# A New Flexible Sigmoidal Growth Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Parameter Estimation

_{i}the ith observation of biomass, $\hat{y}$

_{i}the predicted biomass, n the sample size, and k the number of parameters in the growth model.

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data Accessibility

## References

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**Figure 1.**Fitted growth curves of the new growth model for the datasets of six plant species. Black points represent the actual biomass observed.

**Figure 2.**Fitted growth curves of the Richards growth model for the datasets of six plant species. Black.points represent the actual biomass observed.

**Figure 3.**Fitted growth curves of the Gompertz growth model for the datasets of six animal species. Black points represent the actual biomass observed.

**Figure 4.**Fitted growth curves of the ontogenetic growth model for the datasets of six animal species. Black points represent the actual biomass observed.

**Figure 5.**Fitted growth curves of the new growth model for the datasets of six animal species. Black points represent the actual biomass observed.

**Figure 6.**Fitted growth curves of the logistic growth model for the datasets of six animal species. Black points represent the actual biomass observed.

**Figure 7.**Fitted growth curves of the logistic growth model for the datasets of six plant species. Black points represent the actual biomass observed.

**Figure 8.**Fitted growth curves of the ontogenetic growth model for the datasets of six plant species. Black points represent the actual biomass observed.

**Figure 9.**Fitted growth curves of the Gompertz growth model for the datasets of six plant species. Black points represent the actual biomass observed

**Table 1.**Lower and upper bounds of parameters of each model optimized by differential evolution method. NSG and OGM represent the new growth model and the ontogenetic growth model, respectively.

Model | Parameter | Lower Bound | Upper Bound |
---|---|---|---|

a | 1.00 × 10^{−9} | 1.00 × 10^{−3} | |

b | 0 | 5 | |

NSG | ${w}_{max}$ | 0 | 300 |

m | −200 | 200 | |

n | 0 | 500 | |

${w}_{max}$ | 0 | 500 | |

Richards | k | 0 | 1 |

${t}_{m}$ | 0 | 500 | |

v | 0 | 1 | |

${w}_{max}$ | 0 | 500 | |

Logistic | k | 0 | 1 |

${t}_{m}$ | 0 | 500 | |

${w}_{max}$ | 0 | 500 | |

Gompertz | k | 0 | 1 |

${t}_{m}$ | 0 | 500 | |

${w}_{max}$ | 0 | 500 | |

OGM | ${w}_{0}$ | 0 | 1 |

a | 0 | 1 |

**Table 2.**Akaike’s information criterion (AIC) and R

^{2}values of the five growth models for the datasets of plants and animals. NSG and OGM represent the new sigmoidal growth model and the ontogenetic growth model, respectively.

Code | Models Name | NSG | Logistic | Gompertz | Richards | OGM | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Species | R^{2} | AIC | ΔAIC | R^{2} | AIC | ΔAIC | R^{2} | AIC | ΔAIC | R^{2} | AIC | ΔAIC | R^{2} | AIC | ΔAIC | |

1 | Black soybean | 0.995 | 18.77 | 15.63 | 0.996 | 11.14 | 8 | 0.993 | 20.91 | 17.77 | 0.998 | 3.14 | 0 | 0.985 | 31.04 | 27.9 |

2 | Kidney bean | 0.993 | −11.08 | 10.48 | 0.991 | −11.89 | 9.67 | 0.996 | −21.56 | 0 | 0.996 | −20.24 | 1.32 | 0.995 | −17.94 | 3.62 |

3 | Adzuki bean | 0.995 | −1.55 | 0.3 | 0.992 | 1.65 | 3.5 | 0.987 | 9.18 | 11.03 | 0.995 | −1.85 | 0 | 0.953 | 28.95 | 30.8 |

4 | Mung bean | 0.998 | −11.75 | 11.5 | 0.998 | −4.21 | 19.04 | 0.993 | 12.88 | 36.13 | 0.999 | −23.25 | 0 | 0.978 | 29.21 | 52.46 |

5 | Cotton | 0.994 | 40.29 | 9.64 | 0.993 | 37.98 | 7.33 | 0.99 | 44.77 | 14.12 | 0.996 | 30.65 | 0 | 0.966 | 62.58 | 31.93 |

6 | Sweet sorghum | 0.996 | 58.22 | 3.31 | 0.993 | 63.47 | 8.56 | 0.986 | 73.23 | 18.32 | 0.996 | 54.91 | 0 | 0.949 | 93.06 | 38.15 |

7 | Guppy | 0.993 | −16.84 | 5.89 | 0.994 | −22.73 | 0 | 0.993 | −19.8 | 2.93 | 0.995 | −21.54 | 1.19 | 0.991 | −16.28 | 6.45 |

8 | Robin | 0.997 | −16.61 | 0.47 | 0.995 | −17.08 | 0 | 0.992 | −11.29 | 5.79 | 0.99 | −5.85 | 5.44 | 0.991 | −9.26 | 7.82 |

9 | Shrew | 0.992 | −44.98 | 3.76 | 0.992 | −48.74 | 0 | 0.989 | −44.68 | 4.06 | 0.992 | −46.96 | 1.78 | 0.987 | −42.53 | 6.21 |

10 | Rabbit | 0.994 | −22.52 | 22.03 | 0.998 | −44.55 | 0 | 0.997 | −36.5 | 8.05 | 0.998 | −42.57 | 1.98 | 0.996 | −32.82 | 11.73 |

11 | Florida scrub jay | 0.989 | 50.13 | 3.51 | 0.988 | 47.39 | 0.77 | 0.989 | 46.62 | 0 | 0.989 | 48.21 | 1.59 | 0.988 | 47.63 | 1.01 |

12 | Western scrub jay | 0.999 | −10.43 | 0 | 0.999 | −8.05 | 2.38 | 0.997 | 5.09 | 15.52 | 0.999 | −5.92 | 4.51 | 0.996 | 11.03 | 21.46 |

**Table 3.**Estimated parameter values of the five growth models for the datasets of plants and animals. NSG and OGM represent the new growth model and the ontogenetic growth model, respectively.

Species | NSG | Richards | Logistic | Gompertz | OGM | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | m | n | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | k | ${\mathit{t}}_{\mathit{m}}$ | v | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | k | ${\mathit{t}}_{\mathit{m}}$ | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | k | ${\mathit{t}}_{\mathit{m}}$ | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{w}}_{0}$ | a | |

Black soybean | 3.24 × 10^{−4} | 4.79 | 50.76 | 30.36 | 83.23 | 52.77 | 0.25 | 72.76 | 3.58 | 67.31 | 0.09 | 72.14 | 147.76 | 0.03 | 86.00 | 510.94 | 1.00 × 10^{−25} | 0.19 |

Kidney bean | 6.77 × 10^{−6} | 1.23 | 14.20 | −26.41 | 72.41 | 17.75 | 0.04 | 39.00 | −0.20 | 14.80 | 0.10 | 44.60 | 16.79 | 0.05 | 40.33 | 20.07 | 1.03 × 10^{−24} | 0.29 |

Adzuki bean | 7.64 × 10^{−5} | 1.49 | 22.91 | 29.55 | 73.55 | 22.94 | 0.35 | 59.72 | 3.67 | 23.59 | 0.15 | 55.99 | 24.93 | 0.09 | 52.03 | 67.98 | 2.43 × 10^{−24} | 0.21 |

Mung bean | 2.44 × 10^{−4} | 3.96 | 35.10 | 26.54 | 77.34 | 35.28 | 0.28 | 65.34 | 3.32 | 38.43 | 0.13 | 62.40 | 46.57 | 0.06 | 60.23 | 305.78 | 5.72 × 10^{−24} | 0.18 |

Cotton | 8.40 × 10^{−4} | 8.91 | 88.95 | 29.86 | 80.31 | 90.05 | 0.54 | 74.05 | 7.52 | 111.70 | 0.11 | 69.94 | 180.64 | 0.04 | 73.99 | 521.88 | 1.00 × 10^{−25} | 0.24 |

Sweet sorghum | 2.33 × 10^{−3} | 11.02 | 185.78 | 27.48 | 74.06 | 186.95 | 0.43 | 65.45 | 4.71 | 201.58 | 0.14 | 61.72 | 236.38 | 0.07 | 59.17 | 524.10 | 1.05 × 10^{−24} | 0.35 |

Guppy | 1.37 × 10^{−7} | 0.66 | 13.68 | −246.67 | 68.91 | 14.15 | 0.08 | 26.84 | 0.64 | 14.02 | 0.09 | 28.80 | 14.56 | 0.06 | 22.02 | 14.82 | 0.26 | 0.38 |

Robin | 7.90 × 10^{−4} | 0.86 | 18.38 | −59.23 | 11.30 | 24.11 | 0.19 | 3.88 | −0.27 | 19.75 | 0.44 | 5.20 | 21.79 | 0.26 | 4.17 | 22.87 | 0.89 | 1.87 |

Shrew | 8.18 × 10^{−5} | 0.82 | 3.37 | −40.70 | 14.47 | 3.50 | 0.41 | 6.98 | 1.37 | 3.54 | 0.36 | 6.60 | 3.74 | 0.22 | 5.04 | 3.85 | 0.13 | 1.05 |

Rabbit | 5.17 × 10^{−5} | 0.61 | 12.95 | −20.15 | 21.01 | 13.27 | 0.23 | 7.44 | 0.96 | 13.26 | 0.23 | 7.51 | 13.47 | 0.17 | 5.07 | 13.55 | 1.06 | 1.15 |

Florida scrub jay | 1.11 × 10^{−4} | 0.72 | 76.97 | −141.30 | 21.55 | 86.55 | 0.19 | 8.57 | 0.32 | 81.51 | 0.26 | 9.57 | 90.91 | 0.15 | 7.91 | 96.25 | 2.30 | 1.54 |

Western scrub jay | 3.18 × 10^{−4} | 1.07 | 59.99 | −226.33 | 17.12 | 67.06 | 0.28 | 8.76 | 0.83 | 65.99 | 0.30 | 8.91 | 78.65 | 0.16 | 7.93 | 87.04 | 1.76 | 1.47 |

**Table 4.**The ratio of estimated to observed values of maximum biomass. NSG and OGM represent the new growth model and the ontogenetic growth model, respectively.

Species | Observed (g) | NSG | Richards | Logistic | Gompertz | OGM |
---|---|---|---|---|---|---|

Black soybean | 50.20 | 1.01 | 1.05 | 1.34 | 2.94 | 10.18 |

Kidney bean | 14.35 | 0.99 | 1.24 | 1.03 | 1.17 | 1.40 |

Adzuki bean | 23.60 | 0.97 | 0.97 | 1.00 | 1.06 | 2.88 |

Mung bean | 35.60 | 0.99 | 0.99 | 1.08 | 1.31 | 8.59 |

Cotton | 90.80 | 0.98 | 0.99 | 1.23 | 1.99 | 5.75 |

Sweet sorghum | 191.50 | 0.97 | 0.98 | 1.05 | 1.23 | 2.74 |

Guppy | 0.15 | 0.94 | 0.98 | 0.97 | 1.00 | 1.02 |

Robin | 18.40 | 1.00 | 1.31 | 1.07 | 1.18 | 1.24 |

Shrew | 3.55 | 0.95 | 0.99 | 1.00 | 1.05 | 1.08 |

Rabbit | 1335.00 | 0.97 | 0.99 | 0.99 | 1.01 | 1.01 |

Florida scrub jay | 80.00 | 0.96 | 1.08 | 1.02 | 1.14 | 1.20 |

Western scrub jay | 59.75 | 1.00 | 1.12 | 1.10 | 1.32 | 1.46 |

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Cao, L.; Shi, P.-J.; Li, L.; Chen, G.
A New Flexible Sigmoidal Growth Model. *Symmetry* **2019**, *11*, 204.
https://doi.org/10.3390/sym11020204

**AMA Style**

Cao L, Shi P-J, Li L, Chen G.
A New Flexible Sigmoidal Growth Model. *Symmetry*. 2019; 11(2):204.
https://doi.org/10.3390/sym11020204

**Chicago/Turabian Style**

Cao, Liying, Pei-Jian Shi, Lin Li, and Guifen Chen.
2019. "A New Flexible Sigmoidal Growth Model" *Symmetry* 11, no. 2: 204.
https://doi.org/10.3390/sym11020204