In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting is proposed and analysed. The interaction between prey, middle predator and top predator population is investigated. In order to clarify the characteristics of the proposed model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed model are examined. Some sufficient conditions that ensure the local and global stability of equilibrium points are obtained. By using stability analysis of the fractional-order system, it is proved that if the basic reproduction number

Mathematical analysis is one of the important tools for understanding and interpreting different interactions in the environment around us. The food chain model system is attractive to researchers in theoretical ecology because it helps to understand the relationships between populations and describe the behavior of the ecosystem. Hastings et al. [

The paper is arranged as follows: In Section 2, the mathematical model is described. Some preliminary results, such as existence, uniqueness, nonnegativity and boundedness are presented in Section 3. The local and global stability of equilibrium points of the fractional-order food chain model is analyzed in Section 4. With the help of Sotomayor’s theorem, the transcritical bifurcation of the proposed model is investigated in Section 5. Section 6 extends the deterministic fractional- order food chain model to the stochastic fractional-order model. In Section 7, some numerical simulations are presented to verify the obtained theoretical results. Finally, the conclusions are given in Section 8.

Recently, Nath et al. [

with initial values

In the above model

Introducing dimensionless

Making the above substitutions in the model (1). Then the system yields the following form

The dimensionless parameters in the food chain model (2) are defined as

Following [

where,

In this section, the existence and uniqueness of the solutions of the fractional-order system (3) are investigate in the region

for sufficiently large

For any

where

Hence,

The following results show the non-negativity of the solutions of the fractional-order system (3). According to [

where,

where

Hence all the solutions of fractional-order a tri-trophic food chain model (3) that start in

One can also prove that

The fractional-order system (3) has the following equilibrium points:

1)

2) The middle predator and top predator free equilibrium point

3) Following [

4) The top predator extinction equilibrium point

5) The top predator extinction equilibrium point

6) The coexistence positive equilibrium point

7) and

The locally and globally asymptotically stable of equilibrium points of fractional-order food chain model (3) are now investigated. The Jacobian matrix is given as follows:

The eigenvalues of

The eigenvalues of

The time derivative of

According to Theorem 1,

Choosing

According to generalized Lyapunov–Lasalle’s invariance principle [

Hence the equilibrium point

The global stability of the equilibrium point

when

The stability of top predator extinction equilibrium point

where

The above equation has the following eigenvalue

By calculating the time derivative of

Choosing

The stability of coexistence equilibrium point

where

where

If we choose

Following [

has a similar effect as the real part of the eigenvalue in the integer order system. If

By calculating the time derivative of

Choosing

In this section we will investigate the local bifurcation near the free predator equilibrium point of the food chain model (2) with the help of Sotomayor’s theorem [

The eigenvector corresponding to

then

Thus, according to Sotomayor’s theorem, the food chain model (2) has a transcritical bifurcation at

This section extends the deterministic fractional-order food chain model (3) to the following stochastic fractional-order model.

where

where

According to [

where

where

If

Now, the fractional-order stochastic Hastings-Pwoell model (11) in Grunwald-Letnikov sense can be written as

where,

In this part, the numerical simulations of the mathematical model (3) will be examined, and the focus will be on the effect of harvesting parameters. The numerical results will be compared with the theorems formulated in the previous sections. The interactions between prey, middle predator and the top predator will be simulated by the following parameters:

In order to show the effects of fractional derivative

For better understand the effect of the quadratic harvesting of the middle predator

The effect of prey harvesting rate

If we increase the value of middle predator death rate

In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting has been proposed. It is shown that the proposed model has bounded and non-negative solution as desired in any population dynamics. By using stability analysis of fractional-order system, we have proved that if the basic reproduction number