Dimensionally homogeneous fractional order Rosenzweig–MacArthur model: a new perspective of paradox of enrichment and harvesting

Abstract

The classical (ODE framework) Rosenzweig–MacArthur (RM) predator–prey model is a well-established ecological system. This model is modified further by incorporating time delay, diffusion, stochasticity, etc., in exploring several ecological aspects. However, investigations of this RM model in the fractional order differential equations are limited in the literature. More importantly, we have proposed a dimensionally consistent Caputo fractional order RM model subject to harvesting. The existence and uniqueness of solutions are established first in a feasible domain. Then, non-negativity, boundedness, and positivity of the solutions are demonstrated. Later, the local and global stability of all the equilibria is studied, and proper co-relations between stability zones are established in a two-parameter space. The theory of paradox of enrichment is investigated, and we present our insightful perspective on this ecological theory. The smaller carrying capacity shows a decreased mean population, which can be referred to as a paradox, but further increasing carrying capacity in our model has proved to be beneficial for the ecosystem. Hence, there is no paradox of enrichment in the system in our perspective. In the classical RM model, predator harvesting never changes the stability of a locally stable equilibrium, and the hydra effect is exhibited on predators at unstable coexisting equilibrium. For the first time, we have uncovered that the fractional order model could cause instability of a stable equilibrium through a supercritical Hopf-bifurcation and stabilize it again through a second supercritical Hopf-bifurcation, leading to stability switching and a bubble structure formation. Furthermore, predator harvesting exhibits the hydra effect at the stable state at two non-overlapping effort intervals. This study reveals several hidden results in the fractional order RM model compared to the classical RM model, which acts as the building blocks of ecological communities.

1 Introduction

Population ecology, a sub-field of biological sciences, studies the dynamics of the populations, their size, birth, death, immigration, diffusion, interaction among themselves or with some other species, etc. The effects of environmental factors (e.g. floods, earthquakes, and climate change) on population are well analysed in ecology. Theoretical ecology is a more specific field that systematically investigates ecological systems using mathematical, symbolic, statistical, and simulation models, to name a few. These modelling approaches are well recognised as they are cost-effective (time-saving compared to long-term experiments), help avoid environmental damages, and to improve the economy (without conducting high-budget laboratory experiments).

The ecosystem coexists of multiple tropic levels, and species experience a complex interaction between different tropic levels. Primarily, interactions can be of three types: predator–prey interaction (lizards consume spiders), competition (lions and tigers compete for common prey), and mutualism (trees and birds both benefit). The main emphasis of our study will be on predator–prey interactions. One of the first predator–prey models is the Lotka–Volterra [35], which exhibits non-isolated closed orbits enclosing a unique neutrally stable coexisting equilibrium. Then, the logistic Lotka–Volterra model drew attention, taking the logistic growth of the prey population into account in the Lotka–Volterra model, which shows a unique globally asymptotically stable coexisting equilibrium. Later Rosenzweig and MacArthur [48], incorporating Holling type II [22] functional response, proposed a predator–prey model. This model is popularly known as the Rosenzweig–MacArthur (RM) model, and it exhibits either a unique globally asymptotically stable coexisting equilibrium or a unique isolated closed orbit (limit cycle) enclosing an unstable focus.

Many researchers have studied the classical RM model in various contexts. Bezykin formed the building block of ecological modelling by proposing important models and explaining the dynamics of RM-type models [5]. Recently Adhikary et al. [2], investigated the Bezykin model under harvesting, demonstrating the potential existence of hydra effects in the population model at a stable state. Rinaldi and Scheffer [45] explored the slow-fast dynamics in the RM model. A trade-off between quality and quantity of additional food to the predator in the classical RM model is studied by Srinivasu [53]. Kar [24] examined the impact of prey refuge, revealing that increased refuge may increase prey population and result in predator extinction. Furthermore, it is proved that the system exhibits a single stable limit cycle when the positive equilibrium is unstable. Sieber and Hilker [52] conducted a detailed analysis on the hydra effect in Gause-type predator–prey models, which includes the RM model. From the viewpoint of feedback control, Matsuda and Abrams [30] analyzed the classical RM model and presented novel findings. Ghosh et al. [15] studied the RM model in establishing a relation between sustainability and maximum sustainable yield (MSY), while Tromeur and Loeuille [57] extended the study using the same model in exploring ecological resilience. Researches on RM predator–prey patchy models have revealed exciting dynamics [12, 16, 56] while deploying optimal amount of natural enemies to reduce harmful pests in the agriculture. Studies on the delayed RM model by Xia et al. [59] and Barman and Ghosh [3] have uncovered complex phenomena such as stability switching and Hopf bifurcations. Additionally, Sun and Mai [54] and Barman and Ghosh [4] have investigated delayed dispersal in RM predator–prey patchy models. Many researchers also examine the reaction-diffusion RM model [32, 34, 51] to detect population distribution and pattern formation. Hadeler and Gerstmann [18] was one of the earliest ones to study the discretized RM model to establish stability, positivity, and the existence of chaos. Recently, Rajni and Ghosh [43] explored it with the impact of harvesting and revealed rich, complex dynamics, including various bifurcations, multistability, and chaos in a discrete-time RM model.

Rosenzweig has introduced “paradox of enrichment”, where increasing the food available to the prey caused the ecosystem to be destabilised. Later, May [31] and Gilpin [17] analysed it extensively and showed limit cycle behaviour. Afterwards Abrams and Roth [1] observed the existence of paradox of enrichment in the Hastings–Powell model due to the bottom-up force (sufficient nutrient supply to prey). On the other hand, harvesting has a negative potential on the populations in terms of stock reduction and extinction. In the classical RM model, one of the most significant results in population harvesting is that predator harvesting can stabilise the system, and the stabilised system cannot be destabilised (see [15]). Furthermore, Sieber and Hilker [52] proved that the hydra effect (a phenomenon in which population size grows even when there is an increase in its mortality rate) is not possible for increasing harvesting effort in a stable state.

All the above results are explored in the classical (ODE framework) RM model. Historically, ecological models have predominantly relied on integer-order derivatives to describe the dynamics of physical, biological, and natural systems. However, ecological systems’ inherent complexity and non-linearity often require more refined mathematical frameworks to capture their internal features accurately. Fractional calculus offers a novel approach by introducing non-integer orders of differentiation and integration, enabling a more nuanced representation of ecological processes. In recent years, interest in applying fractional order derivatives to ecological modelling has surged, and researchers have explored its potential across diverse environmental phenomena, including predator–prey dynamics, disease spread, and neuroscience.

A notable study includes the investigation of the fractional-order Rosenzweig–MacArthur model with prey refuge [33], which is one of the highly cited articles. The interior equilibrium is shown to become stable from unstable for reducing the derivative order. Using the same method, it is shown that the paradox of enrichment can also be resolved. Maji [28] explores the impact of fear in fractional-order modelling with a constant prey refuge. The research fills a gap in the literature by considering the dynamics of a fractional-order model incorporating prey refuge and the effect of fear, providing valuable insights into the impact of fear and fractional-order derivatives on the stability of the system. The dynamics of an eco-epidemic model are studied in Panigoro et al. [40], representing the interactions between prey, predators, and infectious populations. A comparison is made between models using Caputo and Atangana–Baleanu–Caputo operators, emphasising differences in dynamical behaviours pertaining to Hopf bifurcation.

However, challenges persist in utilising fractional derivatives in ecological modelling. Many existing fractional-order models are developed from ODE models by replacing ordinary derivatives with fractional-order derivatives. This approach fails to preserve dimensional homogeneity, a critical consideration in any modelling framework. Dimensions of the ordinary derivative and Caputo fractional derivative are not the same. Hence, a more holistic approach to incorporating fractional derivatives into ecological models is essential to settle this discrepancy. Panigoro et al. [40] has successfully addressed and resolved the issue in their model. Motivated by the dimensional inconsistency, we shall develop a fractional order RM model, considering dimensional homogeneity, to find several hidden dynamic results that differ from the existing literature.

We sequentially organise the paper as follows: in Sect. 2, we shall develop a dimensionally homogeneous fractional order RM model. Some qualitative behaviours (e.g. existence uniqueness, non-negativity, boundedness, and positivity) will be systematically discussed in Sect. 3. Although most population dynamics literature maintains such a section in a traditional manner, the information in our section will offer fresh viewpoints that will grab the readers’ interest. Local and global stability will be studied in Sect. 4, followed by analysing the effect of carrying capacity in Sect. 5. Most importantly, the impact of harvesting will be thoroughly studied in Sect. 6. All the results of this article, combined with a thorough comparison to the existing literature, will be drawn in Sect. 7.

2 Model formulation

Here, we are considering a well-established form of the Rosenzweig–MacArthur model, with constant-effort harvesting [7] in both prey population x and predator population y as,

$$\begin{aligned} \begin{aligned} \frac{\textrm{d} x(t)}{\textrm{d} t}&=r x(t)\left( 1-\frac{x(t)}{K}\right) \\&\quad \,-\frac{a x(t) y(t)}{1+a b x(t)} - q_1 e_1 x(t),\\ \frac{\textrm{d} y(t)}{\textrm{d} t}&=c \frac{a x(t) y(t)}{1+a b x(t)}-d y(t) - q_2 e_2 y(t), \end{aligned} \end{aligned}$$

(1)

where all the information of the parameters and variables is given in Table 1.

Table 1 System parameters, variables, their description and dimension

In system (1), we have considered a widely incorporated Holling type II functional response form for interaction between predator and prey [22]. This form considers prey handling time b for predators, which will play a vital role in formulating our fractional order model. Generally, a population can be estimated in terms of biomass, numbers, or species density. Frank et al. [10], have considered population biomass in terms of Kg exploring Caplin fish dynamics and shown its biomass decline in their logistic Lotka–Volterra model. In the context of harvesting Clark and Clark [7], have assessed fish stock population as metric tons. In our modelling, we assume population in terms of biomass rather than population number. Hence dimension of x(t) and y(t) is [Mass]. As mentioned in Clark and Clark [7] (see pp-36), the dimension of harvesting effort is standardised vessel units (SVUs) (e.g. ships, nets, human resources, food supply to attract fishes, etc.), and the dimension of catchability coefficient is $\text {[Time}^{-1}]\text {(SVU)}^{-1}$ (see Table 1). Combining, we get the dimension of per capita harvesting rate (eq) as [$\text {Time}^{-1}$].

We now examine the dimensional consistency of the model (1). The dimension of $\displaystyle \frac{\textrm{d}x}{\textrm{d}t}$ is clearly [Mass][$\text {Time}^{-1}$]. Table 1 suggests that the dimension of the right-hand side of the first equation of system (1) is also [Mass][$\text {Time}^{-1}$]. Thus the first equation (similarly the second equation) is dimensionally consistent. Consequently, the system (1) is dimensionally homogeneous.

In recent decades, fractional order systems grabbed many researchers’ attention due to their memory effect and usefulness in real-world problems. To obtain the fractional counterpart of the ODE systems, many researchers have replaced the classical first-order time derivative with a useful fractional order derivative of order $\alpha$ (generally, Caputo fractional derivative). Following this approach, the fractional Rosenzweig–MacArthur model with prey refuge is studied by Moustafa et al. [33], while Xie and Zhang [60] have investigated the dynamics of a predator–prey model under the Allee effect and fear effect. Such fractional order modelling is also explored in the context of a contagious disease [41], and eco-epidemiological dynamics [19]. Likewise, fractional order population model with ratio-dependent functional response under linear harvesting is analysed by Suryanto et al. [55], and fractional order competition model dynamics is examined by Rivero et al. [46]. However, unlike others, Das et al. [8] considered a non-dimensionalized Hastings–Powell type ODE model, and replaced ordinary derivative by Caputo derivative in order to formulate a fractional order population model. Hence dimension issues do not appear in their analysis. Following a similar approach, we first consider a fractional Rosenzweig–MacArthur model corresponding to the system (1) as:

$$\begin{aligned} \begin{aligned} {}^{C} D^{\alpha }_{t} x(t)&=r x(t)\left( 1-\frac{x(t)}{K}\right) \\&\quad -\frac{a x(t) y(t)}{1+a b x(t)} - q_1 e_1 x(t), \\ {}^{C}D^{\alpha }_{t} y(t)&=c \frac{a x(t) y(t)}{1+a b x(t)}-d y(t)- q_2 e_2 y(t), \end{aligned} \end{aligned}$$

(2)

where Caputo fractional derivative [6], applied on population biomass x(t) and y(t), is denoted by ${}^{C} D^{\alpha }_{t}$ and defined as,

$$\begin{aligned} {}^{C}D^{\alpha }_{t} x(t)= & {} \frac{1}{\Gamma ({1-\alpha })}\int _0^t \frac{\frac{\textrm{d}x(s)}{\textrm{d}s}}{(t-s)^\alpha } d s;\nonumber \\ {}{} & {} \quad 0<\alpha <1. \end{aligned}$$

(3)

Now, we verify if the fractional order model (2) is dimensionally consistent. Dimension of $\displaystyle \frac{\textrm{d}x(s)}{\textrm{d}s},\text { }(t\!-\!s)^\alpha , \text { and }\textrm{d}s$ are, respectively, [Mass][$\text {Time}^{-1}$], [$\text {Time}^\alpha ],$ and [Time]. Together we obtain the dimension of the above Caputo derivative ${}^{C} D^{\alpha }_{t}x(t)$ as [Mass][$\text {Time}^{-\alpha }$]. However the dimension of the right hand side of the first equation (and second equation) of system (2) is [Mass][$\text {Time}^{-1}$], as we have determined before. Consequently,

$$\begin{aligned}{}[\text {Mass}][\text {Time}^{-\alpha }] \ne [\text {Mass}][\text {Time}^{-1}]. \end{aligned}$$

(4)

Then the fractional order models studied by Moustafa et al. [33], Xie and Zhang [60], Partohaghighi and Akgül [41], Hasan et al. [19], Suryanto et al. [55], Rivero et al. [46], and many more existing articles, following this approach are not dimensionally homogeneous. Therefore, to convert the ODE Rosenzweig–MacArthur model (1) into fractional order system, replacement of the classical first order time derivative with the Caputo derivative is not sufficient.

Here, we would like to develop a dimensionally homogeneous Rosenzweig–MacArthur model subject to harvesting. We have observed from the inequality (4) that the problem is primarily with the dimension of time, not with the dimension of mass. Keeping this point in mind, we propose a dimensionally homogeneous fractional order model corresponding to the integer order system (1) as,

$$\begin{aligned} \begin{aligned} {}^{C} D^{\alpha }_{t} x(t)&=r^\alpha x(t)\left( 1-\frac{x(t)}{K}\right) \\&\quad -\frac{a b^{1-\alpha } x(t) y(t)}{1+a b x(t)} - (q_1 e_1)^\alpha x(t), \\ {}^{C} D^{\alpha }_{t} y(t)&=c \frac{a b^{1-\alpha } x(t) y(t)}{1+a b x(t)}\\&\quad -d^\alpha y(t) - (q_2 e_2)^\alpha y(t). \end{aligned} \end{aligned}$$

(5)

Now, we thoroughly examine the dimensions (dimensional consistency) of the system (5). In the first equation of the system, we can easily observe that the dimension of the logistic growth term is now $\text {[Mass][Time}^{-\alpha }]$. Furthermore, the harvesting term $(q_1 e_1)^\alpha x(t)$ possesses the same dimension. We cannot resolve the dimensional inhomogeneity in the predator–prey interaction term $\displaystyle \frac{a x(t) y(t)}{1+a b x(t)}$ by taking the similar transformations, such as, $a \rightarrow a^\alpha$, or $ab \rightarrow (ab)^\alpha$, or both $a \rightarrow a^\alpha$ and $ab \rightarrow (ab)^\alpha$, because of the parameter a involves mass. Here b is the only parameter which purely depends on the dimension of time. Now, we will take the following systematic manipulations. Note that, $\displaystyle \frac{a x y}{1+a b x}=\frac{1}{b} \frac{x y}{\frac{1}{ab}+x}$. Let’s apply the transformation $\displaystyle \frac{1}{b} \rightarrow \frac{1}{b^\alpha }$ and we obtain the interaction term as $\displaystyle \frac{1}{b^\alpha } \frac{x y}{\frac{1}{ab}+x}=\frac{a b^{1-\alpha } x y}{1+a b x}$, which is of dimension $[\text {Mass}][\text {Time}^{-\alpha }]$. The first equation of system (5) is now dimensionally consistent. Following the similar technique, the second equation is also dimensionally homogeneous. Finally, we can conclude that system (5) is dimensionally consistent.

Panigoro et al. [39] proposed fractional order Rosenzweig–MacArthur model with threshold harvesting. Considering constant effort harvesting and applying a slightly different approach, we have formulated the dimensionally homogeneous fractional order Rosenzweig–MacArthur model with constant-effort harvesting. However, our investigation will be different than Panigoro et al. [39].

3 Existence, non-negativity, and boundedness

For differential equations, first, we address the existence of a solution. If a solution exists, we establish its uniqueness. Also, a negative or unbounded solution of a model is not feasible in describing population dynamics. Some research articles proved the positivity of solutions before discussing the existence of the same (see [49]). Ghani et al. [14] established boundedness of solutions even before proving the existence and non-negativity of it. Moreover, they only proved the upper bound of solutions. Thus, without showing positivity (i.e., lower bounds), claiming boundedness remained incomplete. While establishing any qualitative behaviour of solutions of a system, we should prove the existence and uniqueness of the solutions first.

In this context, we will follow a systematic order to establish the existence and uniqueness of solutions of the model (5), followed by non-negativity, boundedness, and positivity which makes this section interesting compared to many other existing literature.

3.1 Existence and uniqueness

First, we state a well-known theorem on the existence and uniqueness of solutions of fractional order differential equations. We recall Podlubny [42], for the existence and uniqueness criterion for any fractional order system. Let us consider the following initial-value problem

$$\begin{aligned} { { }^C D_t^\alpha {\tilde{x}}(t) = f(t, {\tilde{x}}),\,{\tilde{x}}(t_0) = {\tilde{x}}_0 }, \end{aligned}$$

(6)

where ${\tilde{x}} \in \Psi \subseteq {\mathbb {R}}^n, \, f: G \rightarrow {\mathbb {R}} \, (G \subseteq {\mathbb {R}} \times {\mathbb {R}}^n),$ be a vector-valued continuous function, defined on a region G containing the initial condition. Let f satisfies the Lipschitz condition on G with respect to ${\tilde{x}}$, i.e.,

$$\begin{aligned} \Vert f\left( t, {\tilde{x}}_1\right) -f\left( t, {\tilde{x}}_2\right) \Vert \le L\Vert {\tilde{x}}_1-{\tilde{x}}_2\Vert , \end{aligned}$$

where ${\tilde{x}}_1,{\tilde{x}}_2 \in \Psi$ and L is a positive constant. Moreover, if there exist some $M > 0$ such that

$$\begin{aligned} \Vert f(t, {\tilde{x}})\Vert \le M<\infty , \quad \forall (t, {\tilde{x}}) \in G, \end{aligned}$$

then there exist unique and continuous solution ${\tilde{x}}(t)$ of system (6) in a region $\Omega \subseteq G$.

To check the same for any mathematical model, we need a continuous function defined on a compact set $\Psi$ to verify if the Lipchitz condition is satisfied. One may consider an ecological model (for instance, predator–prey systems) consisting of Holling type I functional response of the form ax, where x(t) describes prey population. Then f has no point of singularity in any term (see [50]). On the other hand, in some models, there may exist singularity in functional response terms, e.g., the Holling type II or the Beddington-DeAngelis type functional response. For the systems (1) and (2) the Holling type II functional response of the form $\displaystyle \frac{ax}{1+abx}$ has a singularity at $\displaystyle x=-\frac{1}{ab}$, whereas the Beddington-DeAngelis type functional response of the form $\frac{x}{1+ax+by}$ has a singularity on $ax+by=-1$ [13], where y(t) denotes predator population. A special attention should be paid while choosing a compact set for investigation. Note that we initially treat system (5) as a purely mathematical model, rather than a predator–prey system.

After observing this difficulty in research articles, we appropriately choose the compact set for our model. Since the system (5) has a singularity for $\displaystyle x=-\frac{1}{ab}$, we construct a compact set as

$$\begin{aligned} \Psi= & {} \left\{ (x, y) \in {\mathbb {R}}^2: -\frac{1}{\gamma ab} \le \right. x \le \eta _1,\left. |y| \le \eta _2,\right. \\ {}{} & {} \left. \gamma >1 \right\} , \end{aligned}$$

with positive constants $\eta _1$ and $\eta _2$. Now we validate the existence and uniqueness theorem for our system (5) with initial conditions $x(t_0)=x_0$ and $y(t_0)=y_0$ as follow.

Theorem 1

For each initial condition in $\Psi$, the fractional order system (5) exhibits a unique solution.

Proof

Recall the system (5) and set

$$\begin{aligned} \begin{aligned} F_1(x, y)&=r^\alpha x\left( 1-\frac{x}{k}\right) -\frac{a b^{1-\alpha } x y}{1+a b x}-q_{1}^\alpha e_{1}^\alpha x, \\ F_2(x, y)&=c \frac{a b^{1-\alpha } x y}{1+a b x}-d^\alpha y-q_{2}^\alpha e_{2}^\alpha y, \end{aligned} \end{aligned}$$

with $F=\left[ F_1, F_2 \right]$, a column vector. We apply the sufficient condition for existence and uniqueness of the solutions of the fractional order system (5) in the region [0, $T] \times \Psi$, which is a subset of $G \subseteq {\mathbb {R}} \times {\mathbb {R}}^n$. $\square$

For $X=(x,y)$ and ${\bar{X}}=({\bar{x}},{\bar{y}})$,

$$\begin{aligned}&\Vert F(X)-F({\bar{X}})\Vert _1 \\&\quad =\left| F_1(x, y)-F_1({\bar{x}}, {\bar{y}})\right| \\&\qquad +\left| F_2(x, y)-F_2({\bar{x}}, {\bar{y}})\right| \\&\quad =\left| r^\alpha x\left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } x y}{1+a b x}-q_{1}^\alpha e_{1}^\alpha x\right. \\&\qquad \left. -r^\alpha {\bar{x}}\left( 1-\frac{{\bar{x}}}{K}\right) +\frac{a b^{1-\alpha } {\bar{x}} {\bar{y}}}{1+a b {\bar{x}}}+q_{1}^\alpha e_{1}^\alpha {\bar{x}}\right| \\&\qquad +\left| c \frac{a b^{1-\alpha } x y}{1+a b x}-\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) \right. \\&\qquad \left. -c \frac{a b^{1-\alpha } {\bar{x}}{\bar{y}}}{1+a b {\bar{x}}}+\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) {\bar{y}}\right| \\&\quad =\left| r^\alpha (x-{\bar{x}})-\frac{r^\alpha }{K} \left( x^2-{\bar{x}}^2\right) -q_{1}^\alpha e_{1}^\alpha (x-{\bar{x}})\right. \\&\qquad \left. -a b^{1-\alpha }\left( \frac{x y}{1+a b x}-\frac{{\bar{x}} {\bar{y}}}{1+a b {\bar{x}}}\right) \right| \\&\qquad +\left| c a b^{1-\alpha }\left( \frac{x y}{1+a b x}-\frac{{\bar{x}} {\bar{y}}}{1+a b {\bar{x}}}\right) \right. \\&\qquad -\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) (y-{\bar{y}})\bigg | \\&\quad =\left| r^\alpha (x-{\bar{x}})-\frac{r^\alpha }{K}(x-{\bar{x}})(x+{\bar{x}})-q_{1}^\alpha e_{1}^\alpha (x-{\bar{x}})\right. \\&\qquad \left. -a b^{1-\alpha }\left( \frac{x y+a b x {\bar{x}} y-{\bar{x}} {\bar{y}}-a b x {\bar{x}} {\bar{y}}}{(1+a b x)(1+a b {\bar{x}})}\right) \right| \\&\qquad +\left| c a b^{1-\alpha }\left( \frac{x y+a b x {\bar{x}} y-{\bar{x}} {\bar{y}}-a b x {\bar{x}} {\bar{y}}}{(1+a b x)(1+a b {\bar{x}})}\right) \right. \\&\qquad -\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) (y-{\bar{y}})\bigg | \\&\quad \leqslant r^\alpha |x-{\bar{x}}|+\frac{r^\alpha }{K}|x-{\bar{x}}|(|x|+|{\bar{x}}|)+q_{1}^\alpha e_{1}^\alpha |x-{\bar{x}}|\\&\qquad +\frac{a b^{1-\alpha }|x y-{\bar{x}} {\bar{y}}|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) } + \frac{a^2 b^{2-\alpha }|x||{\bar{x}}||y-{\bar{y}}|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) } \\&\qquad +\frac{c a b^{1-\alpha } |x y-{\bar{x}} {\bar{y}}|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) }+\frac{c a^2 b^{2-\alpha }|x||{\bar{x}}||y-y|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) } \\&\qquad +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) |y-{\bar{y}}|\\&\qquad =r^\alpha |x-{\bar{x}}|+\frac{r^\alpha }{K}|x\\&\qquad -{\bar{x}}|(|x|+|{\bar{x}}|)+q_{1}^\alpha e_{1}^\alpha |x-{\bar{x}}|\\&\qquad +\frac{(1+c)a b^{1-\alpha }|x y-{\bar{x}} {\bar{y}}|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) } \\&\qquad +\frac{(1+c) a^2 b^{2-\alpha }|x||{\bar{x}}||y-y|}{\left( 1+a bx\right) \left( 1+a b{\bar{x}}\right) } \\&\qquad +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) |y-{\bar{y}}|. \end{aligned}$$

Using the properties of the set $\Psi \text { we get, } x, \, {\bar{x}} \ge -\frac{1}{\gamma a b}$, which implies

$$\begin{aligned}{} & {} 1+abx \ge \left( 1 - \frac{1}{\gamma } \right) >0 \text { and}~\frac{1}{1+abx},\\{} & {} \frac{1}{1+ab{\bar{x}}} \le \left( \frac{\gamma }{\gamma -1} \right) . \end{aligned}$$

$\text {Moreover } x \le \eta _1 \text { and } |y| {\le } \eta _2 \text { implies } |x|, \, |{\bar{x}}| {\le } \eta _1 \text { and } |y|, \, |{\bar{y}}| \le \eta _2$ respectively.

Therefore,

$$\begin{aligned}&\Vert F(X)-F({\bar{X}})\Vert _1\leqslant \left( r^\alpha +\frac{2 r^\alpha \eta _1}{K}+q_{1}^\alpha e_{1}^\alpha \right) |x-{\bar{x}}|\\&\qquad + (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 ab^{1-\alpha }|x y-{\bar{x}} y+{\bar{x}} y -{\bar{x}} {\bar{y}}|\\&\qquad + (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 a^2 b^{2-\alpha } \eta _1^2|y-{\bar{y}}|\\&\qquad +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) |y-{\bar{y}}|,\\&\quad \leqslant \left( r^\alpha +\frac{2 r^\alpha \eta _1}{K}+q_{1}^\alpha e_{1}^\alpha \right) |x-{\bar{x}}| +(1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 \\&\qquad \times a b^{1-\alpha }|y||x-{\bar{x}}| +(1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2\\&\qquad \times a b^{1-\alpha }|{\bar{x}}||y-{\bar{y}}|+(1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2\\&\qquad \times a^2 b^{2-\alpha } \eta _1^2|y-{\bar{y}}| +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) |y-{\bar{y}}| \\&\quad \leqslant \left( r^\alpha +\frac{2 r^\alpha \eta _1}{K}+q_{1}^\alpha e_{1}^\alpha + (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 a b^{1-\alpha } \eta _2 \right) \\&\qquad \times |x-{\bar{x}}| +\left( a b^{1-\alpha } \eta _1 (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 + a^2 b^{2-\alpha } \eta _1^2 \right. \\&\qquad \times \left. (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) \right) |y-{\bar{y}}| \\&\quad \leqslant M \Vert X-{\bar{X}}\Vert _1 \end{aligned}$$

where $M=\text {max}\left( M_1 \,, \, M_2\right)$ with

$M_1=r^\alpha +\frac{2 r^\alpha \eta _1}{K}+q_{1}^\alpha e_{1}^\alpha + (1+c) \left( \frac{\gamma }{\gamma -1} \right) ^2 a b^{1-\alpha } \eta _2$, and

$M_2\!=\!a b^{1-\alpha } \eta _1 (1\!+\!c) \left( \frac{\gamma }{\gamma -1} \right) ^2 \!+\!a^2 b^{2-\alpha } \eta _1^2 (1\!+\!c) \left( \frac{\gamma }{\gamma -1} \right) ^2 +\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right)$.

Thus, F(X) satisfies the Lipschitz condition. Also,

$$\begin{aligned} \Vert F(X) \Vert _1=\left| F_1(x,y) \right| + \left| F_2(x,y) \right| \end{aligned}$$

is bounded as each component of F(X) is continuous on the closed and bounded region $\Psi$. Consequently, the existence and uniqueness of the fractional order system (5) follow from the abovementioned theorem.

3.2 Non-negativity

In population modelling, non-negativity is one of the most critical aspects, as this property assures that the biological species do not take negative values. On the other hand, zero state variable implies population extinction. To establish the non-negativity of solutions, we will first establish invariant sets under the motion of the system (5).

For any initial condition $(x_0,0)$, the corresponding solution (x(t), y(t)) of the system (5) will lie on the x-axis. This implies that x-axis is an invariant manifold. Similarly, for any initial condition $(0,y_0)$, the solution (x(t), y(t)) will lie on the y-axis, and correspondongly y-axis is another invariant manifold. Thus for any initial condition $(x_0,y_0) \in \Omega$, where $\Omega =\{(x,y): x \ge 0, y \ge 0\}$, the corresponding solution remain non-negative.

3.3 Boundedness

From the non-negative property, we established that the populations are bounded below in $\Omega$. This sub-section examines upper bound of the populations. To show the same, we define a function: $V=x+\frac{1}{c} y$.

Now,

$$\begin{aligned}&{}^{C} D^\alpha _t V+\left( d^\alpha +q_2^\alpha e_2^\alpha \right) V\\&\quad =r^\alpha x\left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } x y}{1+a b x}-q_1^\alpha e_1^\alpha x\\&\qquad +\frac{a b^{1-\alpha } x y}{1+a b x}-\frac{d^\alpha }{c} y-\frac{q_1^\alpha e_1^\alpha }{c} y \\&\qquad +d^\alpha x+\frac{d^\alpha }{c} y+q_2^\alpha e_2^\alpha x+\frac{q_2^\alpha e_2^\alpha }{c} y \\&\quad =A_1 x-\frac{r^\alpha }{K} x^2, \\&\qquad \text{ where } A_1=r^\alpha +d^\alpha +q_2^\alpha e_2^\alpha -q_1^\alpha e_1^\alpha \\&\quad =-\frac{r^\alpha }{K}\left( x^2- \frac{A_1K}{r^\alpha } x\right) \\&\quad =-\frac{r^\alpha }{K}\left( x^2-2 \frac{A_1 K}{2 r^\alpha } x+\frac{A_1^2 K^2}{4 r^{2 \alpha }}-\frac{A_1^2 K^2}{4 r^{2 \alpha }}\right) \\&\quad =-\frac{r^\alpha }{K}\left( x-\frac{A_1 K}{2 r^\alpha }\right) ^2+\frac{A_1^2 K}{4 r^\alpha } \\&\quad \leqslant A, \, \text{ where } A=\frac{A_1^2 K}{4 r^\alpha }. \\ \end{aligned}$$

Therefore by standard comparison theory (Section 4, [25]) and solution of linear fractional order differential equation using Laplace transform we obtain

$$\begin{aligned} \begin{aligned}&V(t) \leqslant V(0) E_\alpha \left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) \\&+A t^\alpha E_{\alpha , \alpha +1}\left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) . \end{aligned} \end{aligned}$$

(7)

Using the properties of Mittag-Leffler function $E_\alpha$ as $t \rightarrow \infty$ we obtain

$$\begin{aligned} E_\alpha (-\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha ) \rightarrow 0. \end{aligned}$$

(8)

Using the series expansion of $E_{\alpha ,\alpha +1}$ and (8) we eventually end up at

$$\begin{aligned}&t^\alpha E_{\alpha ,\alpha +1} \left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) \\&\quad = t^\alpha \sum _{k=0}^{\infty } \frac{\left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) ^k}{\Gamma (\alpha k +\alpha +1)}\\&\quad = -\frac{1}{\left( d^\alpha +q_2^\alpha e_2^\alpha \right) } \left[ \sum _{k=0}^{\infty } \frac{\left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) ^k}{\Gamma (\alpha k+1)} - 1 \right] \\&\quad = -\frac{1}{\left( d^\alpha +q_2^\alpha e_2^\alpha \right) } \left[ E_\alpha (-\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha ) - 1 \right] \end{aligned}$$

Using (8) this clearly implies

$$\begin{aligned}{} & {} t^\alpha E_{\alpha , \alpha +1}\left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right) \nonumber \\{} & {} \rightarrow \frac{1}{\left( d^\alpha +q_2^\alpha e_2^\alpha \right) } \text { as } t \rightarrow \infty . \end{aligned}$$

(9)

Consequently,

$$\begin{aligned} V(t) \rightarrow \frac{A}{\left( d^\alpha +q_2^\alpha e_2^\alpha \right) }, \, \text {as} \, \, t \rightarrow \infty . \end{aligned}$$

(10)

We know that the above Mittag-Leffler functions $E_\alpha$ and $E_{\alpha ,\alpha +1}$ are continuous. Moreover, $E_\alpha$ is a decreasing function $\forall t \ge 0$, with lower bound 0. Using these observations we can clearly see that $E_\alpha$ attains it’s maximum value at $t=0$, i.e., $E_\alpha (0)=1$. Continuity property of $E_{\alpha , \alpha +1}$ with finite initial value and observing (8), we conclude that $t^\alpha E_{\alpha ,\alpha +1}$ is bounded and hence has a finite maximum. Finally, using (7)–(9) we arrive at

$$\begin{aligned} V(t)= & {} x(t) + \frac{1}{c} y(t) \le V(0)\\{} & {} +AB=C \, \text {(say)}, \, \forall t \ge 0, \end{aligned}$$

where $B=\max _{\forall t \geqslant 0} t^\alpha E_{\alpha , \alpha +1}\left( -\left( d^\alpha +q_2^\alpha e_2^\alpha \right) t^\alpha \right)$. This proves the population remains bounded in $\Omega$, provided the initial condition also belongs there.

3.4 Positivity

In this subsection we prove the positivity of the system which implies that for any positive initial condition the solution of the system will remain positive. Non-negativity and boundedness together imply: $0 \le x \le C; \, 0 \le y \le cC.$ Using these inequalities, we obtain

$$\begin{aligned} \begin{aligned} {}^{C} D^\alpha _t x(t)&=r^\alpha x\left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } x y}{1+a b x}-q_1^\alpha e_1^\alpha x \\&\geqslant r^\alpha x\left( 1-\frac{C}{K}\right) -c a b^{1-\alpha } C x-q_1^\alpha e_1^\alpha x \\&=D x, \, \text{ where } \\&\quad D= r^\alpha \left( 1-\frac{C}{K}\right) -c a b^{1-\alpha } C - q_1^\alpha e_1^\alpha . \\ \end{aligned} \end{aligned}$$

This implies $x(t) \geqslant \, x_0 E_\alpha (D t^\alpha ) > 0, \, \forall t \ge 0.$

Similarly

$$\begin{aligned} \begin{aligned} {}^{C} D^\alpha _t y(t)&=\frac{c a b^{1-\alpha } x y}{1+a b x} - d^\alpha y - q_2^\alpha e_2^\alpha y \\&\geqslant - (d^\alpha +q_2^\alpha e_2^\alpha ) y. \\ \end{aligned} \end{aligned}$$

This implies $y(t) {\geqslant } \, y_0 E_\alpha (- (d^\alpha +q_1^\alpha e_1^\alpha ) t^\alpha ) {>} 0, \forall t {\ge } 0.$ Therefore $x(t),y(t)>0, \, \forall t \ge 0$, which proves the positivity of the system (5).

This section addressed the solution’s existence, uniqueness, and qualitative behaviours in the following order: non-negativity, boundedness, and positivity. Here, we have used the boundedness property of solutions to establish the positivity of the same. For a system of ordinary differential equations (ODEs), it is possible to prove positivity without using boundedness by applying the integral form of the ODE system and properties of the exponential functions (See [49]). One can see the integral form in fractional contexts [42], proving positivity, which is difficult without using the boundedness of solutions. As a result, it would be very interesting to show the positivity of the solutions without assuming the boundedness.

4 Stability analysis and bifurcations

  Equilibrium analysis is essential for mathematical modelling in many domains, including physics, biology, medical sciences, economics, and engineering. Finding analytical solutions is not always possible for non-linear models, and hence a stable equilibrium (steady state) is an alternate hope to estimate several aspects and physical measures. In the following section, we determine the equilibrium points of the system (5) and their stability. Equilibrium points are obtained by setting ${}^{C} D^\alpha _t x=0 \text { and } {}^{C} D^\alpha _t y=0$, likewise ordinary differential equation (ODE) modelling framework. We obtain the following equilibrium points:

• i. The extinction point $E^0=(0,0)$,

• ii. The predator extinction equilibrium $E^1=({\bar{x}}, 0)$, where ${\bar{x}}=K\left( 1-\left( \frac{q_{1} e_{1}}{r}\right) ^\alpha \right) ,$ which exists if $r>q_{1} e_{1}$,

• iii. The co-existing equilibrium $E^*=\left( x^*, y^*\right) ,$ where

  $$\begin{aligned} x^*= & {} \frac{b^{\alpha -1}(d^\alpha +q_{2}^\alpha e_{2}^\alpha )}{a\left( c-b^\alpha \left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) \right) }, \\ y^*= & {} \left( \frac{1+abx^*}{ab^{1-\alpha }}\right) \left( r^\alpha \left( 1-\frac{x^*}{K}\right) -q_{1}^\alpha e_{1}^\alpha \right) . \end{aligned}$$

  Both the components $x^*$ and $y^*$ exist if $c>b^\alpha \left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right)$ and $r^\alpha \left( 1-\frac{x^*}{K}\right) > q_{1}^\alpha e_{1}^\alpha$.

One can observe that the predator extinction equilibrium depends on the order of derivative $\alpha$ of the system. The existence condition of it is the same as its ODE system framework (1) and dimensionally inhomogeneous counterpart (2). On the other hand, the condition for existence and the expression of the co-existing steady state are functions of the order of derivative, $\alpha$. A substantial difference from the dimensionally inhomogeneous equivalent of the system (3) is the fact that co-existing steady state varies with every alteration in $\alpha$. In the forthcoming subsections, we first systematically study the local stability using eigenvalues and Matignon conditions [29], then the global stability of all the steady states will be established with the help of Lyapunov functions.

4.1 Local stability of steady states

  We examine the three steady states’ local stability using the system’s Jacobian matrix, corresponding eigenvalue analysis with Matignon conditions. The conditions can be stated as:

An equilibrium point of a system is

• Locally asymptotically stable if all eigenvalues $\lambda _{j}$ of the Jacobian matrix evaluated at the equilibrium point satisfy $\left| {\text {Arg}} \left( \lambda _{j}\right) \right| >\alpha \frac{\pi }{2}$,

• A saddle-point if there exists i, j such that for $j \ne i$, $|{\text {Arg}} \left( \lambda _{j}\right) |>\alpha \frac{\pi }{2}$ and $|{\text {Arg}} \left( \lambda _{i}\right) |<\alpha \frac{\pi }{2}$,

• Unstable if all the eigenvalues satisfy $| {\text {Arg}} \left( \lambda _j \right) | < \alpha \frac{\pi }{2}$.

Jacobian matrix of the system (5) at an arbitrary point E(x, y) is

$$\begin{aligned} J^E{=} \left( \!\begin{array}{cc} r^\alpha \left( 1{-}\frac{2 x}{K}\right) {-}\frac{a b^{1{-}\alpha } y}{(1{+}a b x)^2}{-}q_{1}^\alpha e_{1}^\alpha &{} -\frac{a b^{1{-}\alpha } x}{1{+}a b x} \\ \frac{a b^{1-\alpha } c y}{(1+a b x)^2} &{} -d^\alpha {+}\frac{a b^{1{-}\alpha } c x}{1{+}a b x}{-}q_{2}^\alpha e_{2}^\alpha \end{array}\!\right) . \end{aligned}$$

4.1.1 Local stability of the extinction equilibrium

At the extinction point $E^0=(0,0)$, the Jacobian matrix is

$$\begin{aligned} J^0=\left( \begin{array}{cc} r^\alpha - q_{1}^\alpha e_{1}^\alpha &{} 0 \\ 0 &{} -d^\alpha -q_{2}^\alpha e_{2}^\alpha \end{array}\right) . \end{aligned}$$

Eigenvalues of $J^0$ are $\lambda _1 = -d^\alpha -q_{2}^\alpha e_{2}^\alpha$ and $\lambda _2 = r^\alpha - q_{1}^\alpha e_{1}^\alpha$. Since, $\lambda _{1}<0$ for all parameter values, ${\text {Arg}}\left( \lambda _1\right) =\pi >\alpha \frac{\pi }{2}$ (Here, Arg stands for the principle argument). Now, $\lambda _2>0$ iff $r > q_{1} e_{1}$ i.e., ${\text {Arg}} \left( \lambda _2\right) =0<\alpha \frac{\pi }{2}$ makes the extinction equilibrium a saddle point, and $\lambda _2 <0$ iff $r < q_{1} e_{1} \text { i.e., } {\text {Arg}} \left( \lambda _2\right) =\pi >\alpha \frac{\pi }{2}$ makes the extinction point stable. It can be noticed that the condition needed for the existence of $E^1$ is identical to the criterion of instability for $E^0$. This demonstrates that the predator extinction equilibrium emerges when the extinction equilibrium turns unstable (saddle), which is also the case for its ODE modelling framework.

4.1.2 Local stability of the predator extinction equilibrium

The predator extinction point $E^1=({\bar{x}}, 0)$ leads to the Jacobian matrix as

$$\begin{aligned} J^1=\left( \begin{array}{cc} -q_{1}^\alpha e_{1}^\alpha + r^\alpha \left( 1- \frac{2 {\bar{x}} }{K} \right) &{} -\frac{a b^{1-\alpha } {\bar{x}} }{1+a b {\bar{x}} } \\ 0 &{} -d^\alpha +\frac{a b^{1-\alpha } c {\bar{x}} }{1+a b {\bar{x}} }-q_{2}^\alpha e_{2}^\alpha \end{array}\right) . \end{aligned}$$

One eigenvalue of $J^1$ is $\lambda _1 = -q_{1}^\alpha e_{1}^\alpha + r^\alpha \left( 1- \frac{2 {\bar{x}} }{K} \right) = q_{1}^\alpha e_{1}^\alpha - r^\alpha <0$ through the positivity condition of ${\bar{x}}$, and hence ${\text {Arg}} \left( \lambda _1\right) =\pi >\alpha \frac{\pi }{2} \text{. }$ Another eigenvalue is $\lambda _2=-d^\alpha +\frac{a b^{1-\alpha } c {\bar{x}} }{1+a b {\bar{x}} }-q_{2}^\alpha e_{2}^\alpha$. Now, $\lambda _2<0 \text { i.e., } d^\alpha + q_{2}^\alpha e_{2}^\alpha > \frac{a b^{1-\alpha } c {\bar{x}} }{1+a b {\bar{x}} }.$

So, the predator extinction equilibrium is stable if

$$\begin{aligned}d^\alpha + q_{2}^\alpha e_{2}^\alpha > \frac{a b^{1-\alpha } c {\bar{x}} }{1+a b {\bar{x}} } \text {,}\end{aligned}$$

and saddle point if

$$\begin{aligned}d^\alpha + q_{2}^\alpha e_{2}^\alpha < \frac{a b^{1-\alpha } c {\bar{x}} }{1+a b {\bar{x}} } \text {.}\end{aligned}$$

4.1.3 Local stability of the co-existing equilibrium

The Jacobian matrix at the coexisting equilibrium point $E^*=\left( x^*, y^*\right)$ is

$$\begin{aligned} J^*=\left( \begin{array}{cc} \frac{b^{\alpha -1}\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) \left( c\left( (1-a b K) r^\alpha +a b K q_{1}^\alpha e_{1}^\alpha \right) +b^\alpha \left( (1+a b K) r^\alpha -a b K q_{1}^\alpha e_{1}^\alpha \right) \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \right) }{a c K\left( -c+b^\alpha \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \right) } &{} -\frac{d^\alpha + q_{2}^\alpha e_{2}^\alpha }{c} \\ c\left( r^\alpha - q_{1}^\alpha e_{1}^\alpha \right) -\frac{b^{\alpha -1}\left( (1+a b K) r^\alpha -a b K q_{1}^\alpha e_{1}^\alpha \right) \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) }{a K} &{} 0 \end{array}\right) . \end{aligned}$$

Characteristic equation of $J^*$ is $\lambda ^2-T \lambda +D=0$, where

$$\begin{aligned} \begin{aligned}&T= \frac{b^{\alpha -1}\left( d^\alpha +q_{2}^\alpha e_{2}^\alpha \right) \left( c\left( (1-a b K) r^\alpha +a b K q_{1}^\alpha e_{1}^\alpha \right) +b^\alpha \left( (1+a b K) r^\alpha -a b K q_{1}^\alpha e_{1}^\alpha \right) \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \right) }{a c K\left( -c+b^\alpha \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \right) }, \\&D= -\frac{\left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \left( -a b c K \left( r^\alpha - q_{1}^\alpha e_{1}^\alpha \right) +b^\alpha r^\alpha \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) +a b^{1+\alpha } K\left( r^\alpha - q_{1}^\alpha e_{1}^\alpha \right) \left( d^\alpha + q_{2}^\alpha e_{2}^\alpha \right) \right) }{a b c K }. \end{aligned} \end{aligned}$$

Therefore, eigenvalues of $J^*$ are

$$\begin{aligned} \lambda _{1,2}=\frac{T \pm \sqrt{\Delta }}{2},\, \text {with } \Delta =T^2-4 D. \end{aligned}$$

If

• i. $T<0, \, 0<D \leqslant \frac{T^2}{4}$ or,

• ii. $T>0,\ T^2<4D,$ and $\tan ^{-1}\left( \frac{\sqrt{\mid \Delta \mid }}{T}\right) > \alpha \frac{\pi }{2}$ (using Matignon conditions),

then the interior equilibrium is locally asymptotically stable since $\left| \arg \left( \lambda _{1,2}\right) \right| >\alpha \frac{\pi }{2}$.

On the other hand, if

• i. $D<0$ or,

• ii. $T>0,\ 2\sqrt{D}>T,$ and $\tan ^{-1}\left( \frac{\sqrt{\mid \Delta \mid }}{T}\right) < \alpha \frac{\pi }{2}$ (using Matignon conditions),

then the interior equilibrium becomes unstable.

4.2 Global stability

We now establish the analysis of the system’s global stability for each of its equilibrium points. For this purpose, we choose suitable Lyapunov functions. Following that, we examine the evolution of the Caputo derivative of the function.

4.2.1 Global stability of the extinction equilibrium

For the extinction equilibrium, we consider the following function

$$\begin{aligned} V(x,y)=x+\frac{1}{c} y \end{aligned}$$

with the property that it is zero only at (0, 0) and remains positive $\forall (x,y) \in {\mathbb {R}}^2_+$, which makes V(x, y) a positive definite function in ${\mathbb {R}}^2_+$. Now

$$\begin{aligned} \begin{aligned} {}^C {D_t^\alpha V}&= {}^C{D_t^\alpha } x+\frac{1}{c} {}^C{D_t^\alpha y} \\&= r^\alpha x\left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } x y}{1+a b x}-\left( q_1 e_1\right) ^\alpha x \\&\quad +\frac{a b^{1-\alpha } x y}{1+a b x}-\frac{d^\alpha }{c} y-\frac{\left( q_2 e_2\right) ^\alpha }{c} y \\&= r^\alpha x-\frac{r^\alpha }{K} x^2-(q_1 e_1)^\alpha x-\frac{d^\alpha }{c}y \\&\quad -\frac{\left( q_2 e_2\right) ^\alpha }{c} y \\&=\left( r^\alpha -\left( q_1 e_1\right) ^\alpha \right) x-\frac{r^\alpha }{K} x^2-\left( \frac{d^\alpha }{c}+\frac{q_2^\alpha e_2^\alpha }{c}\right) y \\&<0 \quad \text{ if } r^\alpha <q_1^\alpha e_1^\alpha . \end{aligned} \end{aligned}$$

As a result, ${}^{C} D^\alpha _t V(x, y)<0$ if $r<q_1 e_1$, which indicates that extinction equilibrium is globally asymptotically stable through Lyapunov function for the exact condition of its local stability. Correspondingly, no other steady state exists when the extinction equilibrium is globally stable.

4.2.2 Global stability of the predator extinction equilibrium

In the case of the predator extinction equilibrium, we consider a function

$$\begin{aligned}V(x,y)=\left[ x-{\bar{x}}-{\bar{x}} \ln \left( \frac{x}{{\bar{x}}}\right) \right] +\frac{1}{c} y\end{aligned}$$

with the property that it is zero only at the boundary equilibrium $({\bar{x}},0)$ and remains positive $\forall (x,y) \in {\mathbb {R}}^2_+$, which makes V(x, y) a positive definite function (see [58]). Now,

$$\begin{aligned}&{}^{C} D^\alpha _t V(x, y)\\&\quad \leqslant \left( \frac{x-{\bar{x}}}{x}\right) {}^C{D_t^\alpha } x+\frac{1}{c} {}^C{D_t^\alpha y}\\&\quad =(x-{\bar{x}})\left[ r^\alpha \left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } y}{1+a b x}-\left( q_1 e_1\right) ^\alpha \right] \\&\qquad +\frac{1}{c}\left[ c \frac{a b^{1-\alpha } x y}{1+a b x}-\left( d^\alpha +q_2^\alpha e_2^\alpha \right) y\right] . \end{aligned}$$

Here we use the fact that $({\bar{x}},0)$ satisfies the system (3), and hence we obtain.

$$\begin{aligned}&{}^{C} D^\alpha _t V(x, y)\\&\quad \leqslant (x-{\bar{x}})\left[ \frac{r^\alpha }{K} {\bar{x}}-\frac{r^\alpha }{K} x-\frac{a b^{1-\alpha } y}{1+a b x}\right] \\&\qquad +\frac{a b^{1-\alpha } x y}{1+a b x}-\frac{d^\alpha +q_2^\alpha e_2^\alpha }{c} y \\&\quad =-\frac{r^\alpha }{K}(x-{\bar{x}})^2+\frac{a b^{1-\alpha } {\bar{x}} y}{1+a b x}-\frac{d^\alpha +q_2^\alpha e_2^\alpha y}{c} \\&\quad <-\frac{r^\alpha }{K}(x-{\bar{x}})^2-\frac{d^\alpha +q_2^\alpha e_2^\alpha }{c} y+a b^{1-\alpha } {\bar{x}} y \\&\quad =-\frac{r^\alpha }{K}(x-{\bar{x}})^2-y\left[ \frac{d^\alpha +q_2^\alpha e_2^\alpha }{c}-a b^{1-\alpha } {\bar{x}}\right] . \end{aligned}$$

Hence, ${}^{C} D^\alpha _t V(x, y)<0$ if $d^\alpha +q_2^\alpha e_2^\alpha \geqslant c a b^{1-\alpha } {\bar{x}}$. Under this condition, the predator extinction equilibrium is globally asymptotically stable. Observe that, if $d^\alpha +q_2^\alpha e_2^\alpha \geqslant c a b^{1-\alpha } {\bar{x}}$, then $d^\alpha +q_2^\alpha e_2^\alpha > \frac{c a b^{1-\alpha } {\bar{x}}}{1+ab{\bar{x}}}$, which is the condition for local stability. However, vice versa is not true. Which indicates that when predator extinction equilibrium is globally stable, it is also locally stable and not the other way around. This suggests that the most significant steady state, the co-existing equilibrium $E^*$, does not come into existence when $E^1$ is globally stable.

4.2.3 Global stability of the co-extincting equilibrium

For the co-existing equilibrium, we consider the following function as

$$\begin{aligned} V(x, y)= & {} \left[ x-x^*-x^* \ln \left( \frac{x}{x^*}\right) \right] \\{} & {} \quad +s\left[ y-y^*-y^* \ln \left( \frac{y}{y^*}\right) \right] . \end{aligned}$$

Note that

$$\begin{aligned} V(x^*,y^*)\left\{ \begin{aligned}&= 0 \, \text { if } (x,y)=(x^*,y^*)\\&> 0 \, \text { if } (x,y) \ne (x^*,y^*). \end{aligned} \right. \end{aligned}$$

Thus V(x, y) is positive definite in ${\mathbb {R}}^2_+$. Now,

$$\begin{aligned} \begin{aligned} {}^{C} D^\alpha _t V&\le \left( 1-\frac{x^*}{x}\right) {}^{C}D^\alpha _t x(t)+ s \left( 1-\frac{y^*}{y}\right) {}^{C}D^\alpha _t y(t)\\&=\left( x-x^*\right) \left[ r^\alpha \left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } y}{1+a b x}-q_{1}^\alpha e_{1}^\alpha \right] \\&\quad + s\left( y-y^*\right) \left[ \frac{c a b^{1-\alpha } x}{1+a b x}-d^\alpha -q_{2}^\alpha e_{2}^\alpha \right] . \\&=\left( x-x^*\right) \left[ r^\alpha \left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } y}{1+a b x}\right. \\&\quad \left. -r^\alpha \left( 1-\frac{x^*}{K}\right) +\frac{a b^{1-\alpha } y^*}{1+a b x^*}\right] \\&\quad + s \left( y-y^*\right) \left[ \frac{c a b^{1-\alpha } x}{1+a b x}-\frac{c a b^{1-\alpha } x^*}{1+a b x^*}\right] \\ \end{aligned} \end{aligned}$$

Here we use the fact that $(x^*,y^*)$ satisfies the system (3), and hence we obtain

$$\begin{aligned} {}^{C} D^\alpha _t V&\leqslant \left( x-x^*\right) \left[ -\frac{r^\alpha }{K}\left( x-x^*\right) +ab^{1-\alpha }\right. \\&\quad \left. \times \frac{y^*+a b x y^*-y-a b x^* y}{(1+a b x)\left( 1+a b x^*\right) }\right] \\&\quad +s \left( y-y^*\right) c a b^{1-\alpha }\\&\quad \times \left[ \frac{x+abxx^*-x^*-abxx^*}{(1+a b x)\left( 1+a b x^*\right) }\right] \\&=-\frac{r^\alpha }{K}\left( x-x^*\right) ^2-a b^{1-\alpha }\\&\quad \times \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }+\left( x-x^*\right) \\&\quad \times \frac{a^2 b^{2-\alpha }\left( x y^*-x^* y^*+x^* y^*-x^* y\right) }{(1+a b x)\left( 1+a b x^*\right) } \\&\quad +s c a b^{1-\alpha } \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&=-\frac{r^\alpha }{K}\left( x-x^*\right) ^2-a b^{1-\alpha } \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\quad +\frac{a^2 b^{2-\alpha } y^*\left( x-x^*\right) ^2}{(1+a b x)\left( 1+a b x^*\right) } \\&\quad -\frac{a^2 b^{2-\alpha } x^*\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\quad +\frac{s c a b^{1-\alpha }\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&=-\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a^2 b^{2-\alpha } y^*}{\left( 1+a b x^*\right) (1+a b x)}\right] \\&\quad +a b^{1-\alpha }\left[ s c-1-a b x^*\right] \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) } \\&=-\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a^2 b^{2-\alpha } y^*}{\left( 1+a b x^*\right) (1+a b x)}\right] ; \\&\text{ taking } s=\frac{1}{c}\left( 1+a b x^*\right)>0 \\&=-\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a b}{(1+a b x)}\right. \\&\quad \left. \times \left( r^\alpha \left( 1-\frac{x^*}{K}\right) -q_{1}^\alpha e_{1}^\alpha \right) \right] \\&\leqslant -\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}{-}a b\left( r^\alpha \left( 1{-}\frac{x^*}{K}\right) {-}q_{1}^\alpha e_{1}^\alpha \right) \right] \\&< 0 \, \, \text{ if } \, \, \frac{r^\alpha }{K} > a b\left( r^\alpha \left( 1-\frac{x^*}{K}\right) -q_{1}^\alpha e_{1}^\alpha \right) . \end{aligned}$$

Thus, ${}^{C} D^\alpha V(x,y) < 0$ for $\frac{r^\alpha }{K} > a b\left( r^\alpha \left( 1-\frac{x^*}{K}\right) \right.$$\left. -q_{1}^\alpha e_{1}^\alpha \right)$ and thus the co-existing equilibrium point $E^*$ is globally asymptotically stable under the above mentioned condition. Notably, we have obtained the identical condition with another Lyapunov function, containing a square function instead of a logarithm one (see Appendix).

In this section, we have already determined the sufficient conditions for local and global stability of the equilibria. Also, we have explained how the existence of an equilibrium influences the stability of another one. Similarly, we have analytically shown how the instability of the boundary equilibrium could stabilize the co-existing equilibrium. However, in the full parameter space, we fail to visualize the correlation between the local and global stability of the most significant equilibrium - the positive equilibrium. We shall graphically demonstrate the conditions for the existence, global and local stability regions of the co-existing equilibrium in, for example, Kr-plane. In Fig. 1, blue and green regions correspond to the non-existence and instability of co-existing equilibrium, respectively. While the yellow region represents the global stability (as well as local stability) of the co-existing equilibrium, the green region is the local stability region. One point to note here is that we can guarantee global stability in the yellow region through our choice of Lyapunov function. However, we cannot claim that the positive equilibrium is not globally stable in the green region. It would be interesting to know if, using some other Lyapunov function, we can obtain an extended global stability region embedded in the local stability region.

Fig. 1

Regions in the Kr-plane: blue region represents non-existence of coexisting equilibrium, yellow region represents global stability zone, green region corresponds to local stability, and red region indicates instability of the interior equilibrium. Stability regions for various $\alpha$ values: a $\alpha =0.9$, b $\alpha =0.75$, and c $\alpha =0.6$. Other parameters for simulation: $a=1,\, b=0.5,\, c=0.8,\, d=0.2,\, q_1=1,\, e_1=0.4,\, q_2=1,\, e_2=0.4$. (Color figure online)

Figure 1a shows that the interior equilibrium does not exist for lower values of r and K. For a fixed r (for instance, $r=4$), as K increases, the coexisting equilibrium comes into existence through a transcritical bifurcation with predator extinction equilibrium, and it becomes globally stable. For further increase in K, the system loses stability. To elaborate the impact of $\alpha$ on the regions, we performed simulation for $\alpha =0.9, \, 0.75, \, 0.6$, which is presented in Fig. 1. If we choose $(K,r)=(10,3)$, the equilibrium is in unstable state for $\alpha =0.9$ in Fig. 1a, while with $\alpha =0.75$ the equilibrium becomes locally stable in Fig. 1b. For $\alpha =0.6$ the equilibrium does not exist as in Fig. 1c. An interesting observation from Fig. 1 is that as the order of derivative $\alpha$ decreases, the stability region shifts to the right and the size of the region increases significantly. To be specific, we can visualize the increase in the area of the local stability region as well as the region indicating non-existence. This clearly shows the stabilizing effect of $\alpha$ and the shift of equilibrium points. Another interesting and contrary effect is that the global stability region remains almost the same.

4.3 Hopf bifurcation

  In the field of dynamical systems, the phenomenon in which a stable equilibrium point undergoes a loss of stability, giving rise to a periodic orbit and the system exhibits oscillatory behaviour due to a continuous change in a parameter, is known as a Hopf bifurcation. The parameter responsible for this transformation is called the bifurcating parameter, and the critical value at which this transition occurs is the bifurcation threshold. Now we will present the theoretical conditions that lead to the Hopf-bifurcation in a fractional order system (see [27]).

Consider the following two-dimensional fractional-order system:

$$\begin{aligned} { }^C D^\alpha _t {\tilde{x}}=f({\tilde{x}},p), \, \end{aligned}$$

where $\alpha \in (0,1), \, {\tilde{x}} \in {\mathbb {R}}^2$, and p is a model parameter.

Suppose $E^*$ is an equilibrium of this system, and Df be the usual Jacobian matrix (D represents the ordinary derivative). A Hopf bifurcation at $E^*$ occurs for $p=p^*$ ($p^*$ is a threshold value) if,

• i. The Jacobian matrix has two complex-conjugate eigenvalues $\lambda _{1,2}(p^*)=\zeta (p^*) \pm \iota \omega (p^*)$, where $\zeta (p^*)>0$,

• ii. $\theta _{1,2}\left( \alpha , p^*\right) =0$,

• iii. $\displaystyle {\left. \frac{d \theta _{1,2}}{d p}\right| _{p=p^*}} \ne 0$, where

  $$\begin{aligned} \theta _i(\alpha , p)=\frac{\alpha \pi }{2}-\left| {\text {Arg}} \left( \lambda _i(p)\right) \right| , \, i=1,2. \end{aligned}$$

An interesting point here is that the Matignon stability and Hopf bifurcation conditions are the same. Hence, the separatrix between the green and red region is the Hopf-bifurcation curve. In the subsequent sections, we will examine the presence of Hopf bifurcation in the fractional-order system (5) for various system parameters using the above-mentioned conditions.

5 Paradox of enrichment

  Generally, when prey species are supplied with more nutrients, the ecosystem is expected to be more healthy (in terms of biomass enhancement). From a mathematical modelling perspective, the increasing carrying capacity is equivalent to the supply of more nutrients, a larger habitat, and a more favourable environment for the species’ survival. The Rosenzweig–MacArthur (RM) model [48] is one of the most significant predator–prey systems describing population dynamics. Rosenzweig [47] studied six reasonably viable two-species models of tropic exploitation considered functional responses in exponential forms, [23], and power functions [11]. Analysing each model, he showed that sufficient nutrient supply causes oscillations and species extinction, and concluded that “increasing the supply of limiting nutrients or energy tends to destroy the steady state”. He termed this counter-intuitive ecological phenomenon, beyond our traditional observations, as the paradox of enrichment. Further investigation with limit cycle behaviour in such a model was analysed extensively by May [31] and Gilpin [17]. The Hastings–Powell (HP) food-chain model [20] is the extension of the RM system with three species. The presence of chaos is established therein, and later Abrams and Roth [1] observed the existence of paradox of enrichment in the same model due to the bottom-up force (sufficient nutrient supply to prey).

Notably, Rana et al. [44] presented a fractional order RM model, and they demonstrated that whenever prey enrichment (by increasing carrying capacity) leads to non-equilibrium dynamics. The paradox of enrichment, it is possible to control the oscillations by adjusting the order of model derivative ($\alpha$). In our view, the order of the derivative of an ecological system should be invariant before starting any analysis, with the other ecological parameters. However, we can only change the control parameters, such as carrying capacity K (by nutrient supply), effect of which we are interested to study. We will examine the impact of increasing carrying capacity in the system (5) to verify whether the paradox of enrichment will continue to exist or it can be controlled.

Fig. 2

Eigenvalue plot with varying K, where blue and black represents the real and imaginary part of eigenvalues respectively. (Color figure online)

To begin our analysis, we assess the system’s eigenvalues of the Jacobian matrix at the co-existing equilibrium point. We consider a parameter set without harvesting as $\alpha = 0.85, r = 1.5, a = 0.9, b = 0.6, c = 0.3, d = 0.3$. Using the conditions for the existence of the co-existing equilibrium $E^*=\left( x^*, y^*\right)$, it exists through a transcritical bifurcation at $K = 6.41525$ and becomes stable. We obtain the expression of eigenvalues in terms of carrying capacity K as

$$\begin{aligned} \lambda _{1,2} = \frac{-8.04089+0.547656 K \pm 0.431625 \sqrt{-32.7721+K} \sqrt{-10.5899+K}}{K}, \end{aligned}$$

The expression mentioned above clearly suggests that the eigenvalues are negative up to $K \approx 10.5899$, and the system is stable (as we can see the same in Fig. 2). Implementing the Matignon condition, we obtain that $K = 16.552$ is the critical value, crossing which the co-existence steady state loses its stability and emerges a closed curve. When the carrying capacity passes through the critical value $K = 32.7721$, the complex conjugates with positive real parts merge to a set of positive eigenvalues. Observing the expression of the eigenvalues again, we understand that the eigenvalues will remain positive for further increase in K and the system retains the closed curve revolution behaviour in the system (5). Hence, a significant eigenvalue behaviour here to observe is that from $K=7 \text { to } K=35$, the system completes an entire circle in eigenvalue space starting from the negative real line through complex conjugates in the left and right half plane; it finally reaches the positive real axis. Figure 2 represents the same: from the variation of eigenvalues with increasing K, we can easily see that before $K = 10.5899$ eigenvalues were negative, after that imaginary part emerges (in the boxed region) and after $K = 32.7721$, again eigenvalues merge to become positive.

Rosenzweig [47] showed the existence of cyclic behaviour (limit cycle) in a class of the predator–prey systems for increasing nutrient supply to prey species. Further, he reported that the species go to extinct due to sufficient nutruent supply. Keeping extinction in mind, he stated, “Man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in great abundance.” In the mean time, May [31] also obtained oscillatory behaviour (limit cycle), but he claimed that extinction is not possible in Rosenzweig [47]’ systems. Destabilization (oscillatons) of a predator–prey system due to prey species enrichment is popularly known as “paradox of enrichment” according to Rosenzweig [47] and May [31]. Rana et al. [44] seems to be the first to explore the paradox of enrichment (destabilization) in a dimensionally non-homogeneous fractional-order predator–prey system. In our homogeneous predator–prey model, we will increase the carrying capacity to establish new principle while enriching the bottom species.

Fig. 3

Time series depicting the impact of varying parameter K on oscillation behavior. a For $K=17$, oscillation is observed. b At $K=18$, amplitude of the oscillation increases noticeably. c When $K=19$, the amplitude is highest among the three

The impact of increment in carrying capacity is shown in Fig. 3, which depicts oscillatory behaviour after K crossing the bifurcation threshold ($K = 16.552$). The destabilization happens with the nutrient enrichment of prey, which is the paradox of enrichment. In the three cases shown in Fig. 3a–c, the oscillation amplitude rise, and the minimum population biomass during the oscillations decreases with increasing K. One may be curious to explore if the oscillation amplitude continues to grow indefinitely, and if the minimum population biomass can reach zero level for sufficient enrichment of K.

Fig. 4

Plot for varying K, where red and black dots represent the equilibrium and mean biomass, respectively. The blue lines depict the maximum and minimum of a prey and b predator. For a stable steady state, a single line, and two branches come out for oscillation. (Color figure online)

Since we have considered a specialist predator–prey systems with finite carrying capacity to prey, extinction (if possible) would occur to predator species only. We have shown (Fig. 3) that populations do not extinct for limited carrying capacities using time series. This observation holds true even for larger K values, because in Sect. 3.4 we have highlighted that the solutions when initiated with positive initial conditions, remains positive. Consequently, while the minimum population size decreases, it never reaches zero. So, our result is well coherent with May’s observation even in the fractional order system. Further one may be interested to know if the unstable equilibrium could be stabilized by increasing K sufficiently. Connecting the eigenvalue analysis presented above, we know that for sufficient increment in K, the set of eigenvalues of the Jacobian matrix at the co-existing equilibrium remains positive. As a result, the system continues to exhibit oscillatory behavior.

As the prey species is enriched, we can expect a healthy ecosystem in terms of higher abundance of both species. If this is the case, then enrichment is beneficial in order to achieve higher food yield. We now estimate population stock for increasing K. When trajectories converge towards an equilibrium (stable equilibrium), predicting population size over an extended period becomes relatively straightforward. However, in the case of non-equilibrium dynamics, assessing the stock level is challenging. In such instances, a practical approach is to consider a time-averaged stock as a reasonable estimate for quantifying the population level.

In this context, we will address an essential concept of mean density [43, 52] in non-equilibrium mode. Mean density is the average population biomass over a certain period, vital for analyzing population dynamics, sustainability as it provides an average measure of distribution of population over a long time.

Consider a continuous system,

$$\begin{aligned} \frac{\textrm{d} X}{\textrm{d} t}=f(X,p) \text{, } \end{aligned}$$

where $f: D \subseteq {\mathbb {R}}^m \times {\mathbb {R}} \rightarrow {\mathbb {R}}^m$, and p is a model parameter. If X(t) be the parameter-dependent solution of the system with initial condition $X(0)=X_0$, then the mean population is defined as

$$\begin{aligned} \phi \left( X_0,p\right) =\lim _{t \rightarrow \infty } \frac{1}{t} \int _0^t X(s) \textrm{d} s. \end{aligned}$$

Here $\phi : M \subseteq {\mathbb {R}}^m \times {\mathbb {R}} \rightarrow {\mathbb {R}}^m$, and M is the subset of initial conditions for which the limit exists.

The time-average population for the Lotka–Volterra model consistently equals its equilibrium, which is not true for the Rosenzweig–MacArthur model in cyclic mode. As per the findings in Sieber and Hilker [52], they highlighted that a boost in carrying capacity can be advantageous for the overall ecological community in ODE-based Gause-type models. Can we expect similar result in fractional cases? For the first time, we will address it for fractional order population model.

We perform numerical experiments to estimate the mean stock for both species for increasing carrying capacity. The black dots show the mean density, and the red dots indicate the unstable equilibrium in Fig. 4. Just after the bifurcation threshold, the equilibrium gets destabilized, and the mean stock of the predator population decreases (as we can see in the zoomed part in Fig. 4b), while the prey mean stock increases. So, in terms of predator’s stock decline, we can refer to it as a paradox of enrichment, which is a new observation. Interestingly, after sufficiently increasing K, the mean density increases again for predators and remains increasing. The increment even surpasses the unstable equilibrium after sufficiently increasing K. The prey mean density kept increasing from the bifurcation point. Hence, ultimately, the mean population size of both species is beneficial for increasing carrying capacity, and we should not call it a paradox of enrichment in the context of mean population stock. Although in the view of Rosenzweig [47], destabilization of the system is often coined as the paradox of enrichment, one should, in our opinion, should redefine paradox of enrichment in terms of mean population size. To the best of our knowledge, this mean stock pattern has never been reported earlier even in the classical RM model. We have presented a detailed information on the paradox of enrichment in terms of mean density for the classical RM model in Appendix B. 8 A comparison of this paradoxical phenomenon among these two models is also drawn therein.

6 Harvesting results

  Harvesting has a significant impact on population dynamics. A system may exhibit several complex behaviours based on harvest rate and strategy. One can maximise sustainability yield with minimal effort through the harvesting approach, it can stabilise (destabilize) an unstable (a stable) equilibrium (via Hopf bifurcation into the system). While harvesting prey can cause predator extinction in a model [26]. In some specific models, it can also introduce hydra effects while harvesting prey [38] and predator [37]. Harvesting in a tri-trophic food chain model can induced stability in a chaotic system [36].

We have seen that prey enrichment causes destabilization and shows paradox of enrichment. Can population harvesting has potential to stabilize (destabilize) an unstable (stable) equilibrium, or after destabilization if the coexisting equilibrium, could harvesting reestablish stability (where carrying capacity cannot reinstall stability)? We’ll also explore if harvesting could establish any paradoxical result such as hydra effect.

6.1 Prey harvesting

Here our focus would be to check the possibility of Hopf-bifurcation and hydra effects under prey harvesting. The coexisting equilibrium under prey harvesting (i.e., $e_2=0$, $q_2=0$) satisfies

$$\begin{aligned} \begin{aligned}&r^\alpha \left( 1-\frac{x^*}{K}\right) -\frac{a b^{1-\alpha } y^*}{1+a b x^*} - (q_1 e_1)^\alpha =0, \\&\quad c \frac{a b^{1-\alpha } x^*}{1+a b x^*}-d^\alpha =0. \end{aligned} \end{aligned}$$

(11)

Fig. 5

Variation of population w.r.t. $e_1$. Mean density (black dots) and unstable equilibrium (red dots) are shown. Biomass (blue line) of a prey, and b predator w.r.t. $e_1$. Unstable equilibrium, mean density, and species biomass are shown in red, black, and blue color, respectively. Parameters are taken as $\alpha = 0.85, \, r = 0.5, \, K=11, \, a = 0.9, \, b = 1.1, \, c = 0.7, \, d = 0.25, \, q_1=1.$. (Color figure online)

Differentiating both sides of the system (11) with respect to $e_1$ we obtain

$$\begin{aligned} \begin{aligned} a_{11} \frac{\textrm{d} x^*}{\textrm{d} e_1}+a_{12} \frac{\textrm{d} y^*}{\textrm{d} e_1}&=\alpha q_1^\alpha e_1^{1-\alpha } \\ a_{21} \frac{\textrm{d} x^*}{\textrm{d} e_1}+a_{22} \frac{\textrm{d} y^*}{\textrm{d} e_1}&=0, \end{aligned} \end{aligned}$$

where

$$\begin{aligned}{} & {} \left( \begin{array}{cc} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \end{array}\right) =\left( \begin{array}{cc} -\frac{r^\alpha }{K}+\frac{a^2 b^{2-\alpha } y^*}{(1+a b x^*)^2} &{} -\frac{a b^{1-\alpha }}{1+a b x^*} \\ \frac{a b^{1-\alpha } c}{(1+a b x^*)^2} &{} 0 \end{array}\right) . \end{aligned}$$

Solution of the above linear system of equations, by elimination method, reveals

$$\begin{aligned} \frac{\textrm{d} x^*}{\textrm{d} e_1}= & {} \frac{a_{22} \alpha q_1^\alpha e_1^{1-\alpha }}{a_{11} a_{22}-a_{12} a_{21}} \text { and}\\ \, \frac{\textrm{d} y^*}{\textrm{d} e_1}= & {} -\frac{a_{21} \alpha q_1^\alpha e_1^{1-\alpha }}{a_{11} a_{22}-a_{12} a_{21}}. \end{aligned}$$

We note that $a_{22}=0, \, \text {and } a_{21}<0$. Consequently, $a_{11} a_{22}-a_{12} a_{21}=\displaystyle {\frac{ca^2 b^{2-2\alpha }}{(1+a b x^*)^3}}>0$. Hence,

$$\begin{aligned} \frac{\textrm{d} x^*}{\textrm{d} e_1}=0 \text{ and } \frac{\textrm{d} y^*}{\textrm{d} e_1}<0. \end{aligned}$$

The findings indicate that an increase in prey harvesting rates does not result any change in prey biomass at the stable state. Instead, the harvesting solely impacts the predator biomass, causing a decrease in biomass. Since, the prey equilibrium does not increase, no hydra effect could occur at the stable state. Now consider the unharvested system in an unstable mode. We will examine whether harvesting can stabilize the unstable equilibrium, and the mean stock will remain the same at the equilibrium, or even it can increase (decrease) at the non-equilibrium dynamics.

When the unharvested system oscillates, increasing prey harvesting effort stabilized the system through a Hopf bifurcation (Fig. 5). The blue lines represent the maximum and minimum population size in the case of oscillatory behaviour, and the minimum and maximum merge at the stable steady state. Hence, prey harvesting has a stabilizing effect on the system (5). After the bifurcation threshold, the system remains stable for further increments of harvesting effort. We have computed the mean population stock (indicated by black dots) when the system is in unstable equilibrium (red dots). The mean population stock of both species is greater than its unstable equilibrium, and it decreases with increasing harvesting effort. Most importantly, although prey is harvested, equilibrium remains constant, but it is more likely that prey will be affected. The effect is realized through the decline of the mean stock. Hence, there is no hydra effect in neither stable nor in unstable mode.

Fig. 6

Phase portrait for a $e_2=0$, b $e_2=0.1$, c $e_2=0.2$, and d $e_2=0.3$. Blue stars represent the initial conditions; black stars depict the equilibrium points. For a and d, the interior equilibrium is stable; for b and c, the equilibrium is unstable. Blue and orange trajectories depict phase portraits of two different initial conditions, one outside and one inside of the stable oscillation (i.e., the black closed orbit). Parameters are taken as $\alpha = 0.85, \, r = 0.5, \, K=11, \, a = 0.9, \, b = 1.1, \, c = 0.7, \, d = 0.1, \, \text {and} \, q_2=1.$. (Color figure online)

Fig. 7

Variation of population w.r.t. $e_2$. Mean density (black dots) and unstable equilibrium (red dots) are shown. Biomass (blue line) of a prey, and b predator w.r.t. $e_2$. Unstable equilibrium, mean density, and species biomass are shown in red, black, and blue color, respectively. Parameter set is same as Fig. 6. (Color figure online)

6.2 Predator harvesting

In the classic RM predator–prey model, it is analytically established that predator harvesting has the potential to stabilize an unstable equilibrium, and a stable steady state remains stable for further increase of effort [15]. Furthermore, in the classic RM model, it is proven that hydra effects are not possible to occur at a stable state [37] for predator harvesting, but existence of it is established in terms of mean density while the system is in cyclic mode [52]. In fractional order modelling, detecting a hydra effect is yet to be uncovered. This subsection will explore the potential consequences of implementing predator harvesting with the following questions in mind:

• i. Wheather predator harvesting has a potential to destabilize a stable equilibrium?

• ii. Could multiple Hopf bifurcations occur under predator harvesting?

• iii. Are hydra effects possible at the stable state under predator harvesting?

The coexisting equilibrium under predator harvesting (i.e., $e_1=0, \, q_1=0$) satisfies

$$\begin{aligned} \begin{aligned} r^\alpha \left( 1-\frac{x^*}{K}\right) -\frac{a b^{1-\alpha } y^*}{1+a b x^*}&=0, \\ c \frac{a b^{1-\alpha } x^*}{1+a b x^*}-d^\alpha - (q_2 e_2)^\alpha&=0. \end{aligned} \end{aligned}$$

(12)

Differentiating both sides of the system (12) with respect to $e_2$ we obtain

$$\begin{aligned} \begin{aligned}&a_{11} \frac{\textrm{d} x^*}{\textrm{d} e_2}+a_{12} \frac{\textrm{d} y^*}{\textrm{d} e_2}= 0, \\&a_{21} \frac{\textrm{d} x^*}{\textrm{d} e_2}+a_{22} \frac{\textrm{d} y^*}{\textrm{d} e_2}=\alpha q_2^\alpha e_2^{1-\alpha }. \end{aligned} \end{aligned}$$

Solving the above linear system of equations we obtain

$$\begin{aligned} \frac{\textrm{d} x^*}{\textrm{d} e_2}=-\frac{a_{12} \alpha q_2^\alpha e_2^{1-\alpha }}{a_{11} a_{22}-a_{12} a_{21}} \text { and} \, \frac{\textrm{d} y^*}{\textrm{d} e_2}=-\frac{a_{11} \alpha q_2^\alpha e_2^{1-\alpha }}{a_{11} a_{22}-a_{12} a_{21}}. \end{aligned}$$

Since $a_{12}<0$, we obtain $\displaystyle {\frac{\textrm{d} x^*}{\textrm{d} e_2}>0}$. Now

$$\begin{aligned} \frac{\textrm{d} y^*}{\textrm{d} e_2}>0&\text { if } \frac{r^\alpha }{K} > \frac{a^2 b^{2-\alpha } y^*}{(1+a b x^*)^2}, \end{aligned}$$

(13a)

$$\begin{aligned} \frac{\textrm{d} y^*}{\textrm{d} e_2}<0&\text { if } \frac{r^\alpha }{K} < \frac{a^2 b^{2-\alpha } y^*}{(1+a b x^*)^2}. \end{aligned}$$

(13b)

When we consider the rate of change of predator equilibrium w.r.t. harvesting effort, the condition (13a) suggests that if the system parameter set satisfies it, the predator equilibrium stock (stable or unstable) will increase with increasing harvesting effort. Hence, we must investigate if (13a) can be satisfied for a certain parameter set.

Let us consider the unharvested system (i.e., $e_2=0$) in a stable steady state. In this case, the trajectory converges to the stable state spirally (Fig. 6a). For increasing harvesting effort, the equilibrium becomes an unstable focus, and oscillatory behaviour appears in phase portrait. We illustrate it for $e_2=0.1$ and $e_2=0.2$ (Fig. 6b, c). For further increase of harvesting effort, the co-existing equilibrium regains stability (Fig. 6d). Hence, for increasing harvesting effort, we can observe the change in the stability, at least twice, of the co-existing equilibrium, i.e., stable to unstable and unstable to stable. Now, we will provide a detailed analysis on how stability is changing in the interval $e_2 \in (0,0.3)$ through a bifurcation diagram. Also, we will define two important dynamical aspects, stability switching and bubble formation, along with some ecological phenomena.

Hopf-bifurcation refers to the smooth change in qualitative behaviour through oscillations at the equilibrium due to a parameter change. Figure 7a, b depict the Hopf bifurcation diagram of prey and predator population, respectively, for varying predator harvesting efforts. The blue lines represent the maximum and minimum population size of oscillations, and we plot the stable steady state in the same colour (Fig. 7). Unstable equilibrium (red dots) and corresponding mean population size (black dots) are shown in the same figure window. The stable equilibrium becomes unstable under predator harvesting at $e_2=0.05$ through a Hopf bifurcation (HB1), and equilibrium gains stability at superior effort $e_2=0.262$ through a second Hopf bifurcation (HB2).

Interestingly, both the Hopf bifurcations are supercritical in nature because whenever we take an initial condition outside or inside of the limit cycle (black closed curve in Fig. 6b, c), the trajectories comes back towards the limit cycle. Hence, from the bifurcation diagram (Fig. 7), we observed that the equilibrium was stable first in the effort interval $e_2 \in (0,0.05)$, then shows stable oscillation in the effort interval $e_2 \in (0.05,0.262)$, and finally got stabilized for further increment of harvesting effort. Hence, equilibrium stability changes from stable to unstable, and then to stable again for changing harvesting efforts. In delay-induced predator–prey systems, such stability changes and a series of stability and instability changes can occur [54] due to time delay. This is referred to as stability switching. In our paper, due to the harvesting, we can obtain stability switching, which didn’t occur in the classical RM model. In the Gause map, a period-doubling bifurcation followed by a period halving bifurcation resulted in the formation of a bubble-like structure ([21], pp-195). In our model as well, we found this structure on the bifurcation diagram (Fig. 7a–b). This is possible because of the two supercritical Hopf bifurcations, with the first (HB1) leading to system destabilization and the second (HB2) resulting in system stability, and together they give rise to a bubble-like structure.

Now, we examine the existence of hydra effects in the fractional order RM model. Prey biomass at equilibrium increases at a stable state, a well coherent harvesting principle observed in the ODE RM model. On the other hand, predator biomass increases at a stable state on the intervals $e_2 \in (0,0.05)$ and $e_2 \in (0.262,0.32)$. Hence, hydra effects appear at a stable state on the predator, which never occurs in the RM system. Further, we estimated the mean population size while the equilibrium is unstable. It is found that the mean population of predators (for prey, too) increases within the bubble structure. Hence, the hydra effect appears when the fractional order RM model is in oscillatory mode.

Additionally, the discussion of stability switching in the context of the continuous fractional order Rosenzweig–MacArthur model has not been explored yet. This unique behaviour is attributed to the presence of the memory effect or memory parameter. In contrast, stability switching is lacking when the memory is absent, as evidenced by our experiments with the same parameter sets. Hence, we conclude that while carrying capacity and prey harvesting have destabilizing and stabilizing effects in the system, respectively, predator harvesting can induce both effects.

7 Conclusions

  This investigation delves into the dynamic analysis and potential outcomes of a fractional order Rosenzweig–MacArthur (RM) predator–prey model incorporating harvesting, utilising the Caputo fractional derivative for dimensional consistency. Identifying a gap in existing literature concerning the region in the state space for solution existence, excluding singularities, we first established the existence of solutions in a suitable region before systematically exploring qualitative features such as positivity and boundedness.

Notably, due to dimensional homogeneity, equilibrium points vary with the derivative order $\alpha$, significantly impacting the stability and Hopf bifurcation thresholds. Employing the Matignon stability condition, we explored local stability criteria, primarily focusing on the co-existence equilibrium. We discovered that the extinction equilibrium’s local and global stability regions are identical, whereas other equilibria exhibit local stability regions as supersets of their corresponding global stability regions. We also examined how stability regions change with the derivative order $\alpha$.

We extensively explored the impact of carrying capacity. We found that increased carrying capacity destabilises the stable co-existing equilibrium, leading to increased oscillation amplitude for further increments of K. This destabilisation is called the paradox of enrichment in general. However, from a mean density perspective, a larger carrying capacity maintains a healthier ecosystem, suggesting that prey enrichment does not show paradoxical results. This is a new perspective on the paradox of enrichment.

Further, prey harvesting was found to stabilise the system without causing any stability switching and hydra effect. Predator harvesting revealed novel results, such as two Hopf bifurcations in our fractional RM model at different effort levels and the emergence of a bubble-like structure. The hydra effect, completely absent in the stable state in the classical RM model, was observed in the stable state on two different effort intervals in our fractional order model. Multiple Hopf bifurcations and stability switching were possible only in predator harvesting. These findings, which diverge from classical RM model dynamics, emphasise the importance of investigating fractional differential equation models, particularly the RM model, to deepen our understanding of ecological systems.

We have mostly uncovered several intricate dynamics of the population model. However, we could draw a few implications of this theoretical study in terms of ecosystem conservation and management. One could appropriately deploy nutrients to the bottom species in ecosystems to enhance the abundance of top trophic levels, which could possibly meet the demand for human food. Generally, a stable ecological community, including the fishery, produces predictable, steady yields, and hence, our study could guide the fishery industry in maintaining stock with suitable fishing efforts. While hydra effects appear, fishermen can obtain relatively higher harvested yield with smaller costs (a smaller number of boats, nets, etc.).

8 Future perspective

  Despite our findings, there are some questions yet to be answered as future perspectives.

• (i) We have established the positivity of the solutions through the boundedness. Hence, an independent proof of positivity would still be noteworthy.

• (ii) We found that the global stability regions, in two-parameter space, using two different Lyapunov functions remain the same. It would be interesting to investigate if we can analytically determine a larger region of global stability for the interior equilibrium or if local and global stability regions overlap with each other, with a different Lyapunov function.

• (iii) The Bazykin predator–prey model [2] is the extended version of the classical RM model where predator process intraspecific competition. The predator dynamics includes an additional term $-\gamma y^2$, where $\gamma$ is called intraspecific competition with dimension $\text {[Mass}^{-1}]\text {[Time}^{-1}]$. Therefore, it is now challenging to make the fractional order (Caputo derivative) Bazykin predator–prey model dimensionally consistent.

• (iv) Fractional order RM model revealed several new insights while the predator is harvested. We expect some new results from the three-species well-known Hastings–Powell model [9] in its fractional framework. This is our immediate future plan.

Appendix

1.1 Appendix A: Global stability with another Lyapunov function

For the co-existing equilibrium, we consider the following function as

$$\begin{aligned}V(x, y)=\left( x-x^*\right) ^2+ s \left( y-y^*\right) ^2.\end{aligned}$$

Note that

$$\begin{aligned} V(x^*,y^*)\left\{ \begin{aligned}&= 0 \, \text { if } (x,y)=(x^*,y^*)\\&> 0 \, \text { if } (x,y) \ne (x^*,y^*). \end{aligned} \right. \end{aligned}$$

Thus V(x, y) is positive definite in ${\mathbb {R}}^2_+$. Now,

$$\begin{aligned} \begin{aligned} {}^{C} D^\alpha _t V \le&\, 2x\left( x-x^*\right) {}^{C}D^\alpha _t x(t)+ 2y\left( y-y^*\right) {}^{C}D^\alpha _t y(t)\\ =&\, 2x \left( x-x^*\right) \left[ r^\alpha \left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } y}{1+a b x}-q_{1}^\alpha e_{1}^\alpha \right] \\&\quad + 2 s y \left( y-y^*\right) \left[ \frac{c a b^{1-\alpha } x}{1+a b x}-d^\alpha -q_{2}^\alpha e_{2}^\alpha \right] \\ =&\, 2x \left( x-x^*\right) \left[ r^\alpha \left( 1-\frac{x}{K}\right) -\frac{a b^{1-\alpha } y}{1+a b x}\right. \\&\quad \left. -r^\alpha \left( 1-\frac{x^*}{K}\right) +\frac{a b^{1-\alpha } y^*}{1+a b x^*}\right] \\&\quad + 2 s y \left( y-y^*\right) \left[ \frac{c a b^{1-\alpha } x}{1+a b x}-\frac{c a b^{1-\alpha } x^*}{1+a b x^*}\right] \\ \end{aligned} \end{aligned}$$

Here we use the fact that $(x^*,y^*)$ satisfies the system (3), and hence we obtain

$$\begin{aligned} {}^{C} D^\alpha _t V&\leqslant \, 2x\left( x-x^*\right) \left[ -\frac{r^\alpha }{K}\left( x-x^*\right) \right. \\&\quad \left. +ab^{1-\alpha } \frac{y^*+a b x y^*-y-a b x^* y}{(1+a b x)\left( 1+a b x^*\right) }\right] \\&\quad +2sy \left( y-y^*\right) c a b^{1-\alpha }\\&\quad \times \left[ \frac{x+abxx^*-x^*-abxx^*}{(1+a b x)\left( 1+a b x^*\right) }\right] \\&= -2x\left( x-x^*\right) ^2\frac{r^\alpha }{K} -2xa b^{1-\alpha }\\&\quad \times \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) } +2x\left( x-x^*\right) \\&\quad \times \frac{a^2 b^{2-\alpha }\left( x y^*-x^* y^*+x^* y^*-x^* y\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\quad + 2ys c a b^{1-\alpha } \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) } \\&= -2x\left( x-x^*\right) ^2\frac{r^\alpha }{K} - 2x a b^{1-\alpha }\\&\quad \times \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\quad + 2x\frac{a^2 b^{2-\alpha } y^*\left( x-x^*\right) ^2}{(1+a b x)\left( 1+a b x^*\right) } \\&\quad -\frac{2a^2 b^{2-\alpha } xx^*\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\quad +2y\frac{s c a b^{1-\alpha }\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) }\\&\leqslant -2x\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a^2 b^{2-\alpha } y^*}{\left( 1+a b x^*\right) (1+a b x)}\right] \\&\quad +2 C a b^{1-\alpha }\left[ s c-1-a b x^*\right] \\&\quad \times \frac{\left( x-x^*\right) \left( y-y^*\right) }{(1+a b x)\left( 1+a b x^*\right) } \\&= -2x\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a^2 b^{2-\alpha } y^*}{\left( 1+a b x^*\right) (1+a b x)}\right] ;\\&\quad \text{ taking } s=\frac{1}{c}\left( 1+a b x^*\right)>0 \\&= -2x\left( x-x^*\right) ^2\left[ \frac{r^\alpha }{K}-\frac{a b}{(1+a b x)}\right. \\&\quad \left. \times \left( r^\alpha \left( 1-\frac{x^*}{K}\right) -q_{1}^\alpha e_{1}^\alpha \right) \right] \\&\leqslant -2x\left( x{-}x^*\right) ^2\left[ \frac{r^\alpha }{K}{-}a b\left( r^\alpha \left( 1{-}\frac{x^*}{K}\right) {-}q_{1}^\alpha e_{1}^\alpha \right) \right] \\&< \, 0 \, \, \text{ if } \, \, \frac{r^\alpha }{K} > a b\left( r^\alpha \left( 1-\frac{x^*}{K}\right) -q_{1}^\alpha e_{1}^\alpha \right) . \end{aligned}$$

Thus, ${}^{C} D^\alpha V(x,y) < 0$ for $\frac{r^\alpha }{K} > a b\left( r^\alpha \left( 1-\frac{x^*}{K}\right) \right.$$\left. -q_{1}^\alpha e_{1}^\alpha \right)$ and thus the co-existing equilibrium point $E^*$ is globally asymptotically stable under the above mentioned condition.

1.2 Appendix B: Paradox of enrichment for classical RM model

 

Fig. 8

Plot for varying K, where red and black dots represent the equilibrium and mean biomass, respectively. The blue dots depict the maximum and minimum of predator stock. (Color figure online)

We will discuss the paradox of enrichment in the classical RM model (ODE modelling framework). Therefore, we will consider the same parameter set as discussed in the fractional order modelling (Fig. 4b). For classical RM modelling, after destabilization, the mean density of the system decreases and becomes almost stagnant as indicated by black dots (Fig. 8) with increasing carrying capacity. On the other hand, for fractional-order systems, after destabilization, the mean density of the system first decreases and starts increasing after a sufficient increment of K (Fig. 4b). Hence, we can conclude that enriching prey is not beneficial for predator species in the classical RM model but is beneficial for the fractional RM model.