Prediction of dissolved oxygen concentration in aquaculture based on attention mechanism and combined neural network

Wenbo Yang; Wei Liu; Qun Gao; Wenbo Yang; Wei Liu; Qun Gao

doi:10.3934/mbe.2023046

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 998-1017. doi: 10.3934/mbe.2023046

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Research article

Prediction of dissolved oxygen concentration in aquaculture based on attention mechanism and combined neural network

1.
School of Information and Electronic Engineering, Shandong Technology and Business University, Yantai 264005, China
2.
Key Laboratory of Sensing Technology and Control in University of Shandong, Yantai 264005, China

Academic Editor: Weining Wu

Received: 30 June 2022 Revised: 25 August 2022 Accepted: 07 October 2022 Published: 21 October 2022

As an essential water quality parameter in aquaculture ponds, dissolved oxygen (DO) affects the growth and development of aquatic animals and their feeding and absorption. However, DO is easily influenced by external factors. It is not easy to make scientific and accurate predictions of DO concentration trends, especially in long-term predictions. This paper uses a one-dimensional convolutional neural network to extract the features of multidimensional input data. Bidirectional long and short-term memory neural network propagated forward and backward twice and thoroughly mined the before and after attribute relationship of each data of dissolved oxygen sequence. The attention mechanism focuses the model on the time series prediction step to improve long-term prediction accuracy. Finally, we built an integrated prediction model based on convolutional neural network (CNN), bidirectional long and short-term memory neural network (BiLSTM) and attention mechanism (AM), which is called CNN-BiLSTM-AM model. To determine the accuracy of the CNN-BiLSTM-AM model, we conducted short-term (30 minutes, one hour) and long-term (6 hours, 12 hours) experimental validation on real datasets monitored at two aquaculture farms in Yantai City, Shandong Province, China. Meanwhile, the performance was compared and visualized with support vector regression, recurrent neural network, long short-term memory neural network, CNN-LSTM model and CNN-BiLSTM model. The results show that compared with other comparative models, the proposed CNN-BiLSTM-AM model has an excellent performance in mean absolute error, root means square error, mean absolute percentage error and determination coefficient.

Keywords:

Citation: Wenbo Yang, Wei Liu, Qun Gao. Prediction of dissolved oxygen concentration in aquaculture based on attention mechanism and combined neural network[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 998-1017. doi: 10.3934/mbe.2023046

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Abstract

1. Introduction

In a tritrophic food chain, successive predation occurs between three species in the order prey-mesopredator-superpredator, in which there is a transfer of energy and nutrients when one organism eats another. The importance of the coexistence of species in the food chain, lies in avoiding overpopulation and extinction of species. There are several interesting results on the study of these systems involving Holling-type functional responses, and among them we mention: Rao and Narayan ^[1] studied the stability of the interior equilibrium point using the Routh-Hurwitz criterion and the global stability of a three-species food chain model with harvesting by constructing an appropriate Lyapunov function. Mammat et al. ^[2] studied an ecological model with a trophic chain with a classical Lotka-Volterra functional response and found a parameter space where a Hopf bifurcation occurs. Moreover, it is possible to find bifurcation points analytically and to show that the system has periodic solutions around these points. Rihand et al. ^[3] studied the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there is a direct competition between them. Castillo et al. ^[4] demonstrated the existence of a limit cycle, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, for an asymmetric intragremial food web model with Holling type Ⅱ functional response for intermediate and top predators and logistic growth for (common) prey. Cheng et al. ^[5] reviewed the existence and local stabilities of all equilibria of the classical Holling type Ⅱ model of the three-species food chain. They obtained the existence of a single limit cycle when the limit equilibrium loses its stability, and they also demonstrated the global stability of the limit cycle in $\mathbb{R}^3$ . Alsakaji et al. ^[6] studied the dynamics of a delay differential model of predator-prey system involving teams of two-prey and one-predator, with Monod-Haldane and Holling type Ⅱ functional responses, and a cooperation between the two-teams of preys against predation. Blé et al. ^[7] demonstrated that a tritrophic chain model with Holling type Ⅲ functional response has an equilibrium point where it presents a supercritical Hopf bifurcation independently of the prey growth rate. In the logistic case, they demonstrate the existence of at least three equilibrium points in the positive octant and one of them presents a supercritical Hopf bifurcation.

On the other hand, the theory of fractional order differential equations has gained great popularity in several scientific areas; such as biomathematics, control theory, and financial mathematics, among others (see ^[8,9,10,11]). In particular, in the case of biomathematics, tritrophic models with fractional derivative have been studied; for example, Maria et al. ^[12] calculated the equilibrium points and analyzed their stability to exhibit the dynamic behavior of a fractional order prey-predator model (3-Species). Mondal et al. ^[13] studied the dynamics of a three-dimensional discrete fractional-order eco-epidemiological model with Holling type Ⅱ functional response. They determined analytical conditions for the local stability of different fixed points using the Jury criterion and showed that the stability of the fractional-order discrete system depends strongly on the step size and the fractional order. More specifically, the critical value of the step size, at which stability switching occurs, decreases as the order of the fractional derivative decreases.

In the literature, we did not find results concerning the study of fractional order tritrophic systems with Holling type Ⅲ functional response.

In this work, we study the stability for the positive solution of the fractional tritrophic food chain models:

$\begin{eqnarray} {D}_{t_{0}}^{\alpha}x& = &\rho x-k_{1}f_{i}(x)y, \\ {D}_{t_{0}}^{\alpha}y& = &c_{1}f_{i}(x)y-k_{2}g_{i}(y)z-c_{2}y,\\ {D}_{t_{0}}^{\alpha}z& = &c_{3}g_{i}(y)z-dz, \end{eqnarray}$

(1.1)

where $x$ , $y$ , $z$ represent the density of the prey, mesopredator, and superpredator, respectively; ${D}_{t_{0}}^{\alpha}$ is the Caputo fractional derivative, $\alpha$ $\in (0, 1]$ , and all parameters are nonnegative. The parameters $c_1$ and $c_3$ represent the benefits of food consumption, $k_1$ and $k_2$ are the predator rates of the mesopredator, and superpredator, respectively, $c_2$ and $d$ are the mortality rate of the corresponding predators, and $\rho$ is the growth rate of the prey in the absence of predators. Since system (1.1) is an ecological model, our region of interest is the positive octant $\Omega = \{(x, y, z)\in \mathbb{R}^3 | x > 0, y > 0, z > 0 \}.$ Also, $f_{2}$ , $g_{2}$ and $f_{3}$ , $g_{3}$ are the types Ⅱ and Ⅲ functional responses, respectively, defined as follows:

$f_{i}(x) = {x^{i-1} \over x^{i-1}+a_{1}}, \ \ g_{i}(y) = {y^{i-1} \over y^{i-1}+a_{2}}, \ \ i = 2,3.$

The type Ⅱ functional response is used to describe predator organisms which take some time capture and ingest their prey. On the other hand, type Ⅲ functional response is characteristic of predator organisms which do not capture prey intensively below a certain level of threshold density; however, above that density level, predator organisms increase their feeding rates until some saturation level is reached.

2. Preliminaries

We begin this section by defining the Hurwitz polynomial as follows.

Definition 2.1. A polynomial $p$ is said to be a Hurwitz polynomial or Hurwitz stable if all the roots $s_{i}$ of $p$ lie in the open left half plane $\mathbb{C}_{-}$ .

We assume that $p(s)$ is the following polynomial:

$\begin{equation} p(s) = A_{0} +A_{1} s + A_{2} s^{2} +A_{3} s^{3}, \end{equation}$

(2.1)

where $A_{i}\in\mathbb{R}$ , $i = 0, 1, 2, 3$ and $A_{3} > 0$ , and $A_{0}\neq 0$ . Using the coefficients of $p(s)$ , we construct the Table 1 (see ^[15]).

Table 1. The Routh table.

$A_{3}$	$A_{1}$	0
$A_{2}$	$A_{0}$	0
$A_{1}-\dfrac{A_{0}A_{3}}{A_{2}}$	0	0
$A_{0}$	0	0

| Show Table

DownLoad: CSV

Theorem 2.1. Consider $p(s)$ given in (2.1). Then, $p(s)$ is Hurwitz if, and only if, each element of the first column of the Routh table is positive, i.e., $A_{3} > 0$ , $A_{2} > 0$ , $A_{1}-\dfrac{A_{0}A_{3}}{A_{2}} > 0$ , and $A_{0} > 0$ .

Proof. The proof can be found in ^[15].

Theorem 2.2. Consider $p(s)$ given in (2.1). Suppose when calculating the Routh table that no element in the first column is zero. Then, the number of sign changes in the first column of the Routh table is the number of open right half-plane zeros of $p(s)$ .

Proof. The proof can be found in ^[15].

Now, we give some definitions and properties of fractional calculus theory.

Definition 2.2. (See ^[11]) Let $x(t)\in L_{1}(\mathbb{R}^{+})$ . The Riemann-Liouville fractional integral of order $\alpha\in (0, 1]$ is defined by

$(I_{t_{0}}^{\alpha}x)(t) = {1\over \Gamma(\alpha)}\int_{t_{0}}^t {x(\tau)\over (t-\tau)^{1-\alpha}} > d\tau, \ \ \ \ t > t_{0}\geq 0,$

where $\Gamma$ is the Gamma function.

Definition 2.3. (See ^[11]) The Caputo fractional derivative of order $\alpha\in (0, 1]$ and $x\in AC(\mathbb{R}^{+})$ , the space of absolutely continuous functions on $\mathbb{R}^{+}$ , is defined by

$(D_{t_{0}}^{\alpha}x)(t) = (I_{t_{0}}^{1-\alpha}Dx)(t),$

where $D = d/dt$ .

The autonomous nonlinear differential system in the sense of Caputo is given as follows:

$\begin{equation} D_{t_{0}}^{\alpha} X(t) = F(X(t)), \end{equation}$

(2.2)

where $X(t)\in \mathbb{R}^3$ , $X(t_{0}) = X_{t_{0}}$ , $F(X)$ is continuous.

Definition 2.4. Suppose that $P$ is an equilibrium point of system (2.2) and that all the eigenvalues $\lambda$ of the linearized matrix $J(P) = \partial F / \partial X |_{X = P}$ evaluated at $P$ satisfy: $|\lambda| \neq 0$ and $|arg(\lambda)|\neq \dfrac{\pi \alpha}{2}$ , then we call $P$ a hyperbolic equilibrium point.

Theorem 2.3. If $P$ is a hyperbolic equilibrium point of (2.2), then vector field $F(x)$ is topologically equivalent with its linearization vector field $J(P)x$ in the neighborhood $\delta (P)$ .

Proof. The proof can be found in ^[16].

Theorem 2.4. We consider the fractional-order system

$\begin{equation} D^{\alpha}X(t) = F(X(t)), \end{equation}$

(2.3)

where $0 < \alpha < 1$ . System (2.3) is asymptotically stable at the equilibrium point $P$ if, and only if,

$| arg(\lambda)| > {\pi \alpha \over 2}$

for all roots $\lambda$ of the following equation $\det(\lambda I - J(P)) = 0$ .

Proof. The proof can be found in ^[17].

3. Main result

In this section, we present two results obtained by analyzing the stability around the coexistence equilibrium points of three species of the tritrophic system (1.1). First, we consider the system (1.1) with Holling type Ⅱ functional response, which has only one unstable equilibrium point in the first octant.

Theorem 3.1. We assume $\alpha = 1$ , $i = 2, \ c_{3} > d, \ a_{2}dk_{1} > a_{1}\rho(c_{3}-d), \ (c_{1}-c_{2})a_{2}dk_{1} > a_{1}c_{1}\rho (c_{3}-d)$ . Then, there is only one equilibrium point $P$ in the first octant for the system (1.1),

$P = \Big( {k_{1}a_{2}d_-\rho a_{1}(c_{3}-d) \over \rho (c_{3}-d)}, {a_{2}d \over c_{3}-d}, {c_{3}((c_{1}-c_{2})k_{1}a_{2}d-\rho a_{1}c_{1}(c_{3}-d))\over dk_{1}k_{2}(c_{3}-d)} \Big).$

Also, the system (1.1) is unstable around the equilibrium point $P$ .

Proof. Let

$\begin{eqnarray} F_{1}(x, y)& = &\rho x-{k_{1}xy \over x+a_{1}}, \\ F_{2}(x, y, z)& = &{c_{1}xy \over x+a_{1}}-{k_{2}yz \over y+a_{2}}-c_{2}y,\\ F_{3}(y, z)& = &{c_{3}yz \over y+a_{2}}-dz. \end{eqnarray}$

(3.1)

The equilibrium point $P \in \Omega$ for system (1.1) when $i = 2$ is obtained by solving system $F_{1}(x, y) = F_{2}(x, y, z) = F_{3}(y, z) = 0$ .

On the other hand, we obtain the Jacobian matrix $J$ of (3.1)

$J = \begin{pmatrix} \rho-\dfrac{a_{1}k_{1}y}{(x+a_{1})^2} & -\dfrac{k_{1}x}{x+a_{1}} & 0 \\ \dfrac{a_{1}c_{1}y}{(x+a_{1})^2} & \dfrac{c_{1}x}{x+a_{1}}-\dfrac{a_{2}k_{2}z}{(y+a_{2})^2}-c_{2} & \dfrac{-k_{2}y}{y+a_{2}} \\ 0 & \dfrac{a_{2}c_{3}z}{(y+a_{2})^2} & \dfrac{c_{3}y}{y+a_{2}}-d \end{pmatrix}.$

Now, evaluate $P$ in $J$

$J(P)$ =

$\begin{pmatrix} \dfrac{\rho(a_{2}dk_{1}-a_{1}\rho(c_{3}-d))}{a_{2}dk_{1}} & \dfrac{\rho a_{1}(c_{3}-d)-k_{1}a_{2}d}{a_{2}d} & 0 \\ \dfrac{a_{1}c_{1}\rho^{2}(c_{3}-d)}{a_{2}dk_{1}^{2}} & \dfrac{a_{2}dk_{1}(c_{1}-c_{2})-a_{1}c_{1}\rho (c_{3}-d)}{ a_{2}c_{3}k_{1}} & \dfrac{-dk_{2}}{ c_{3}} \\ 0 & \dfrac{(c_{3}-d)((c_{1}-c_{2})a_{2}dk_{1}-a_{1}c_{1}\rho (c_{3}-d))}{ a_{2}dk_{1}k_{2}} & 0 \end{pmatrix}$ .

We denote by $J(P) = (m_{jk})$ , $j, k = 1, 2, 3$ . The characteristic polynomial is

$\begin{equation} p(\lambda) = \det[J(P)-\lambda I], \end{equation}$

(3.2)

where $I$ is the identity matrix 3x3. That is,

$\begin{equation} p(\lambda) = A_3 \lambda^{3}+A_2 \lambda^{2}+A_1 \lambda+A_0, \end{equation}$

(3.3)

with

$A_3 = 1, \ A_2 = -m_{11}-m_{22}, \ A_1 = m_{11}m_{22}-m_{12}m_{21}-m_{23}m_{32}, \ A_0 = m_{11}m_{23}m_{32}.$

Let $P = (x_0, y_0, z_0)$ be the equilibrium point in the first octant. We reduce $m_{11}$ and $m_{32}$ :

$m_{11} = \frac{\rho^2 x_{0}}{k_{1}y_{0}}, \ \ m_{32} = \frac{a_{2}d^2 z_{0}}{c_{3}y_{0}^2}.$

Then

$A_{0} = - \frac{a_{2}d^{3}k_{2}\rho^{2}x_{0}z_{0}}{c_{3}^{2}k_{1}y_{0}^{3}} < 0.$

Therefore, by Theorem 2.2, the system is unstable.

Now, we consider the system (1.1) with Holling type Ⅲ functional response, which has two equilibrium points in the first octant, one stable and the other unstable.

Theorem 3.2. Assuming $i = 3, \ k_{1} = 2\rho \sqrt{r}$ , $c_{3} = rd$ , $c_{1} = rc_{2}$ , $a_{1} = a_{2}$ , and $r > 4$ , the system (1.1) has points of equilibrium in the first octant:

$\begin{eqnarray*} P_{1}& = &\bigg({ \sqrt{a_{1}} (\sqrt{r}+1) \over \sqrt{r-1}}, \sqrt{{a_{1} \over r-1}}, { \sqrt{a_{1}} c_{2} r (r+\sqrt{r}-2) \over 2 k_{2} \sqrt{r-1}}\bigg),\\ P_{2}& = &\bigg({ \sqrt{a_{1}} (\sqrt{r}-1) \over \sqrt{r-1}}, \sqrt{{a_{1} \over r-1}}, { \sqrt{a_{1}} c_{2} r (r-\sqrt{r}-2) \over 2 k_{2} \sqrt{r-1}}\bigg). \end{eqnarray*}$

Then, the system (1.1) is unstable around the equilibrium point $P_{1}$ and asymptotically stable around $P_{2}$ .

Proof. Let

$\begin{eqnarray} G_{1}(x, y)& = &\rho x-{2\rho \sqrt{r} x^{2}y \over x^2 +a_{1}}, \\ G_{2}(x, y, z)& = &{c_{2}rx^2 y \over x^2 +a_{1}}-{k_{2}y^2 z \over y^2 +a_{1}}-c_{2}y,\\ G_{3}(y, z)& = &{dry^2 z \over y^2 +a_{1}}-dz. \end{eqnarray}$

(3.4)

Then, the Jacobian matrix $J$ of system (3.4) is

$J$ =

$\begin{pmatrix} \rho-\dfrac{4 a_{1} \rho \sqrt{r} x y}{(x^2 +a_{1})^2} & \dfrac{-2 \rho \sqrt{r}x^2}{ x^2 +a_{1}} & 0 \\ \dfrac{2 a_{1}c_{2} xy}{(x^2 +a_{1})^2} & \dfrac{c_{2}rx^2}{ x^2 +a_{1}}-\dfrac{2a_{1}k_{2}yz }{(y^2 +a_{1})^2}-c_{2} & \dfrac{-k_{2}y^2}{ y^2 +a_{1}} \\ 0 & \dfrac{2a_{1}dryz}{(y^2 +a_{1})^2} & \dfrac{dry^2}{y^2 +a_{1}}-d \end{pmatrix}$ .

We evaluate $J$ in $P_{1}$ ,

$J(P_{1})$ =

$\begin{pmatrix} \dfrac{\rho }{\sqrt{r}} & -\rho (\sqrt{r}+1) & 0 \\ \dfrac{c_{2} (\sqrt{r}-1)}{ 2} & c_{2}\Big(-\dfrac{r }{ 2}-\dfrac{\sqrt{r} }{ 2}+\dfrac{1 }{ \sqrt{r}}-\dfrac{2 }{ r}+2 \Big) & -\dfrac{k_{2} }{ r} \\ 0 & \dfrac{c_{2}d }{ k_{2}} \Big( r^2 +r^{3/2}-3r-\sqrt{r}+2\Big) & 0 \end{pmatrix}$ .

Let $J(P_{1}) = (m_{jk})$ , $j, k = 1, 2, 3$ . We calculate the characteristic polynomial:

$p(\lambda) = \det[J(P_{1})-\lambda I].$

That is,

$\begin{equation} p(\lambda) = A_{3} \lambda^{3}+A_{2} \lambda^{2}+A_{1} \lambda+A_{0}, \end{equation}$

(3.5)

where

$\begin{eqnarray*} A_{0}& = &-{c_{2}d \rho \over r^{3/2}} (r-1)(r+\sqrt{r}-2),\\ A_{1}& = &{c_{2}\rho \over \sqrt{r}}\Big( {r^{3/2} \over 2}-\frac{r}{2}-\sqrt{r}+\frac{1}{\sqrt{r}}-\frac{2}{r}+2\Big)+{c_{2}d \over r} \Big( r^2 +r^{3/2}-3r-\sqrt{r}+2\Big),\\ A_{2}& = &c_{2}\Big({r \over 2}+{\sqrt{r} \over 2}-2-{1 \over \sqrt{r}}+{2 \over r} \Big)-{\rho \over \sqrt{r}},\\ A_{3}& = &1. \end{eqnarray*}$

We note that $A_{0} < 0$ , when $r > 4$ . Then, by Theorem 2.1, $p(\lambda)$ is not a Hurwitz polynomial.

On the other hand,

$J(P_{2})$ =

$\begin{pmatrix} -\dfrac{\rho }{ \sqrt{r}} & -\rho (\sqrt{r}-1) & 0 \\ \dfrac{c_{2} (\sqrt{r}+1)}{ 2} & c_{2}\Big(-\dfrac{r }{ 2}+\dfrac{\sqrt{r} }{ 2}+2-\dfrac{1 }{ \sqrt{r}}-\dfrac{2}{ r} \Big) & -\dfrac{k_{2} }{ r} \\ 0 & \dfrac{c_{2}d }{ k_{2}} \Big( r^2 -r^{3/2}-3r+\sqrt{r}+2\Big) & 0 \end{pmatrix}$ .

The characteristic polynomial has the form (3.5), where $A_{i}$ are

$\begin{eqnarray*} A_{0}& = &{c_{2}d \rho \over r^{3/2}} (r-1)(r-\sqrt{r}-2),\\ A_{1}& = &{c_{2}\rho \over \sqrt{r}}\Big( {r^{3/2} \over 2}+\frac{r}{2}-\sqrt{r}-2+\frac{1}{\sqrt{r}}+\frac{2}{r}\Big)+{c_{2}d \over r} (r-1)(r-\sqrt{r}-2), \\ A_{2}& = &c_{2}\Big({r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} \Big)+{\rho \over \sqrt{r}},\\ A_{3}& = &1. \end{eqnarray*}$

Also, $r > 4$ , from which we obtain

$\begin{eqnarray} & & r-\sqrt{r}-2 > 0, \\ \dfrac{r^{3/2}}{2}+\dfrac{r}{2}-\sqrt{r}-2+\dfrac{1}{\sqrt{r}}+\dfrac{2}{r}& > &{r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} > \biggr({\sqrt{r} \over 2}-{2 \over \sqrt{r}} \biggr)^{2}+\dfrac{r}{4}-\dfrac{\sqrt{r}}{2} > 0. \end{eqnarray}$

(3.6)

For (3.6), we have $A_{i} > 0$ , for each $i = 0, 1, 2$ . Now, we calculate $A_{1}A_{2}-A_{0}$ .

$\begin{equation*} A_{1}A_{2}-A_{0} = {c_{2}\rho A_{1} \over \sqrt{r}}\Big( {r^{3/2} \over 2}+\frac{r}{2}-\sqrt{r}-2+\frac{1}{\sqrt{r}}+\frac{2}{r}\Big)+{c_{2}^{2}d \over r} (r-1)(r-\sqrt{r}-2)\Big({r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} \Big). \end{equation*}$

By (3.6) and $A_{1} > 0$ , we have $A_{1}A_{2}-A_{0} > 0$ . Then, by Theorem 2.1, $p(\lambda)$ is a Hurwitz polynomial. Therefore, the system (1.1), when $\alpha = 1$ , is stable around $P_2$ .

Now, let us consider $\alpha \in (0, 1)$ . By Theorem 2.3, the system (1.1) and its linearization vector field $J(P_{2})X$ are topologically equivalent. By Theorem 2.4, $D^{\alpha}X = J(P_{2})X$ is stable around $P_2$ . Therefore, the system (1.1) is stable around $P_2$ .

4. Analytical solution

In this section, we present the construction of the analytical solution for the system (1.1), with Holling type Ⅲ functional response, $k_{1} = 2\rho \sqrt{r}$ , $c_{3} = rd$ , $c_{1} = rc_{2}$ , and $a_{1} = a_{2}$ , which are obtained by applying the multistage homotopy perturbation method (see ^[18]):

First, let's define a regular partition of the interval $[0, T]$ by $t_{0} = 0 < t_{1} < t_{2} < \dots < t_{m} = T$ and the following family of fractional-order systems

$\begin{eqnarray} & &{D}_{t_{k-1}}^{\alpha}x_k = \rho x_k-{2\rho \sqrt{r}x_{k}^{2}y_k \over x_k^{2}+a_{1}},\\ & & {D}_{t_{k-1}}^{\alpha}y_k = \frac{c_{2} r x_{k}^2 y_k}{x_{k}^2 +a_{1}}-{k_{2}y_k^{2}z_k \over y_k^{2}+a_{1}}-c_{2}y_{k},\\ & & {D}_{t_{k-1}}^{\alpha}z_k = \frac{dr y_{k}^2 z_{k}}{y_{k}^2 +a_{1}}-dz_{k}, \end{eqnarray}$

(4.1)

with $t \in [t_{k-1}, t_{k}]$ and initial conditions $(x_{0}, y_{0}, z_{0}) = (x_{1}(t_0), y_{1}(t_0), y_{1}(t_0))$ and $(x_k(t_{k-1}), y_k(t_{k-1}), z_k(t_{k-1})) = (x_{k-1}(t_{k-1}), y_{k-1}(t_{k-1}), z_{k-1}(t_{k-1}))$ for $k = 1, \dots, m$ .

Now, we define the homotopy for each $k$ :

$\begin{eqnarray} & &(1-p)\big({D}_{t_{k-1}}^{\alpha}u_k-{D}_{t_{k-1}}^{\alpha}x_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}u_k- \rho u_k+{2\rho \sqrt{r}u_{k}^{2}v_k \over u_k^{2}+a_{1}}\bigg) = 0, \\ & &(1-p)\big({D}_{t_{k-1}}^{\alpha}v_k-{D}_{t_{k-1}}^{\alpha}y_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}v_k- \frac{c_{2} r u_{k}^2 v_k}{x_{k}^2 +a_{1}}+{k_{2}v_k^{2}w_k \over v_k^{2}+a_{1}}+c_{2}v_{k}\bigg) = 0,\\ & &(1-p)\big({D}_{t_{k-1}}^{\alpha}w_k-{D}_{t_{k-1}}^{\alpha}z_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}w_k+cw_k-{dv_k^{2} w_k \over v_k^{2}+e}\bigg) = 0, \end{eqnarray}$

(4.2)

where $p\in[0, 1]$ . Let's suppose that the solution for (4.2) is given by

$\begin{eqnarray} u_{k}& = &u_{0k}+pu_{1k}+p^{2}u_{2k}+p^{3}u_{3k}+\cdots, \\ v_{k}& = &v_{0k}+pv_{1k}+p^{2}v_{2k}+p^{3}v_{3k}+\cdots,\\ w_{k}& = &w_{0k}+pw_{1k}+p^{2}w_{2k}+p^{3}w_{3k}+\cdots, \end{eqnarray}$

(4.3)

where $u_{ik}$ , $v_{ik}$ , $w_{ik}$ , $i = 1, 2, \dots$ , are functions to be determined. Then, the solution for $t\in[t_{k-1}, t_{k}]$ is

$\begin{eqnarray} x_{k} = \lim\limits_{p \rightarrow 1 } u_{k}& = &u_{0k}+u_{1k}+u_{2k}+u_{3k}+\cdots, \\ y_{k} = \lim\limits_{ p \rightarrow 1 }v_{k}& = &v_{0k}+v_{1k}+v_{2k}+v_{3k}+\cdots,\\ z_{k} = \lim\limits_{ p \rightarrow 1 }w_{k}& = &w_{0k}+w_{1k}+w_{2k}+w_{3k}+\cdots. \end{eqnarray}$

(4.4)

Therefore, the analytic solution for $t\in [0, T]$ is given by

$\begin{eqnarray} x(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} u_{jk}(t), \\ y(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} v_{jk}(t), \\ z(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} w_{jk}(t), \end{eqnarray}$

(4.5)

where $I_{[t_{k-1}, t_k]}$ is the characteristic function, $t_{k} = k\dfrac{T}{m}$ , and $\bigcup\limits_{k = 1}^{m} [t_{k-1}, t_k] = [0, T]$ . We show the first addends of the series (4.4),

$\begin{eqnarray*} u_{1k}(t)& = & \dfrac{\rho u_{0k} (a_{1} + u_{0k}(u_{0k} - 2 \sqrt{r} v_{0k}))}{(a_{1}+ u_{0k}^2)\Gamma(1 + \alpha)}(t-t_{k-1} )^{\alpha}, \nonumber \\ v_{1k}(t)& = & - \dfrac{v_{0k}(c_{2}(a_{1}-(r-1)u_{0k}^2)(a_{1}+v_{0k}^2) +k_{2}(a_{1}+u_{0k}^2)v_{0k}w_{0k})}{\alpha (a_{1}+u_{0k}^2)(a_{1}+v_{0k}^2)\Gamma(\alpha)}(t-t_{k-1} )^{\alpha}, \\ w_{1k}(t)& = & - \dfrac{d w_{0k}(a_{1}-(r-1)v_{0k}^2)}{(a_{1}+v_{0k}^2)\Gamma(1+\alpha)}(t-t_{k-1} )^{\alpha}, \nonumber \\ u_{2k}(t)& = & \dfrac{\rho u_{0k}}{(a_{1}+u_{0k}^2)^{3}\Gamma^2 (1+\alpha)}(a_{1}\rho (a_{1}+u_{0k}^2)(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))+3\rho u_{0k}^2 (a_{1}+u_{0k}^2)(a_{1}+u_{0k} \nonumber \\ & & \times (u_{0k} -2\sqrt{r}v_{0k}))-4\rho \sqrt{r}u_{0k}v_{0k}(a_{1}+u_{0k}^2)(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))-2\rho u_{0k}^2 (a_{1}+u_{0k} \nonumber \\ & & \times (u_{0k}-2\sqrt{r}v_{0k}))^2+\dfrac{2\sqrt{r}u_{0k}v_{0k}}{a_{1}+v_{0k}^2}(a_{1}+u_{0k}^2)(c_{2}(a_{1}-(r-1)u_{0k}^2)(a_{1}+v_{0k}^2)+k_{2}(a_{1}+u_{0k}^2) \nonumber \\ & & \times v_{0k}w_{0k}))(t-t_{k-1} )^{2\alpha}, \nonumber \\ v_{2k}(t)& = &\dfrac{v_{0k}}{(a_{1}+u_{0k}^2 )^3 (a_{1}+v_{0k}^2 )^3 \Gamma^2 (1+\alpha)}((a_{1}+u_{0k}^2 )((a_{1}+v_{0k}^2 )c_{2}(a_{1}-(r-1)u_{0k}^2 )+k_{2}(a_{1}+u_{0k}^2 ) \nonumber \\ & &\times v_{0k}w_{0k})(c_{2}(a_{1}-(r-1)u_{0k}^2 )(a_{1}+v_{0k}^2 )^2 +2a_{1}k_{2}(a_{1}+u_{0k}^2 )v_{0k}w_{0k})+(a_{1}+v_{0k}^2 )(2a_{1}c_{2}\rho \nonumber \\ & & \times u_{0k}^2 r(a_{1}+ v_{0k}^2 )^2(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))+dk_{2}(a_{1}+u_{0k}^2 )^3 v_{0k}(a_{1}-(r-1)v_{0k}^2 )w_{0k})) \nonumber \\ & & \times(t-t_{k-1} )^{2\alpha}, \nonumber \\ w_{2k}(t)& = &\dfrac{dw_{0k}}{(a_{1}+u_{0k}^2 )(a_{1}+v_{0k}^2 )^3 \alpha \Gamma^2 (1+\alpha)}(d (a_1 + u_{0k}^2) (a_1 + v_{0k}^2 ) (a_{1} - (r-1) v_{0k}^2 )^2 \alpha - 2 a_1 r v_{0k}^2 (c_2 \nonumber \\ & & \times (a_1 - (r-1) u_{0k}^2) (a_1 + v_{0k}^2 ) + k_2 (a_1 + u_{0k}^2 ) v_{k0} w_{0k} ) \alpha)(t-t_{k-1})^{2\alpha}. \nonumber \end{eqnarray*}$

5. Examples

In this section, we illustrate the results obtained in Theorems 3.1 and 3.2, through some particular examples, where its graphs show the dynamics of the solutions around the equilibrium points. Furthermore, we can observe that as the order of the derivative $\alpha$ moves away from $1$ , the length of the trajectories increases. For this purpose, we use the analytical solution given in (4.5).

Example 5.1. (See ) For $i = 2$ , fix $a_{1} = c_{2} = \rho = r = 1$ , $c_{1} = d = k_{1} = k_{2} = 2$ , $c_{3} = 2.5$ , $a_{2} = 5$ , $m = 50$ , $t \in [0, 0.5]$ , $(16, 8, 5)$ is the initial condition, and $P = (39, 20, 11.875)$ is an unstable equilibrium point.

Figure 1. Unstable equilibrium point

$P$ .

DownLoad: Full-Size Img PowerPoint

Example 5.2. (See ) For $i = 3$ :

Figure 2. Unstable equilibrium point

$P_{1}$ and asymptotic stability around

$P_{2}$ .

DownLoad: Full-Size Img PowerPoint

● Let $a_{1} = c_{2} = d = k_{2} = \rho = r = 1$ , $r = 5$ , $m = 100$ , $t \in [0, 1]$ , $(2, 1, 7)$ is the initial condition, and $P_{1} = (1.61803, 0.5, 6.54508)$ is an unstable equilibrium point.

● We take $a_{1} = c_{2} = d = k_{2} = \rho = r = 5$ , $m = 100$ , $t \in [0, 1]$ , $(1, 1, 2)$ is the initial condition, and $P_{2} = (1.38197, 1.11803, 2.13525)$ is a locally asymptotically stable equilibrium point.

6. Conclusions

In this work, the local stability around the coexistence equilibrium points of two tritrophic fractional models were studied, with Holling type Ⅱ and Ⅲ functional responses, respectively. Under certain conditions of the parameters, it was found that only the second model has a stable equilibrium point $P_2$ , when $\alpha \in (0, 1]$ . The multistage homotopic perturbation method was used to obtain the solution of the tritrophic fractional model with Holling type Ⅲ function response, and it was observed that the trajectories tend faster toward the equilibrium point when $\alpha < 1$ . These results reveal the possibility of tritrophic coexistence when the interaction is Holling type Ⅲ. A future work is to use the analytical solution, obtained via the homotopic perturbation method, to solve the inverse problem associated to system (1.1); that is, to estimate the parameters involved in the system, from a Bayesian analysis perspective, which is important to model experimental problems in ecology.

Author contributions

Anel Esquivel-Navarrete: investigation, formal analysis, writing – review $\And$ editing; Jorge Sanchez-Ortiz: investigation, supervision, writing – review $\And$ editing; Gabriel Catalan-Angeles: formal analysis, writing- original draft preparation, visualization; Martin P. Arciga-Alejandre: investigation, methodology, writing – review $\And$ editing.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	Agriculture Organization of the United Nations, Fisheries Department, The State of World Fisheries and Aquaculture, Food & Agriculture Org, 2000.
[2]	X. Li, J. Li, Y. Wang, L. Fu, Y. Fu, B. Li, et al., Aquaculture industry in china: Current state, challenges, and outlook, Rev. Fish. Sci., 19 (2011), 187–200. https://doi.org/10.1080/10641262.2011.573597 doi: 10.1080/10641262.2011.573597
[3]	J. Huan, H. Li, M. Li, B. Chen, Prediction of dissolved oxygen in aquaculture based on gradient boosting decision tree and long short-term memory network: A study of Chang Zhou fishery demonstration base, China, Comput. Electron. Agric., 175 (2020), 105530. https://doi.org/10.1016/j.compag.2020.105530 doi: 10.1016/j.compag.2020.105530
[4]	S. Midilli, D. Çoban, M. Güler, S. Küçük, Gas bubble disease in nile tilapia and hybrid red tilapia (cichlidae, oreochromis spp.) under culture conditions, J. Fish. Aquat. Sci., 36 (2019), 285–291. http://dx.doi.org/10.12714/egejfas.2019.36.3.09 doi: 10.12714/egejfas.2019.36.3.09
[5]	C. E. Boyd, E. L. Torrans, C. S. Tucker, Dissolved oxygen and aeration in ictalurid catfish aquaculture, J. World Aquacult. Soc., 49 (2018), 7–70. https://doi.org/10.1111/jwas.12469 doi: 10.1111/jwas.12469
[6]	A. Sentas, L. Karamoutsou, N. Charizopoulos, T. Psilovikos, A. Psilovikos, A. Loukas, The use of stochastic models for short-term prediction of water parameters of the thesaurus dam, River Nestos, Greece, Proceedings, 2 (2018), 634. https://doi.org/10.3390/proceedings2110634 doi: 10.3390/proceedings2110634
[7]	M. Valera, R. K. Walter, B. A. Bailey, J. E. Castillo, Machine learning based predictions of dissolved oxygen in a small coastal embayment, J. Mar. Sci. Eng., 8 (2020), 1007. https://doi.org/10.3390/jmse8121007 doi: 10.3390/jmse8121007
[8]	C. Xu, X. Chen, L. Zhang, Predicting river dissolved oxygen time series based on stand-alone models and hybrid wavelet-based models, J. Environ. Manage., 295 (2021), 113085. https://doi.org/10.1016/j.jenvman.2021.113085 doi: 10.1016/j.jenvman.2021.113085
[9]	A. Sorjamaa, J. Hao, N. Reyhani, Y. Ji, A. Lendasse, Methodology for long-term prediction of time series, Neurocomputing, 70 (2007), 2861–2869. https://doi.org/10.1016/j.neucom.2006.06.015 doi: 10.1016/j.neucom.2006.06.015
[10]	M. Längkvist, L. Karlsson, A. Loutfi, A review of unsupervised feature learning and deep learning for time-series modeling, Pattern Recognit. Lett., 42 (2014), 11–24. https://doi.org/10.1016/j.patrec.2014.01.008 doi: 10.1016/j.patrec.2014.01.008
[11]	L. Karamoutsou, A. Psilovikos, Deep learning in water resources management: The case study of Kastoria lake in Greece, Water, 13 (2021), 3364. https://doi.org/10.3390/w13233364 doi: 10.3390/w13233364
[12]	Q. Ye, X. Yang, C. Chen, J. Wang, River water quality parameters prediction method based on LSTM-RNN model, in 2019 Chinese Control And Decision Conference (CCDC), (2019), 3024–3028. https://doi.org/10.1109/CCDC.2019.8832885
[13]	J. Yan, Y. Gao, Y. Yu, H. Xu, Z. Xu, A prediction model based on deep belief network and least squares svr applied to cross-section water quality, Water, 12 (2020), 1929. https://doi.org/10.3390/w12071929 doi: 10.3390/w12071929
[14]	L. Sheng, J. Zhou, X. Li, Y. Pan, L. Liu, Water quality prediction method based on preferred classification, IET Cyber-Phys. Syst. Theory Appl., 5 (2020), 176–180. https://doi.org/10.1049/iet-cps.2019.0062 doi: 10.1049/iet-cps.2019.0062
[15]	J. Wu, Z. Wang, A hybrid model for water quality prediction based on an artificial neural network, wavelet transform, and long short-term memory, Water, 14 (2022), 610. https://doi.org/10.3390/w14040610 doi: 10.3390/w14040610
[16]	B. Lim, S. Zohren, Time-series forecasting with deep learning: A survey, Philos. Trans. A Math. Phys. Eng. Sci., 379 (2021), 20200209. https://doi.org/10.1098/rsta.2020.0209 doi: 10.1098/rsta.2020.0209
[17]	H. Ismail Fawaz, G. Forestier, J. Weber, L. Idoumghar, P. A. Muller, Deep learning for time series classification: A review, Data Min. Knowl. Discovery, 33 (2019), 917–963. https://doi.org/10.1007/s10618-019-00619-1 doi: 10.1007/s10618-019-00619-1
[18]	P. Shi, G. Li, Y. Yuan, G. Huang, L. Kuang, Prediction of dissolved oxygen content in aquaculture using clustering-based softplus extreme learning machine, Comput. Electron. Agric., 157 (2019), 329–338. https://doi.org/10.1016/j.compag.2019.01.004 doi: 10.1016/j.compag.2019.01.004
[19]	W. Li, H. Wu, N. Zhu, Y. Jiang, J. Tan, Y. Guo, Prediction of dissolved oxygen in a fishery pond based on gated recurrent unit (GRU), Inf. Process. Agric., 8 (2021), 185–193. https://doi.org/10.1016/j.inpa.2020.02.002 doi: 10.1016/j.inpa.2020.02.002
[20]	C. Li, Z. Li, J. Wu, L. Zhu, J. Yue, A hybrid model for dissolved oxygen prediction in aquaculture based on multi-scale features, Inf. Process. Agric., 5 (2018), 11–20. https://doi.org/10.1016/j.inpa.2017.11.002 doi: 10.1016/j.inpa.2017.11.002
[21]	Q. Ren, X. Wang, W. Li, Y. Wei, D. An, Research of dissolved oxygen prediction in recirculating aquaculture systems based on deep belief network, Aquacult. Eng., 90 (2020), 102085.
[22]	J. Huang, S. Liu, S. G. Hassan, L. Xu, C. Huang, A hybrid model for short-term dissolved oxygen content prediction, Comput. Electron. Agric., 186 (2021), 106216. https://doi.org/10.1016/j.compag.2021.106216 doi: 10.1016/j.compag.2021.106216
[23]	H. Liu, R. Yang, Z. Duan, Wind speed forecasting using a new multi-factor fusion and multi-resolution ensemble model with real-time decomposition and adaptive error correction, Energy Convers. Manage., 217 (2020), 112995. https://doi.org/10.1016/j.enconman.2020.112995 doi: 10.1016/j.enconman.2020.112995
[24]	J. Bi, Y. Lin, Q. Dong, H. Yuan, M. Zhou, An improved attention-based lstm for multi-step dissolved oxygen prediction in water environment, in 2020 IEEE International Conference on Networking, Sensing and Control (ICNSC), (2020), 1–6. https://doi.org/10.1109/ICNSC48988.2020.9238097
[25]	Y. Liu, Q. Zhang, L. Song, Y. Chen, Attention-based recurrent neural networks for accurate short-term and long-term dissolved oxygen prediction, Comput. Electron. Agric., 165 (2019), 104964. https://doi.org/10.1016/j.compag.2019.104964 doi: 10.1016/j.compag.2019.104964
[26]	X. Yang, B. Liu, Uncertain time series analysis with imprecise observations, Fuzzy Optim. Decis. Making, 18 (2018), 263–278. https://doi.org/10.1007/s10700-018-9298-z doi: 10.1007/s10700-018-9298-z
[27]	T. Wang, M. Wang, Communication network time series prediction algorithm based on big data method, Wireless Pers. Commun., 102 (2017), 1041–1056. https://doi.org/10.1007/s11277-017-5138-7 doi: 10.1007/s11277-017-5138-7
[28]	Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, et al., Backpropagation applied to handwritten zip code recognition, Neural Comput., 1 (1989), 541–551. https://doi.org/10.1162/neco.1989.1.4.541 doi: 10.1162/neco.1989.1.4.541
[29]	Á. Zarándy, C. Rekeczky, P. Szolgay, L. O. Chua, Overview of cnn research: 25 years history and the current trends, in 2015 IEEE International Symposium on Circuits and Systems (ISCAS), (2015), 401–404. https://doi.org/10.1109/ISCAS.2015.7168655
[30]	L. Alzubaidi, J. Zhang, A. J. Humaidi, A. Al-Dujaili, Y. Duan, O. Al-Shamma, et al., Review of deep learning: concepts, cnn architectures, challenges, applications, future directions, J. Big Data, 8 (2021), 53. https://doi.org/10.1186/s40537-021-00444-8 doi: 10.1186/s40537-021-00444-8
[31]	M. Sahu, R. Dash, A survey on deep learning: convolution neural network (CNN), in Intelligent and Cloud Computing, Springer, (2021), 317–325.
[32]	G. Ortac, G. Ozcan, Comparative study of hyperspectral image classification by multidimensional convolutional neural network approaches to improve accuracy, Expert Syst. Appl., 182 (2021), 115280. https://doi.org/10.1016/j.eswa.2021.115280 doi: 10.1016/j.eswa.2021.115280
[33]	X. Song, Y. Liu, L. Xue, J. Wang, J. Zhang, J. Wang, et al., Time-series well performance prediction based on long short-term memory (LSTM) neural network model, J. Pet. Sci. Eng., 186 (2020), 106682. https://doi.org/10.1016/j.petrol.2019.106682 doi: 10.1016/j.petrol.2019.106682
[34]	S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735 doi: 10.1162/neco.1997.9.8.1735
[35]	R. Yu, J. Gao, M. Yu, W. Lu, T. Xu, M. Zhao, et al., LSTM-EFG for wind power forecasting based on sequential correlation features, Future Gener. Comput. Syst., 93 (2019), 33–42. https://doi.org/10.1016/j.future.2018.09.054 doi: 10.1016/j.future.2018.09.054
[36]	X. Wu, J. Li, Y. Jin, S. Zheng, Modeling and analysis of tool wear prediction based on SVD and BiLSTM, Int. J. Adv. Manuf. Technol., 106 (2020), 4391–4399. https://doi.org/10.1007/s00170-019-04916-3 doi: 10.1007/s00170-019-04916-3
[37]	M. Schuster, K. K. Paliwal, Bidirectional recurrent neural networks, IEEE Trans. Signal Process., 45 (1997), 2673–2681. https://doi.org/10.1109/78.650093 doi: 10.1109/78.650093
[38]	S. Siami-Namini, N. Tavakoli, A. S. Namin, The performance of LSTM and BiLSTM in forecasting time series, in 2019 IEEE International Conference on Big Data (Big Data), (2019), 3285–3292. https://doi.org/10.1109/BigData47090.2019.9005997
[39]	Z. Niu, G. Zhong, H. Yu, A review on the attention mechanism of deep learning, Neurocomputing, 452 (2021), 48–62. https://doi.org/10.1016/j.neucom.2021.03.091 doi: 10.1016/j.neucom.2021.03.091
[40]	A. De Santana Correia, E. L. Colombini, Attention, please! A survey of neural attention models in deep learning, Artif. Intell. Rev., 2022 (2022), 1–88. https://doi.org/10.1007/s10462-022-10148-x doi: 10.1007/s10462-022-10148-x

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