# A Machine Learning Strategy for the Quantitative Analysis of the Global Warming Impact on Marine Ecosystems

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## Abstract

**:**

## 1. Introduction

- The RP-LMS neural network is a novel supervised computational paradigm that we designed. It is fast and efficient and requires little computing power.
- For the GAPF model, the original mathematical equations are solved by the RK4 method to prepare the dataset for the RP-LMS neural network. For the convenience of readers, the notations used in this paper are summarized in the abbreviation section.
- The presentation of the designed neural network through RP-LMS for successfully resolving the GAPF model was further validated by mean square error, regression analysis, and histogram convergence plot.
- The correctness and repeatability of the design schemes were further validated using reliability, effectiveness, MSE convergence analysis, correlation analysis, and bar charts.
- Table 1 gives a quick summary of some relevant studies conducted in the past and illustrates how they differ from the suggested approach (RP-LMS).

## 2. Mathematical Formulation

- ${W}_{0}=W\left(0\right)>0$, ${X}_{0}=X\left(0\right)>0$,
- ${Y}_{0}=Y\left(0\right)\ge 0$, ${Z}_{0}=Z\left(0\right)\ge 0$.

- Greenhouse gases that contribute to global warming W(t)
- Atmospheric temperature X(t)
- Planktonic individuals Y(t)
- The Fishing Community Z(t).

_{2}promotes the concentration of maritime phytoplankton, when the density of dissolved CO

_{2}is so high, it limits phytoplankton metabolism by lowering the volume of dissolved O

_{2}, which slows the growth of the aquatic community. The influence of saturated carbon dioxide and dissolved oxygen deficit limit the growth of marine fisheries. As a result, $chi1$ denotes the rise in planktonic inhabitants brought on by $C{O}_{2}$ absorption, $ch{i}_{2}$ reflects the decline in planktonic inhabitants brought on by warming, $ch{i}_{3}$ represents the improvement in planktonic inhabitants brought on by exploitation or expenditure of fishery resources, and $ch{i}_{4}$ denotes the reduction in planktonic inhabitants brought on by acidity. While ${\beta}_{1}$ represents the number of fish that rise as a result of eating plankton, ${\beta}_{2}$ represents the number of fish that decrease amount of absorbed $C{O}_{2}$, and ${\beta}_{3}$ represents the number of fish that decrease due to climate change. Table 2 summarises the parametric descriptions as well as the related values.

## 3. Design Methodology

## 4. Discussion on Symmetry in Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Abbreviation | Description |

ANNs | Artificial Neural Networks |

RP | Reverse Propagated |

LMS | Levenberg–Marquaradt Scheme |

G | Greenhouse Gases |

A | Atmospheric Temperature |

P | Planktonic Population |

F | Fish Population |

NN | Neural Network |

NODEs | Nonlinear Ordinary Differential Equations |

MSE | Mean Square Error |

GHGs | Greenhouse Gases |

Deqs | Differential Equations |

FFN | Feed Forward Network |

MQE | Mean Quadratic Error |

$C{O}_{2}$ | Carbon dioxide gas |

${O}_{2}$ | Oxygen gas |

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**Figure 5.**Mean square error of RP-LMS through NNs for greenhouse gases and ambient temperature for Scenario 1. (

**a**)

**W(t)**; (

**b**)

**X(t)**.

**Figure 6.**Mean square error of RP-LMS through NNs for aquatic population and fish population for Scenario 1. (

**a**)

**Y(t)**, (

**b**)

**Z(t)**.

**Figure 7.**Mean square error of RP-LMS through NNs for greenhouse gases, ambient temperature, aquatic population, and fish population for Scenario 2. (

**a**)

**W(t)**, (

**b**)

**X(t)**, (

**c**)

**Y(t)**, (

**d**)

**Z(t)**.

**Figure 8.**Performance of RP-LMS through SNN in terms of mu, gradient, and validation checks for greenhouse gases, ambient temperature, aquatic population, and fish population for Case 1. (

**a**)

**W(t)**, (

**b**)

**X(t)**, (

**c**)

**Y(t)**, (

**d**)

**Z(t)**.

**Figure 9.**Performance of RP-LMS through SNN in terms of mu, gradient, and validation checks for greenhouse gases and ambient temperature for Case 2. (

**a**)

**W(t)**, (

**b**)

**X(t)**.

**Figure 10.**Performance of RP-LMA through SNN in terms of mu, gradient, and validation checks for aquatic population, and fish population for Case 2. (

**a**)

**Y(t)**, (

**b**)

**Z(t)**.

**Figure 11.**Regression analysis of RP-LMS through SNN for training, validation, testing, and all samples, respectively, for the GAPF model in Case 1. (

**a**)

**W(t)**, (

**b**)

**X(t)**, (

**c**)

**Y(t)**, (

**d**)

**Z(t)**.

**Figure 12.**Regression analysis of RP-LMS through SNN for training, validation, testing, and all samples, respectively, for the GAPF model in Case 2. (

**a**)

**W(t)**, (

**b**)

**X(t)**, (

**c**)

**Y(t)**, (

**d**)

**Z(t)**.

**Figure 13.**Error histogram for the proposed methodology in terms of greenhouse gases, ambient temperature, aquatic population, and fish population for Case Study 1. (

**a**)

**W(t)**, (

**b**)

**X(t)**, (

**c**)

**Y(t)**, (

**d**)

**Z(t)**.

**Figure 14.**Error histogram for the proposed methodology in terms of greenhouse gases and ambient temperature for Case Study 2. (

**a**)

**W(t)**, (

**b**)

**X(t)**.

**Figure 15.**Error histogram for the proposed methodology in terms of aquatic population and fish population for Case Study 2. (

**a**)

**Y(t)**, (

**b**)

**Z(t)**.

**Figure 16.**Comparison between the numerical reference solution and the proposed RP-LMS through SNN for greenhouse gases. (

**a**)

**Impact of ${\Delta}_{2}$ on greenhouse gases**, (

**b**)

**collective analysis of Absolute Error**, (

**c**)

**analysis of case 1’s errors**, (

**d**)

**analysis of case 2’s errors**, (

**e**)

**analysis of case 3’s errors**.

**Figure 17.**Comparison between the numerical reference solution and the proposed RP-LMS through SNN for ambient temperature. (

**a**)

**Impact of ${\beta}_{3}$ on atmospheric temperature**, (

**b**)

**collective analysis of Absolute Error,**(

**c**)

**analysis of case 1’s errors**, (

**d**)

**analysis of case 2’s errors**, (

**e**)

**analysis of case 3’s errors**.

**Figure 18.**Comparison between the numerical reference solution and the proposed RP-LMS through SNN for aquatic population. (

**a**)

**Impact of ${\chi}_{1}$ on planktonic population**, (

**b**)

**collective analysis of Absolute Error**, (

**c**)

**analysis of case 1’s errors**, (

**d**)

**analysis of case 2’s errors**, (

**e**)

**analysis of case 3’s errors**.

**Figure 19.**Comparison between the numerical reference solution and the proposed RP-LMS through SNN for fish population. (

**a**)

**Impact of ${\beta}_{3}$ on fish population**, (

**b**)

**collective analysis of Absolute Error**, (

**c**)

**analyze of error for case 1**, (

**d**)

**analyze of error for case 2**, (

**e**)

**analyze of error for case 3**.

Literature | Parameter Growth Rate of | Effect of Climate Change | Solution | Case | ||||
---|---|---|---|---|---|---|---|---|

Review | Growth and Rate of Decline | on Marine Ecosystem | Type | Study | ||||

Secondary | Measured | Global Warming | Marine Plankton | Fish Community | Exact | Heuristic | ||

Hinners et al. [25] | Yes | No | Yes | Yes | No | Yes | No | No |

Asch et al. [10] | Yes | No | Yes | Yes | Yes | No | No | No |

Mandal et al. [16] | No | Yes | Yes | Yes | No | Yes | No | No |

Speers et al. [8] | Yes | No | Yes | Yes | Yes | Yes | No | No |

Sekerci and | No | Yes | Yes | Yes | No | Yes | No | No |

Petrovskii [13] | ||||||||

This study | No | Yes | Yes | Yes | Yes | Yes | Yes | Yes |

Symbols | Descriptions | Values |
---|---|---|

${n}_{1}$ | GHG levels in the oceans are naturally growing. | 0.00095 kg/km${}^{2}$ |

${n}_{2}$ | Plankton-feeding fish growth during the period | 0.099 ${}^{\circ}$C |

${n}_{3}$ | The normal rate of expansion of the aquatic species | 0.00225 km${}^{-3}$ |

${n}_{4}$ | The perfectly natural increase in the population of fish | 0.0002/1000 |

${\Delta}_{1}$ | The oceanic aquatic demographic’s rate of GHG production | 0.0029 kg/km${}^{2}$ |

${\Delta}_{2}$ | GHG absorption rate by planktonic populations in the oceans | 0.00099 kg/km${}^{2}$ |

${\Delta}_{3}$ | Due to rising temperatures, the rate at which GHGs are emitted | 1.0 $\mathsf{\mu}$ kg/km${}^{2}$ |

${\chi}_{1}$ | Planktonic population growth rate due to $C{O}_{2}$ | 0.00108 km${}^{-3}$ |

${\chi}_{2}$ | Nutrient rates are slowing as a result of global warming | 0.00001 km${}^{-3}$ |

${\chi}_{3}$ | The rate at which fish consume plankton. | 0.0031 km${}^{-3}$ |

${\chi}_{4}$ | Effects of acidity on plankton loss | 10.1 $\mathsf{\mu}$ km${}^{-3}$ |

${\psi}_{1}$ | GHG-induced increase in the rate of surface temperatures | 0.00025 ${\phantom{\rule{4pt}{0ex}}}^{\circ}$C |

${\psi}_{2}$ | Rate of temperature absorption by the planktonic community | 0.00565 ${\phantom{\rule{4pt}{0ex}}}^{\circ}$C |

${\beta}_{1}$ | Plankton feeding/consumption increases the growth rate of fish populations. | $175\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$/1000 |

${\beta}_{2}$ | The fish population declines to owe to acidification caused by GHGs. | $190\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$/1000 |

${\beta}_{3}$ | Global warming is slowing the growth of fish populations | $61\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$/1000 |

$\alpha $ | Stability in the level of saturation | 0.01 |

${A}_{1}$ | Planktonic population’s capacity for sustained growth | 1,000,000 km |

${A}_{2}$ | The population’s ability to sustain themselves | 10,000 km |

Scenarios | Cases | Parameters | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\Delta}_{\mathbf{1}}$ | ${\Delta}_{\mathbf{2}}$ | ${\Delta}_{\mathbf{3}}$ | ${\mathbf{\chi}}_{\mathbf{1}}$ | ${\mathbf{\chi}}_{\mathbf{2}}$ | ${\mathbf{\chi}}_{\mathbf{3}}$ | ${\mathbf{\chi}}_{\mathbf{4}}$ | ${\mathbf{\beta}}_{\mathbf{1}}$ | ${\mathbf{\beta}}_{\mathbf{2}}$ | ${\mathbf{\beta}}_{\mathbf{3}}$ | $\mathbf{\alpha}$ | ${\mathbf{\psi}}_{\mathbf{1}}$ | ${\mathbf{\psi}}_{\mathbf{2}}$ | ||

1 | 1 | 0.0029 | 0.00101 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 |

2 | 0.0029 | 0.00108 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 | |

3 | 0.0029 | 0.00114 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 | |

2 | 1 | 0.0029 | 0.00099 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 |

2 | 0.0029 | 0.00099 | 1 | 0.00114 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 | |

3 | 0.0029 | 0.00099 | 1 | 0.00118 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.01 | 0.00025 | 0.00565 | |

3 | 1 | 0.0029 | 0.00099 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 0.000061 | 0.01 | 0.00025 | 0.00565 |

2 | 0.0029 | 0.00099 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 0.00071 | 0.01 | 0.00025 | 0.00565 | |

3 | 0.0029 | 0.00099 | 1 | 0.0011 | 0.00001 | 0.0031 | 10.1 | 0.000000175 | 0.00000019 | 0.001361 | 0.01 | 0.00025 | 0.00565 |

Index | Description |
---|---|

Number of layers | Three |

Layers structure | One input, one hidden, and one output layer |

Hidden neurons | 20–80 |

Training samples | 875 samples |

Testing samples | 188 samples |

Validation samples | 188 sample |

Learning methodology | Levenberg–Marquaradt Scheme |

Label target data | Created with Adams numerical method |

Maximum iteration | 1000 |

Activation function | Sigmoid Symmetric Transfer Function |

**Table 5.**Numerical analysis of the RP-LMS in terms of mu, gradient, performance, and number of iterations for Scenarios 1 and 2.

Fitness on MSN | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Scenario | Case Index | Neuron Setting | Training | Validation | Testing | Gradient | Performance | Mu | Epochs | R |

w(t) | 80 | 4.67 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 5.61 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.36 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 5.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 7 | 1 | |

1 | X(t) | 80 | 5.61 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.43 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 2.12 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.4817 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.97 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 7 | 1 |

Y(t) | 80 | 3.11 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 3.81 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 3.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 7.91 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1.43 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 9 | 1 | |

Z(t) | 80 | 2.06 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.92 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 4.4005 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 2.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 7 | 1 | |

w(t) | 45 | 4.09 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.97 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.45 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 9.91 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.81 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 234 | 1 | |

2 | X(t) | 45 | 1.24 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 2.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 1.64 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 9.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1.82 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 368 | 1 |

Y(t) | 45 | 1.61 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.82 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.87 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 3.62 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1000 | 1 | |

Z(t) | 45 | 6.29 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 3.75 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 3.73 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.75 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1000 | 1 |

**Table 6.**Statistical analysis of numerical solution and proposed methodology for varying GHG concentration through aquatic inhabitants in coral reefs, as well as error analysis for Greenhouse Gases.

${\mathsf{\Delta}}_{2}$ = 0.00101 | ${\mathsf{\Delta}}_{2}$ = 0.00108 | ${\mathsf{\Delta}}_{2}$ = 0.00114 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | RK4-W(t) | RP-LMS | Absolute Errors | RK4-W(t) | RP-LMS | Absoluten Errors | RK4-W(t) | RP-LMS | Absolute Errors |

0 | 0.04 | 0.040001 | $9.03\times {10}^{-6}$ | 0.04 | 0.039975 | 2.46$\times {10}^{-5}$ | 0.04 | 0.04 | 4.69 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

3.96 | 0.040976 | 0.040969 | 6.57 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.040749 | 0.040753 | $4.19\times {10}^{-6}$ | 0.040468 | 0.040467 | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

7.96 | 0.041994 | 0.041987 | 6.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.041528 | 0.04153 | $1.04\times {10}^{-6}$ | 0.040953 | 0.040954 | $1.08\times {10}^{-6}$ |

11.96 | 0.043052 | 0.043048 | 4.47 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.042335 | 0.042332 | 2.96 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.041454 | 0.041455 | $4.06\times {10}^{-7}$ |

15.96 | 0.044156 | 0.044153 | 2.97 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.043174 | 0.043173 | 9.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.041974 | 0.041972 | 1.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

19.96 | 0.045312 | 0.045309 | 3.15 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.044051 | 0.044057 | $6.90\times {10}^{-6}$ | 0.042517 | 0.042518 | $6.48\times {10}^{-7}$ |

23.96 | 0.04653 | 0.046531 | $1.78\times {10}^{-6}$ | 0.044971 | 0.044968 | 3.16 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.043086 | 0.043089 | $3.24\times {10}^{-6}$ |

27.96 | 0.047816 | 0.047825 | $8.56\times {10}^{-6}$ | 0.045941 | 0.045937 | 4.35 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.043685 | 0.043687 | $2.29\times {10}^{-6}$ |

31.96 | 0.049181 | 0.049189 | $7.98\times {10}^{-6}$ | 0.046968 | 0.046978 | $9.65\times {10}^{-6}$ | 0.044318 | 0.04432 | $2.31\times {10}^{-6}$ |

35.96 | 0.050633 | 0.050642 | $9.26\times {10}^{-6}$ | 0.048059 | 0.048063 | $3.67\times {10}^{-6}$ | 0.044989 | 0.044988 | 8.63 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

39.96 | 0.052184 | 0.052187 | $3.20\times {10}^{-6}$ | 0.049222 | 0.049216 | 6.21 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.045704 | 0.045706 | $1.98\times {10}^{-6}$ |

43.96 | 0.053845 | 0.053845 | $5.21\times {10}^{-6}$ | 0.050465 | 0.050469 | $3.65\times {10}^{-6}$ | 0.046466 | 0.046469 | $2.71\times {10}^{-6}$ |

47.96 | 0.055628 | 0.055624 | 4.63 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.051798 | 0.051803 | $5.63\times {10}^{-6}$ | 0.047283 | 0.047284 | $1.28\times {10}^{-6}$ |

50 | 0.056589 | 0.056578 | 1.08 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.052515 | 0.052517 | $2.28\times {10}^{-6}$ | 0.047722 | 0.04772 | 2.29 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

**Table 7.**Statistical analysis of the numerical solution and the proposed methodology for varying the absorption probability of emission by aquatic inhabitants in seas, as well as the error analysis for ambient temperature, are presented.

${\mathit{\beta}}_{3}$ = 0.00061 | ${\mathit{\beta}}_{3}$ = 0.000711 | ${\mathit{\beta}}_{3}$ = 0.001361 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | RK4-X(t) | RP-LMS | Absolute Errors | RK4-X(t) | RP-LMS | Absolute Errors | RK4-X(t) | RP-LMS | Absolute Errors |

0 | 0.07 | 0.069998 | 1.61 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.07 | 0.07003 | $2.96\times {10}^{-5}$ | 0.07 | 0.069876 | 0.000124 |

3.96 | 0.070063 | 0.070064 | $6.04\times {10}^{-7}$ | 0.070061 | 0.070061 | 4.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.070059 | 0.070058 | 5.92 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

7.96 | 0.070209 | 0.070208 | 1.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.070195 | 0.070196 | $7.51\times {10}^{-7}$ | 0.070177 | 0.070179 | $2.08\times {10}^{-6}$ |

11.96 | 0.070483 | 0.070486 | $2.72\times {10}^{-6}$ | 0.070436 | 0.070437 | $2.28\times {10}^{-7}$ | 0.070377 | 0.070378 | $1.85\times {10}^{-7}$ |

15.96 | 0.070933 | 0.070932 | 1.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.070822 | 0.07082 | 2.34 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.070683 | 0.070679 | 3.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

19.96 | 0.071607 | 0.071607 | $5.90\times {10}^{-8}$ | 0.071391 | 0.07139 | 9.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.071117 | 0.071117 | 1.74 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

23.96 | 0.072556 | 0.072555 | 3.47 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.072181 | 0.072187 | $6.64\times {10}^{-6}$ | 0.071706 | 0.07171 | $4.05\times {10}^{-6}$ |

27.96 | 0.073834 | 0.07383 | 3.55 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.073235 | 0.073232 | 3.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.072478 | 0.072476 | 1.72 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

31.96 | 0.075503 | 0.075507 | $4.80\times {10}^{-6}$ | 0.074602 | 0.074593 | 9.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.073463 | 0.073458 | 4.69 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

35.96 | 0.077632 | 0.077631 | 1.76 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.076335 | 0.076353 | $1.77\times {10}^{-5}$ | 0.074694 | 0.074698 | $4.41\times {10}^{-6}$ |

39.96 | 0.080304 | 0.080302 | 2.12 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.078495 | 0.07852 | $2.49\times {10}^{-5}$ | 0.076209 | 0.07622 | $1.15\times {10}^{-5}$ |

43.96 | 0.083614 | 0.083603 | 1.17 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.081153 | 0.081124 | 2.88 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.078051 | 0.078037 | 1.37 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

47.96 | 0.087678 | 0.087703 | $2.53\times {10}^{-5}$ | 0.084393 | 0.084384 | 8.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.08027 | 0.080262 | 8.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

50 | 0.090085 | 0.089821 | 0.000264 | 0.086303 | 0.086154 | 0.000149 | 0.081565 | 0.081496 | 6.90 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

**Table 8.**Statistical analysis of numerical solution and proposed methodology for varying aquatic rate of growth due to $C{O}_{2}$, as well as error analysis for aquatic population.

${\mathit{\chi}}_{1}$ = 0.00101 | ${\mathit{\chi}}_{1}$ = 0.00108 | ${\mathit{\chi}}_{1}$ = 0.00114 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | RK4-Y(t) | RP-LMS | Absolute Errors | RK4-Y(t) | RP-LMS | Absolute Errors | RK4-Y(t) | RP-LMS | Absolute Errors |

0 | 17.5 | 17.5 | 1.58 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 17.5 | 17.5 | 1.18 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 17.5 | 17.5 | 7.79 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

3.96 | 17.46275 | 17.46275 | $6.97\times {10}^{-7}$ | 17.46603 | 17.46603 | $5.58\times {10}^{-7}$ | 17.47015 | 17.47015 | $2.35\times {10}^{-7}$ |

7.96 | 17.39636 | 17.39636 | $6.79\times {10}^{-7}$ | 17.40947 | 17.40947 | $3.79\times {10}^{-7}$ | 17.42596 | 17.42596 | 6.43 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

11.96 | 17.30152 | 17.30152 | $3.77\times {10}^{-7}$ | 17.33074 | 17.33074 | $4.76\times {10}^{-7}$ | 17.36765 | 17.36765 | 5.78 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

15.96 | 17.17871 | 17.1787 | 4.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 17.23002 | 17.23002 | $2.45\times {10}^{-7}$ | 17.29512 | 17.29512 | $2.48\times {10}^{-7}$ |

19.96 | 17.02842 | 17.02842 | 4.90 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 17.10746 | 17.10746 | 6.59 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 17.20828 | 17.20828 | 2.91 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

23.96 | 16.8512 | 16.8512 | $6.22\times {10}^{-7}$ | 16.96324 | 16.96324 | $6.17\times {10}^{-7}$ | 17.10699 | 17.10699 | 1.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

27.96 | 16.64766 | 16.64766 | 5.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 16.79754 | 16.79754 | $2.25\times {10}^{-7}$ | 16.9911 | 16.9911 | $4.06\times {10}^{-11}$ |

31.96 | 16.41845 | 16.41845 | 2.25 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 16.61057 | 16.61056 | 6.34 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 16.86044 | 16.86044 | $4.60\times {10}^{-7}$ |

35.96 | 16.16432 | 16.16432 | $7.46\times {10}^{-7}$ | 16.40255 | 16.40255 | $5.56\times {10}^{-8}$ | 16.71482 | 16.71482 | 3.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

39.96 | 15.88609 | 15.88609 | $4.01\times {10}^{-7}$ | 16.17379 | 16.17379 | $3.62\times {10}^{-7}$ | 16.55409 | 16.55409 | $5.06\times {10}^{-7}$ |

43.96 | 15.5847 | 15.5847 | 6.51 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 15.92463 | 15.92463 | 9.43 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 16.37806 | 16.37806 | 4.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

47.96 | 15.26118 | 15.26118 | $6.81\times {10}^{-7}$ | 15.6555 | 15.6555 | $4.45\times {10}^{-7}$ | 16.18659 | 16.18659 | 5.13 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

50 | 15.08803 | 15.08803 | $1.39\times {10}^{-6}$ | 15.51071 | 15.51071 | $1.11\times {10}^{-6}$ | 16.08296 | 16.08296 | $1.00\times {10}^{-6}$ |

**Table 9.**Statistical analysis of numerical solution and the proposed methodology for varying hampering rate of fish populations by global warming and also show the error analysis for fish population.

${\mathit{\beta}}_{3}$ = 0.00101 | ${\mathit{\beta}}_{3}$ = 0.001361 | ${\mathit{\beta}}_{3}$ = 0.00071 | |||||||
---|---|---|---|---|---|---|---|---|---|

t | RK4-Z(t) | RP-LMS | Absolute Error | RK4-Z(t) | RP-LMS | Absolute Error | RK4-Z(t) | RP-LMS | Absolute Error |

0 | 7.8 | 7.79998 | 1.95 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 7.8 | 7.799982 | 1.77 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 7.8 | 7.8 | 4.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

3.96 | 6.586341 | 6.58634 | 6.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 6.22928 | 6.229279 | 1.41 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 7.057025 | 7.057025 | 2.74 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

7.96 | 5.553165 | 5.553169 | $4.72\times {10}^{-6}$ | 4.966141 | 4.966142 | $1.03\times {10}^{-6}$ | 6.378245 | 6.378246 | $1.05\times {10}^{-7}$ |

11.96 | 4.687032 | 4.687033 | $5.13\times {10}^{-7}$ | 3.967926 | 3.967924 | 2.17 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.765623 | 5.765623 | 3.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

15.96 | 3.96571 | 3.965706 | 3.45 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.185818 | 3.185819 | $7.15\times {10}^{-7}$ | 5.214275 | 5.214275 | $8.22\times {10}^{-8}$ |

19.96 | 3.369767 | 3.369771 | $4.14\times {10}^{-6}$ | 2.578881 | 2.578882 | $9.97\times {10}^{-7}$ | 4.719984 | 4.719985 | $3.02\times {10}^{-7}$ |

23.96 | 2.881718 | 2.881713 | 4.86 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.112426 | 2.112426 | 3.76 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 4.278938 | 4.278939 | $3.05\times {10}^{-7}$ |

27.96 | 2.485687 | 2.485692 | $5.11\times {10}^{-6}$ | 1.757278 | 1.757278 | 3.88 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.887535 | 3.887535 | 3.45 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ |

31.96 | 2.167335 | 2.167331 | 4.85 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 1.489318 | 1.489318 | 7.67 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.542271 | 3.542271 | 5.84 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

35.96 | 1.913895 | 1.9139 | $5.90\times {10}^{-6}$ | 1.289007 | 1.289007 | 2.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.239676 | 3.239676 | 4.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ |

39.96 | 1.71418 | 1.714177 | 2.61 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 1.140805 | 1.140804 | 4.41 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.976299 | 2.976299 | $2.42\times {10}^{-7}$ |

43.96 | 1.558555 | 1.558548 | 7.45 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 1.032519 | 1.03252 | $1.02\times {10}^{-6}$ | 2.748719 | 2.74872 | $1.76\times {10}^{-7}$ |

47.96 | 1.438823 | 1.438821 | 2.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.954651 | 0.95465 | 9.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 2.553575 | 2.553575 | $2.23\times {10}^{-7}$ |

50 | 1.389308 | 1.389327 | $1.86\times {10}^{-5}$ | 0.924196 | 0.924203 | $7.54\times {10}^{-6}$ | 2.465482 | 2.465494 | $1.17\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Alhakami, H.; Kamal, M.; Sulaiman, M.; Alhakami, W.; Baz, A.
A Machine Learning Strategy for the Quantitative Analysis of the Global Warming Impact on Marine Ecosystems. *Symmetry* **2022**, *14*, 2023.
https://doi.org/10.3390/sym14102023

**AMA Style**

Alhakami H, Kamal M, Sulaiman M, Alhakami W, Baz A.
A Machine Learning Strategy for the Quantitative Analysis of the Global Warming Impact on Marine Ecosystems. *Symmetry*. 2022; 14(10):2023.
https://doi.org/10.3390/sym14102023

**Chicago/Turabian Style**

Alhakami, Hosam, Mustafa Kamal, Muhammad Sulaiman, Wajdi Alhakami, and Abdullah Baz.
2022. "A Machine Learning Strategy for the Quantitative Analysis of the Global Warming Impact on Marine Ecosystems" *Symmetry* 14, no. 10: 2023.
https://doi.org/10.3390/sym14102023