Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions

This study introduces several updated results for the piecewise plant-pathogen-herbivore interactions model with singular-type and nonsingular fractional-order derivatives. A piecewise fractional model has developed to describe the interactions between plants, disease, (insect) herbivores, and their natural enemies. We derive essential findings for the aforementioned problem, specifically regarding the existence and uniqueness of the solution, as well as various forms of Ulam Hyers (U-H) type stability. The necessary results were obtained by utilizing fixed-point theorems established by Schauder and Banach. Additionally, the U-H stabilities were determined based on fundamental principles of nonlinear analysis. To implement the model as an approximate piecewise solution, the Newton Polynomial approximate solution technique is employed. The applicability of the model was validated through numerical simulations both in fractional as well as piecewise fractional format. The motivation of our article is that we have converted the integer order problem to a global piecewise and fractional order model in the sense of Caputo and Atangana-Baleanu operators and investigate it for existence, uniqueness of solution, Stability of solution and approximate solution along with numerical simulation for the validity of our obtained scheme.


Introduction
The interactions between plants, herbivores, and pathogens in nature are complex and multifaceted. According to a reference [1], approximately 50% of the estimated 6 million species of insects are herbivorous. On the other hand, plant pathogenic microbes pose significant threats to plants [2], but their total number has not been accurately estimated. Plants have evolved sophisticated defense mechanisms to detect and respond to multiple attacks by herbivores and pathogens [3]. These mechanisms can be broadly classified into two types: physical and chemical defenses. Regarding chemical defense, plants have the ability to release volatile organic compounds (VOCs) that can attract natural enemies of herbivores. This process can ultimately help to decrease herbivore pressure [4,5]. Predatory mites and parasitic wasps are among the natural enemies of herbivores that are attracted by volatile organic compounds (VOCs) released by plants [6].
A good example of this is when lima bean and apple plants are attacked by spider mites. In response to this damage, the plants release volatile compounds that can attract predatory mites to help control the spider mite population. Similarly, several plant species, including cucumber, corn, and cotton, release herbivore-induced plant volatiles (HIPVs) when attacked by herbivores. The use of HIPVs can help control pests and potentially reduce the need for artificial pesticides. Plants have the ability to release HIPVs, which are lipophilic liquids with high vapor pressures that can be emitted from various parts of a plant, such as its leaves, flowers, and fruits [7]. Through the release of HIPVs, plants may help protect forestry and agriculture by potentially attracting predatory arthropods or repelling herbivores, ultimately promoting plant fitness [8][9][10]. In addition, the release of volatile compounds can also help damaged plants by attracting natural enemies of herbivores [11]. Studies focusing on the interactions between plants and herbivores, as well as plants and pathogens, have been extensively researched. The influence of VOCs in mediating tri-trophic interactions, which occur when plants are attacked by both herbivores and pathogens, has yet to be fully understood. In a recent study by Liu et al. [12], they examined a model that involved plants, herbivores, and natural enemies of herbivores in the context of tri-trophic interactions, but did not take into account the potential pathogenic effects on the plant population. The study indicated that an increase in the strength of plant-induced volatile attraction to natural enemies resulted in a greater fluctuation amplitude of plant biomass and herbivore population.
Previous studies have primarily investigated the interactions between plants and either herbivores or pathogens. However, the extent to which VOCs mediate tri-trophic interactions when plants are attacked by both herbivores and pathogens remains uncertain. In a recent study by Liu et al. [12], they investigated tri-trophic interactions involving plants, herbivores, and natural enemies of herbivores without considering the pathogenic effects on plant populations. They found that an increase in the attraction strength of plant-induced volatiles to natural enemies leads to high fluctuation amplitudes of plant biomass and herbivore populations. Based on the study by Fergola and Wang [13], when the attack strength of natural enemies reaches a certain level, the fluctuation amplitude of plant biomass and herbivore population decreases, and the plant biomass tends to approach its environmental carrying capacity. They enhanced Liu et al.'s model [12] by considering the impact of time delay. Their research indicated that for Volterra-type interactions, chemical attractions do not affect the threshold value for the persistence of herbivore and carnivore populations. In addition, their observations showed that the presence of carnivores could lead to a decrease in herbivore density while increasing plant density. Moreover, the model demonstrated a fold bifurcation when the predation process follows Leslie-type interactions.
Based on the literature discussed above, we investigated a model for plants [14] using the framework of piecewise derivative and integral operators. The model consists of four compartments, including the susceptible plant compartment (S), the infected plant compartment (I), and the herbivore (Y) and natural enemy (Z) populations. The system of ordinary differential equations can be expressed as follows: The parameters used in the above system with details are presented in Table 1.  (1).

Notations Details r
The growth rate of plant k The rate of environmental carrying capacity β The rate of transmission of the disease α Rate of the inhibition effect ω Death rate of the infected plants d 1 Herbivore death rate c 1 , c 2 The rates of conversion µ The rate herbivore natural enemy p 1 , p 2 The rates of predation for plants Fractional derivatives are known for their ability to incorporate memory and genetic effects, making them essential for studying various real-world dynamic problems. The application of fractional-order derivatives in mathematical models of infectious disease spread makes the situation more realistic and plausible. The conventional fractional calculus method involves long memory effects, which can lead to complications when dealing with long-term calculation problems. Moreover, power-law long memory is represented using classical fractional calculus mathematical tools, which consist of fractional order derivatives and integrals. Several researchers have presented different fractional operators and investigated mathematical modeling of linear and nonlinear problems to discuss real-world phenomena [15][16][17][18]. It is worth noting that when modeling different memory phenomena, the memory process is generally split into two stages. The memory process is typically divided into two stages, one involving short-term memory with permanent retention, while the other is governed by a simple fractional derivative model [19].
To tackle the issue of short-term memory, it is necessary to employ equations with piecewise fractional-order derivatives. This approach can enhance performance and efficiency, enabling clearer explanations of physical phenomena (see [20] for further information). Moreover, the idea of using piecewise derivatives of fractional orders has recently been explored in several studies. The concept of piecewise fractional differential equations has been employed by researchers in recent studies, after Atangana and Araz presented in [21], to explore various dynamical systems [22][23][24]. Such operators have proven useful in solving problems that exhibit crossover behavior and are gaining popularity among researchers. For instance, in [25], the CAT-T cells-SARS-2 disease model was investigated using the concept of piecewise fractional differential equations. Similarly, a Covid-19 model was studied using this approach in [26]. Additionally, authors in [27] examined the third wave of the Covid-19 outbreak in Türkiye, Spain, and Czechia by employing piecewise differential and integral operators in their model. In addition, the concept of piecewise operators is being utilized by several authors to obtain numerical results for the food web model (see Ref. [28]). Numerous works have been documented by several researchers regarding piecewise differential and integral operators with various operators [29][30][31][32][33].
As for as the contribution of our article is concerned, we have converted the integer order problem to a global piecewise and fractional order model in the sense of Caputo and Atangana Baleanu operators (PCABC). As fractional operators are generalized because they have an extra degree of freedom and choices. Therefore, we checked the dynamics of different fractional orders lying between 0 and 1, and compared them with the integer order. On small fractional orders, stability is achieved quickly. The piecewise fractional model is investigated for the existence and uniqueness of the solution in the sense of fractional Caputo and Atangana Baleanu operator having a kernel of non-singularity in the form of the exponential function. The approximate or semi-analytical solution is obtained by the technique of piecewise fractional Caputo and ABC derivative. Taking different fractional orders we have simulated the obtained scheme for the first four terms. We have also compared the fractional dynamics with the integer order dynamics. All the quantities of the proposed problem are converging to their equilibrium points showing spectrum dynamics with the removal of singularity as well as the crossover or abrupt dynamics. Motivated by the above-mentioned work, we study the system taken from [14] in the sense of piecewise Caputo and Atangana-Baleanu-Caputo operator as given below: In order to elaborate further, equation (2) can be expressed as follows: CABC 0 In our study, we used piecewise fractional operators to describe the dynamics of the system, as they can capture crossover and abrupt dynamics effectively [34,35]. Specifically, we applied different fractional operators to different intervals of the system and provided a qualitative analysis for each subinterval. To assess stability, we used the UH stability analysis. Furthermore, we investigated the utilization of piecewise terms and fractional orders in the final step of the system to obtain approximate solutions that can accommodate dynamics of both integer and rational orders. The rest of the paper is organized as in the section 2 we include some preliminaries in form of definitions and lemmas. Section 3 comprised with the existence and uniqueness of solution of said model along with the stability of solution. In section 4 we established the numerical scheme for the approximate solution of model in fractional piecewise format. To show the validness of the obtain scheme we provide the graphical representation in the section 5 of numerical simulation. At the end we give the concluding remarks.

Preliminaries
In this section, we recalled some definition from the literature.
Definition 1 [16] The ABC operator of a function X(t ) with the condition X(t ) ∈ H 1 (0, T) is defined as follows: Definition 2 [16] If we consider X(t ) ∈ H 1 (0, T), the ABC integral can be expressed as follows: Definition 3 [15] Let a function X(t ), the Caputo derivative can be defined as: Definition 4 [21] Assume that X(t ) is a function that is differentiable in a piecewise manner. In this case, one can compute the piecewise derivative of X(t ) using Caputo and ABC operator.
The piecewise differential operator PCABC 0 D v t can be used to calculate the piecewise derivative of X(t ). Specifically, the Caputo operator is employed in the interval 0 < t ≤ t 1 , while the ABC operator is used in the interval t 1 < t ≤ T.
Definition 5 [21] Let X(t ) be a function that is piecewise integrable. We can then compute its piecewise derivative using the Caputo and ABC operators PCABC 0 We can represent the piecewise integral operator as PCABC where the Caputo operator is applied in the interval 0 < t ≤ t 1 , and the ABC operator is applied in the interval t 1 < t ≤ T.

Theoretical analysis
This section focuses on establishing the existence results and uniqueness of solution for the proposed system in the context of piecewise functions. We will investigate whether a solution exists for the hypothetical piecewise differentiable function and its particular solution attribute.
For the required solution, we will utilize the model (3) and provide additional clarification as follows: where we consider i = 1, 2, 3, 4 and take 0 < t ≤ t < ∞. Let E 1 represent the space for all piecewise differential function from C[0, t ] to R, obviously it is a complete normed space and thus the Banach We assume the following growth condition: If G is piece-wise continuous on (0, Consider a mapping $ : B → B and using (6) as Upon analyzing the previous equation, it can be concluded that U belongs to the set B. As a result, it follows that $(B) ⊆ B, indicating the closure and completeness of $. To further showcase its complete continuity, let us consider the initial interval in the Caputo sense as Next (9), we obtain t i → t j , and then So, $ is equi-continuous in [0, t 1 ]. Consider t i , t j ∈ [t 1 , T] in the sense of ABC as If t i → t j , then Therefore, it can be concluded that the operator $ exhibits equi-continuity in the interval [t 1 , T], thus making it an equi-continuous map. By applying the Arzelà-Ascoli theorem, it can be inferred that $ is uniformly continuous, continuous, and bounded. Furthermore, based on the Schauder theorem, it can be established that problem (3) has at least one solution in the subintervals.
Moreover, if the operator $ satisfies the condition of being a contraction mapping with assumption (C1), then the proposed system possesses a unique solution. The mapping $ : B → B is characterized by being piece-wise continuous, let us consider two elements U andŪ ∈ B on the interval [0, t 1 ] in the sense of Caputo, as follows: From (11), we have Consequently, it follows that $ satisfies the contraction mapping condition. Thus, according to the Banach fixed-point theorem, the problem at hand possesses a unique solution in the given subinterval. Additionally, it is worth noting that t ∈ [t 1 , T] in the ABC sense: or This implies that $ satisfies the contraction mapping property. Consequently, the problem being considered has a unique solution in the given sub-interval by virtue of the Banach fixed-point theorem. Therefore, taking into account equations (12) and (14), it can be concluded that the proposed problem has a unique solution on each sub-interval.

Analysis of stability
To prove the Ulam-Hyers stability for the considered model, we need to show that small perturbations in the initial conditions or the parameters of the model result in small perturbations in the solution of the model. This can be done by showing that the operator $ is Lipschitz continuous with respect to the initial conditions or the parameters of the model.

Definition 6
The considered system (1) is said to be U-H stable if, for every ℵ > 0, the inequality holds true.
There exists a unique solution U ∈ Z that is constant A > 0, Furthermore, if we consider an increasing function Ψ : [0, ∞) → R + , the inequality described above can be expressed as follows: If Ψ(0) = 0, then the resulting solution is considered to be generalized U-H (G-H-U) stable.

Remark 1.
Assuming that a function Ψ ∈ C(T ) does not depend on U ∈ W and satisfies Ψ(0) = 0, we can conclude that:

Lemma 1 Consider the function
The solution of (17) is (1) is that if

Theorem 1 The implication of Lemma
, the solution to model (2) is H-U stable as well as G-H-U stable.

Proof 2
If U ∈ W is a solution of (2) and U ∈ W is also a unique solution of (2), then we can conclude that Case:1 for t ∈ T , we have On more calculation Case:2 and further calculation we have We use Now, from Eq. (21) and (22), we have Thus, we can conclude that the solution to model (2) is H-U stable. Additionally, if we substitute ℵ with Ψ(ℵ) in (23), we get: Therefore, we can infer that the solution to our proposed model (2) is G-H-U stable based on the fact that Ψ(0) = 0.

Numerical scheme
Here, we will find the numerical scheme for the considered system (2).

Results and discussion
In this section we provide the numerical simulation of all four quantities of the considered piecewise problem on different fractional orders using the data of Table 2 taken from [14]. We also check the dynamics on two different intervals in the sense of Caputo and ABC fractional operators. The biological interpretation and their explanation are given in the article [14]. Further, the graphical results are simulated on different values lying between 0 and 1 which shows the total density of each quantity in the form of a continuous spectrum. Furthermore, chaotic dynamics are present in both divergent and convergent cases. All four quantities in the ecosystem oscillate which shows the dependence of one quantity over another one. The piecewise simulation is also provided on two subintervals the total time showing the sudden change dynamics or the crossover behavior The graphical representation of the four compartments is given in Figures 1a-1d with chaotic behaviors or phase portraits are given in Figures 1e-1f on different fractional orders using data of Table 2  In figures 5a-5f, the graphical representation of the four compartments are given with one another depending behaviors or phase portrait are given on different fractional orders using data of

Conclusion
Fractional-order differential equations with short memory play a crucial role in describing various real-world problems. Based on this premise, the current work explores a four-compartmental fractional-order plant model using the concept of piecewise derivative in both the Caputo and ABC sense. The study has further successfully investigated for the existence results, uniqueness of solution, and stability analysis of the considered problem, utilizing the fixed point concept and nonlinear functional analysis tools. Substantial results were obtained and presented in this work. The numerical solutions of the piecewise fractional model were computed using the Newton Polynomial technique in this study. MATLAB-18 was utilized to depict the numerical results for various fractional orders and time durations in this study. The results indicated that the piecewise data provided additional information that described crossover dynamics for different fractional orders. The graphical results obtained from both the piecewise and fractional order analysis were found to be highly intriguing as such analysis are more informative and generalized in the contrast of other relative work. The convergence and stability results are obtained by using fractional, piece wise fractional aspects. Fractional operators are generalized because they have an extra degree of freedom and choices. Therefore, we checked successfully the dynamics of different fractional orders lying between 0 and 1, and compare them with the integer order. On small fractional orders, stability is achieved quickly. Further, the piecewise fractional model is also tested validly for the existence and uniqueness of the solution in the sense of fractional Caputo and Atangana Baleanu operators having a kernel of non-singularity in the form of an exponential function. The approximate or semi-analytical solution is obtained by the technique of piecewise fractional Caputo and ABC derivative having the fractional parameter which shows the extra degree of freedom. Taking different fractional orders, we have simulated the obtained scheme for the first four terms. We also compare the fractional dynamics with the integer order dynamics. All the quantities of the proposed problem are converging to their equilibrium points showing spectrum dynamics with the removal of singularity as well as the crossover or abrupt dynamics.

Conflicts of interest
The authors declare that they have no conflict of interest.