) holds, the positive fixed point (p∗, z∗) = (γδ/b − δ − ρ, rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(δ + ρ − b)2) remains valid for (13). Furthermore, let
(29)
Then, the point (p∗, z∗) constitutes a stable fixed point if and only if
(30)
(p∗, z∗) is a unstable fixed point if and only if
(31)
(p∗, z∗) represents a if and only if the following condition is satisfied:
(32)
The point (p∗, z∗) becomes nonhyperbolic under the condition:
(33)
and
(34)
Remark 5. System (11) can never experience the period-doubling bifurcation as for all a, b, r, k, γ, δ, ρ, β > 0 and α ≠ 1.
We present the following theorem for validating Remark 5.
Theorem 6. Assuming that condition (13) holds true, let the point (γδ/b − δ − ρ, rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(δ + ρ − b)2) be the sole positive fixed point of system (11).
In addition, consider the expression
(35)
Under these conditions, it follows that holds true for any parameter values specified in (13). Consequently, for all a, b, r, k, γ, δ, ρ, β > 0 and α ≠ 1, the function is decreasing.
Proof. Assume that (13) holds true and (γδ/b − δ − ρ, rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(δ + ρ − b)2) is the one and only positive fixed point of system (11) and
Lemma 7. Assume that I1 = [p1, p2] and I2 = [q1, q2] are intervals of real numbers and that f : I1 × I2⟶I1 and g : I1 × I2⟶I2 are continuous functions. Consider the following system:
(38)
with initial conditions (s0, t0) ∈ I1 × I2. Moreover, let us assume the validity of the following assertions:
(i)
The function f(s, t) exhibits nonincreasing behavior with respect to t and nondecreasing behavior with respect to s.
(ii)
The function g(s, t) demonstrates nondecreasing behavior in both variables s and t.
(iii)
The solution to the system is given by , where f belongs to the Cartesian product of intervals and . In addition, we have
(39)
(iv)
such that a1 = A1 and a2 = A2. Then, there exists precisely one positive fixed point (s∗, t∗) of system (38) such that limn⟶∞(sn, tn) = (s∗, t∗).
A globally stable equilibrium indicates that, regardless of the initial conditions or the size of perturbations, the system will eventually converge to that equilibrium. Moreover, this is an advantageous attribute for the long-term persistence and sustainability of populations in an ecosystem.
Theorem 8. The positive fixed point (p∗, z∗) of system (11) is a global attractor for solutions with positive initial conditions.
Proof. Let us consider the functions F(p, z) = p + sα/αΓ(α)(rp(1 − p/k) − βpz/γ + p) and G(p, z) = z + sα/αΓ(α)(bpz/γ + p − δz − ρpz/γ + p). It is evident that F(p, z) exhibits a nonincreasing behavior with respect to z and a nondecreasing behavior with respect to p. Furthermore, G(p, z) demonstrates a nondecreasing trend with respect to both p and z. Let F(a1, A1, a2, A2) be the solution of the system
(40)
Then, we have
(41)
and
(42)
Moreover, we have
(43)
Finally, from Theorem 5 of [46], it is sufficient to suppose that
This section of the article discusses the analysis of the Neimark–Sacker bifurcation in connection with system (13) around the point (p∗, z∗). The conditions for the occurrence of bifurcation and the positivity of (p∗, z∗) are detailed in Section 4. Bifurcation occurs when a mathematical system loses stability in a dynamical system, leading to significant fluctuations in its properties. This intricate behavior manifests in any system when parameters undergo slight changes in the vicinity of a fixed point.
For a deeper exploration of the bifurcation theory and to comprehend the remarkable behavior of discrete-time systems, readers are referred to [47–49]. In this study, we utilize conventional bifurcation theory to examine the Neimark–Sacker bifurcation of system (13) at (p∗, z∗). Considering ξ1 and ξ2 as the roots of , it is observed that these solutions are complex conjugates with a unit modulus when (p∗, z∗) represents a nonhyperbolic fixed point, meeting the final condition (34) outlined in Proposition 4.
(48)
where s ∈ (0, 1]. Let (α, β, b, k, r, s, γ, δ, ρ) ∈ ℧∗ with s1 = ((b(γ − k) + γ(δ − ρ) + k(δ + ρ))Γ(1 + α)/(δ + ρ − k)(γδ − bk + k(δ + ρ)))1/α, then system (13) can be written as
(49)
Consider the perturbation of (49) by using the parameter . In addition, by selecting as the bifurcation parameter, we derive the following expression for system (49):
(50)
where . Next, we consider that X = p − (γδ/b − δ − ρ), Y = z − (rγ(ρ − b)(kδ − bk + γδ + kρ)/kβ(−b + δ + ρ)2), then the map (49) changes to the following form:
(51)
where
(52)
The equation , formed by the Jacobian matrix of (51) around the point (0, 0), determines the characteristic roots as follows:
(53)
where
(54)
Suppose (α, β, b, k, r, s, γ, δ, ρ) belongs to ℧∗. The complex roots for (23) are determined as follows:
(55)
Then, it easily follows that
(56)
Moreover, we get
(57)
Since |T(0)| < 2 as (α, β, b, k, r, s, γ, δ, ρ) ∈ ℧∗, a simple computation yields that
(58)
Let us assume T(0) is not equal to zero and T(0) is not equal to minus one; this implies
(59)
and
(60)
Assuming the validity of (59) and (60) along with the condition (α, β, b, k, r, s, γ, δ, ρ) ∈ ℧∗, it can be deduced that T(0) is not equal to ±2, 0, −1. In other words, and are not equal to 1 for every m in the set {1, 2, 3, 4} when s1 = 0. As a result, when s1 = 0, both solutions of (53) stay outside the region where the unit circle intersects with the coordinate axes. In addition, we make the assumption that and . At this point, in order to transform (51) into normal form, we consider the following transformation:
(61)
Using transformation (61), the subsequent authoritative representation of system (51) is obtained:
(62)
where
(63)
where
(64)
Hence, we have the following nonzero real quantities:
(65)
where
(66)
Hence, by the analysis mentioned earlier, we can now discuss the existence and direction of Neimark–Sacker bifurcation in the shape of the following theorem [48, 50, 51].
Theorem 9. Let equations (24) and (25) hold true as s varies within a small neighborhood of , where
(67)
In addition, consider the unique positive fixed point of system (13) characterized by
(68)
then this positive fixed point (p∗, z∗) exhibits random motion within invariant closed curves resulting from Neimark–Sacker bifurcation. Moreover, for Ω < 0(Ω > 0), if , an attracting invariant closed curve emerges from the fixed point, whereas if , a repelling invariant closed curve bifurcates from the fixed point.
Note that in (13), we have used the same values of f(p) and g(p), and different parametric relations for bifurcation analysis are used in this section. This allows us to explore a range of scenarios and potential ecological dynamics. This approach enables a more comprehensive investigation of the system’s behavior under various conditions by strengthening the study.
7. Control of Neimark–Sacker Bifurcation
Exploring chaos theory and bifurcation control forms an interdisciplinary field within mathematics, concentrating on the inherent fundamental patterns and categorical laws governing highly intricate dynamical systems. Initially perceived as displaying completely random states of disorder and inconsistency, these systems have become a focal point for research.
Usually, the main factor determining disorder focuses on how even a small deviation in any state of a nonlinear dynamical system can result in notable changes in its advanced state, showcasing a complex reliance on initial conditions [48]. Each disordered attractor encompasses numerous periodic and unstable orbits. Chaotic behavior occurs when the system state approaches any of these regions for a period and then shifts to a nearby periodic and unstable orbit, where it persists for a certain duration, and so on [48]. Chaos and bifurcation control stabilize these irregular, periodic orbits, or invariant closed curves through small structural perturbations. In addition, chaos control methods aim to either stabilize a chaotic attractor or suppress chaos entirely within a system. There are situations where a system’s chaotic behavior is induced by a Neimark–Sacker bifurcation. Consequently, instead of conflicting, controlling a Neimark–Sacker bifurcation can be viewed as a means to steer a system away from chaos and towards more predictable dynamics. This aligns with the overarching objectives of chaos control techniques. Moreover, various established methods have previously emerged for managing chaos in discrete dynamical systems. See Ott et al. [52], Luo et al. [53], and He and Lai [50] for further elaboration on these techniques. Our approach utilizes a hybrid control strategy, as described by Khan [47], Khan et al. [48], and Khan [49]. Particularly, Khan [54] emphasizes parameter perturbation and incorporates a state feedback control method. Implementing this adaptable hybrid control methodology (with control parameter μ ∈ (0, 1]) on system (13) results in
(69)
Where μ is control parameter. Moreover, detailed description on μ is given at the end of the introduction section.
The system described by (69) displays controllability if its equilibrium point
(70)
experiences local asymptotic stability. Moreover, the Jacobian matrix for the system in (69) at its positive equilibrium point (p∗, z∗) is calculated as follows:
(71)
Theorem 10. System (28) possesses a locally asymptotically stable positive fixed point (p∗, z∗) if and only if the specified inequality is satisfied.
(72)
Now, let us examine a generalized hybrid control technique that incorporates state feedback and parameter perturbation:
(73)
Here, ℏ > 0 resides in Z, and μ1 ∈ (0, 1] serves as the parameter controlling the bifurcation in (73). In addition, g(ℏ) represents the kth iterative value of the function g(.). Applying (73) with m = 1 to system (13) yields the following controlled system:
(74)
The controllability of system (69) is ensured when its fixed point (p∗, z∗) exhibits local asymptotic stability.
In addition, the Jacobian matrix for system (74) at its positive fixed point (p∗, z∗) is determined as
(75)
The subsequent theorem outlines the essential conditions for both the necessity and sufficiency concerning the local stability of system (74).
Theorem 11. The local asymptotic stability of the positive fixed point (p∗, z∗) of system (30) is preserved if and only if the subsequent inequality is satisfied:
(76)
8. Numerical Simulations
Firstly, we assume that ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22 and s ∈ (0, 1]. Then, mathematical system (11) takes the following form:
(77)
The system has initial conditions given by p0 = 0.380975609 and z0 = 1.6789467026.
In this situation, Figure 2 demonstrates the graphical representation of both population variables. Figure 2(c) specifically showcases the maximum Lyapunov exponent for the system described in (77). Furthermore, varying s in the interval (0, 1] results in intriguing graphical behaviors for system (13), as shown in Figure 3. Notably, the Neimark–Sacker bifurcation occurs when the parameter s crosses the value s = 0.3267952585158358, as indicated in Figure 3(b).
Graphs representing the behavior of system (31) for the parameters ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ranging from 0 to 1. The system is initiated with p0 = 0.380975609 and z0 = 1.6789467026. (a) Bifurcation diagram for pn. (b) Bifurcation diagram for zn. (c) MLE for system (77).
Graphs representing the behavior of system (31) for the parameters ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ranging from 0 to 1. The system is initiated with p0 = 0.380975609 and z0 = 1.6789467026. (a) Bifurcation diagram for pn. (b) Bifurcation diagram for zn. (c) MLE for system (77).
Graphs representing the behavior of system (31) for the parameters ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ranging from 0 to 1. The system is initiated with p0 = 0.380975609 and z0 = 1.6789467026. (a) Bifurcation diagram for pn. (b) Bifurcation diagram for zn. (c) MLE for system (77).
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
Phase portraits for system (31) for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1] with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) pnzn-plane for s = 0.3367952585. (b) pnzn-plane for s = 0.3267952585. (c) pnzn-plane for s = 0.4567952585. (d) pnzn-plane for s = 0.3667952585. (e) pnzn-plane for s = 0.5667952585. (f) pnzn-plane for s = 0.8567952585.
It is easy to see that varying initial conditions influences short-term dynamics, while changing major parameters, like predation rates or growth rates, can have long-term impact on the stability and balance of phytoplankton-zooplankton ecosystems. In addition, if represents the characteristic equation of system (77) about the constant solution (p∗, z∗) = (0.38097560975, 1.67894670261), then for these parametric values, has the following form:
(78)
On solving (78), one can get ξ1 = −0.901944964908 − 0.431850993139i and ξ2 = −0.901944964908 + 0.431850993139i with |ξ1| = |ξ1| = 1. Moreover, by performing some mathematical calculations, one can obtain
(79)
where X = v12ϕ, and . Furthermore, we have
(80)
Hence, we have Ω < 0, which provides us with numerical proof for the presence of the Neimark–Sacker bifurcation as we have argued in Theorem 9. From an ecological viewpoint, the stable phase portrait in Figure 3(a) illustrates a balanced and resilient ecosystem with consistent population dynamics, capable of absorbing disturbances and maintaining long-term stability. In addition, the unstable phase portrait in Figure 3(b) depicts an ecologically weak state marked by variations and sensitivity to disorders, posing challenges to maintaining equilibrium and sustainability.
Next, we consider the following systems of difference equations:
(81)
and
(82)
In addition, μ ∈ (0, 1], μ1 ∈ (0, 1], and μ∗ ∈ (0, 1] are parameters used for controlling the bifurcation. Furthermore, for both systems (81) and (82), we get (p∗, z∗) = (0.380976, 1.67895), which is the unique positive fixed point of original system (13). Controlled regions for both populations are graphically shown in Figure 4. Furthermore, controlled figures for phytoplankton and zooplankton populations by using models (81) and (82) are, respectively, shown in Figure 5(a)–5(d). Taking the situation from an ecological viewpoint, one can see that the Neimark–Sacker bifurcation plot exposes potential disruptions in ecological balance. On the other hand, the control plots reveal that both modified hybrid methods outperform the traditional hybrid method in managing these disruptions. This suggests that the modifications bring about more effective control, offering a promising possibility for maintaining a stable and balanced phytoplankton-zooplankton ecosystem (see Figure 5).
Controlled regions for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot for system (81) for μ1 ∈ (0, 1]. (b) Plot for system (82) for μ ∈ (0, 1]. (c) Plot for system (83) for μ∗ ∈ (0, 1].
Controlled regions for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot for system (81) for μ1 ∈ (0, 1]. (b) Plot for system (82) for μ ∈ (0, 1]. (c) Plot for system (83) for μ∗ ∈ (0, 1].
Controlled regions for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot for system (81) for μ1 ∈ (0, 1]. (b) Plot for system (82) for μ ∈ (0, 1]. (c) Plot for system (83) for μ∗ ∈ (0, 1].
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
Controlled diagrams for ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, and s ∈ (0, 1], with initial conditions p0 = 0.380975609 and z0 = 1.6789467026. (a) Plot of pn for system (82). (b) Plot of zn for system (82). (c) Plot of pn for system (81). (d) Plot of zn for system (81). (e) Plot of pn for system (83). (f) Plot of zn for system (83).
The controlled figures for phytoplankton and zooplankton populations by using the hybrid method are shown in Figure 5(e) and 5(f), respectively. The following two-dimensional mathematical system is obtained by applying the old hybrid method proposed by Luo et al. [53] to (13):
(83)
where μ∗ ∈ (0, 1] is a control parameter. The bifurcation diagrams for controlled systems (33)–(35) are, respectively, shown in Figure 5. Finally, it can be easily experienced that by using generalized schemes, the stability of the initial system (13) is gloriously reclaimed for an enormous range of control parameter μ (see Figure 5). Finally, let ρ = 0.599, b = 2.459, k = 3.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84 and δ = 0.22. Then, from system (3), we get (p∗, z∗) = (0.380976, 1.917895). Moreover, by taking the initial conditions p0 and z0 in the least neighborhood of (p∗, z∗), an unstable behavior of continuous-time mathematical system (3) can be seen easily. In addition, the unstable plots for p(t) and z(t) are given in Figure 6(a) and 6(b), respectively, and the Hopf curve for the corresponding system is given in Figure 6(c).
Graphs illustrating the behavior of system (2) with parameters ρ = 0.599, b = 2.459, k = 3.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, and δ = 0.22, along with initial conditions p0 = 0.380975609 and z0 = 1.91789467026. (a) Plot of p(t) for system (3). (b) Plot of z(t) for system (3). (c) Phase portrait for system (3).
Graphs illustrating the behavior of system (2) with parameters ρ = 0.599, b = 2.459, k = 3.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, and δ = 0.22, along with initial conditions p0 = 0.380975609 and z0 = 1.91789467026. (a) Plot of p(t) for system (3). (b) Plot of z(t) for system (3). (c) Phase portrait for system (3).
Graphs illustrating the behavior of system (2) with parameters ρ = 0.599, b = 2.459, k = 3.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, and δ = 0.22, along with initial conditions p0 = 0.380975609 and z0 = 1.91789467026. (a) Plot of p(t) for system (3). (b) Plot of z(t) for system (3). (c) Phase portrait for system (3).
Therefore, it is evident that our continuous-time system (3) undergoes the Hopf bifurcation with parameter values same to those that have been utilized for system (77), underscoring the consistency of our model’s discretization technique.
9. Conclusions
We examine various dynamic characteristics of a model representing the interaction between phytoplankton and zooplankton (11). In Table 1, certain parameter values for (13) are assumed, while the rest are computed. In addition, Table 1 provides the parameter ranges for the entire system (13). Over the recent years, extensive investigations have been carried out on the dynamic analysis of the model (3) in both discrete and continuous forms (see [13–17]). However, the original form of the model cannot be examined using the fractional-order discretization method.
Table 1.
Parameter values used for numerical simulations.
Parameters
Default values
Reported ranges
r (maximum p(t) growth rate)
1.992(m−1day−1)
1.5 − 2.5(m−1day−1) assumed
β (maximum z(t) growth rate)
2.999(day−1)
2.5 − 3.5(m−1day−1) assumed
b (z(t) growth efficiency)
2.549
2.4 − 2.8 calculated
δ (natural death rate of z(t))
0.22(m−1day−1)
0.22 − 0.25(m−1day−1) calculated
k (environmental carrying capacity of p(t))
3.77(m−1)
1.5 − 4.5(m−1) calculated
ρ (rate of toxin production per p(t))
0.5999(m−1day−1)
0.1999 − 0.9999(m−1day−1)
γ (grazing half-saturation coefficient for z(t))
2.84(gCm−3day−1)
2.3 − 3.5(gCm−3day−1) assumed
Applying the fractional-order discretization method to any continuous system can lead to surprising changes in its dynamical behavior when converted into its discrete counterpart. The discretized system may exhibit chaos, or there may be different types of bifurcation that the original continuous system could never experience.
While studying any continuous-time system alongside its discrete counterpart, one can observe that the fractional parameter α plays a crucial role in the dynamical study of the discretized system. For α < 1, the system may remain consistent, but for α = 1, it reduces to Euler’s forward method, which generates a chaotic system after discretization. Applying a fractional-order discretization technique yielded a discrete-time rendition of the mathematical model put forth by [19]. Subsequently, we deduced mathematical findings concerning the presence of a positive fixed point. Furthermore, an examination of the local stability of the resultant mathematical system (13) was conducted around each fixed point. The proof established the global asymptotic stability of our system.
We presented a mathematical proof for the presence of the Neimark–Sacker bifurcation at a distinct positive equilibrium point, illustrating the intricate dynamics of the mathematical system (13). In addition, we provided noteworthy numerical examples. Our investigations indicate that the Neimark–Sacker bifurcation occurs in system (13) over a broad range of the bifurcation parameter s. Furthermore, we established that the continuous-time system (3) experiences the Hopf bifurcation for corresponding parameter values (see Figure 6).
To effectively control the Neimark–Sacker bifurcation, we utilized three different chaos-controlling methods. Our numerical studies have shown that the generalized hybrid methods (81) and (82) are superior to the hybrid method proposed in [53]. Methods utilizing feedback control are implemented to reinstate the system’s stability across a wide spectrum of parameters. Through numerical analysis, it’s apparent that the generalized hybrid methods (81) and (82) excel in controlling the Neimark–Sacker bifurcation.
Finally, we compared the effectiveness of the modified strategies with the old hybrid control technique [53]. As shown in Figure 5, the generalized techniques are much more effective than the old hybrid techniques. The comparison is also presented in Tables 2 and 3. In these tables, I1 represents the controlled interval for system (83), while the controlled intervals for systems (81) and (82) are denoted by I2 and I3, respectively.
Table 2.
Contrast between the updated hybrid approach (33) and its predecessor for s ∈ (0, 1], with parameter values ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22, along with initial coordinates x0 = 0.380976 and y0 = 1.67895.
Value of s
I1
I2
Length of I1
Length of I2
0.595258515
μ∗ ∈ (0, 0.84037928282)
μ1 ∈ (0, 0.997982102)
0.84037928282
0.9979821023
0.695258515
μ∗ ∈ (0, 0.80337356026)
μ1 ∈ (0, 0.997982103)
0.80337356026
0.9979821034
0.795258515
μ∗ ∈ (0, 0.77266733280)
μ1 ∈ (0, 0.883032240)
0.77266733280
0.8830322403
0.895258515
μ∗ ∈ (0, 0.74657752698)
μ1 ∈ (0, 0.841697792)
0.74657752698
0.8416977923
0.995258515
μ∗ ∈ (0, 0.72399998114)
μ1 ∈ (0, 0.809583311)
0.72399998114
0.8095833112
Table 3.
Comparison of the updated hybrid technique (34) with the previous hybrid method for s ∈ (0, 1] and ρ = 0.599, b = 2.459, k = 1.77, β = 2.999, r = 1.992, α = 0.29, γ = 2.84, δ = 0.22 using initial coordinates x0 = 0.380976, y0 = 1.67895.
Value of s
I1
I3
Length of I1
Length of I3
0.595258515
μ∗ ∈ (0, 0.84037928282)
μ ∈ (0, 0.94368078582)
0.84037928282
0.94368078582
0.695258515
μ∗ ∈ (0, 0.80337356026)
μ ∈ (0, 0.92962082676)
0.80337356026
0.92962082676
0.795258515
μ∗ ∈ (0, 0.77266733280)
μ ∈ (0, 0.91762277446)
0.77266733280
0.91762277446
0.895258515
μ∗ ∈ (0, 0.74657752698)
μ ∈ (0, 0.90717617862)
0.74657752690
0.90717617862
0.995258515
μ∗ ∈ (0, 0.72399998114)
μ ∈ (0, 0.89793765683)
0.72399998114
0.89793765683
It can be observed from these tables that the modified models (81) and (82) have longer controlled intervals compared to the old hybrid system (83). In addition, the controlled regions for systems (81)–(83) are illustrated in Figure 4(a)–4(c), respectively, when controlling techniques (81) and (82) are applied to (13). Hence, it is evident from the graphical and tabular comparisons of the control methods that the feasibility and effectiveness of the newly designed hybrid techniques can be easily seen.
Additional Points
Confirmation of Noninvolvement of AI or Third-Party Assistance. We confirm that no AI or third-party software was utilized in the research or preparation of this manuscript.
Ethical Approval
The procedures adhered to ethical standards outlined in the Helsinki Declaration of 1975, revised in 2000.
Consent
All authors voluntarily agreed to participate in this research. All authors have consented to publication, and there are no legal barriers to publishing the data included in the manuscript.
Conflicts of Interest
The authors declare there are no conflicts of interest.
Authors’ Contributions
Muhammad Salman Khan and Rizwan Niaz conceived the idea, conducted the computations, and drafted the initial version. Both authors further investigated the findings and finalized the manuscript. After the revision, Mohammed Ahmed Alomair and Mohamed Hussien provided extensive consultations on mathematical aspects and contributed to writing sections of the manuscript. In addition, they took charge of resolving technical intricacies. Moreover, in revision, they oversaw and improved the manuscript’s English language, grammar, and mathematical and statistical components. They also enriched the methodology section and enhanced the readability of the results and discussion. These collaborative efforts significantly bolstered the manuscript’s quality and clarity, warranting the inclusion of all authors in the manuscript. Furthermore, all listed authors have made a significant scientific contribution to the manuscript, and agreed to be an author.
Acknowledgments
The authors extend their gratitude to the Deanship of Scientific Research, Vice Presidency at King Faisal University, Saudi Arabia, for financially supporting this work for Graduate Studies and Scientific Research under grant no. 5220.
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