# Validation of a Mathematical Model for Green Algae (Raphidocelis Subcapitata) Growth and Implications for a Coupled Dynamical System with Daphnia Magna

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## Abstract

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## 1. Introduction

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Asymptotic Theory

`fmincon`in MATLAB (Mathworks, 2015b, Natick, MA, USA, 2015) with function and step tolerances of ${10}^{-20}$ and 1000 iterations.

#### 2.3. Boostrapping

- First estimate ${\widehat{\theta}}^{0}$ from the entire sample ${\left\{{y}_{i}\right\}}_{i=1}^{n}$ using OLS.
- Using this estimate, define the standardized residuals$${\overline{r}}_{i}=\sqrt{\frac{n}{n-p}}({y}_{i}-f({t}_{i},{\widehat{\theta}}^{0}))$$
- Create a bootstrapping sample of size n using random sampling with replacement from the data (realizations) $\{{\overline{r}}_{1},\cdots ,{\overline{r}}_{n}\}$ to form a bootstrapping sample $\{{\overline{r}}_{1}^{m},\cdots ,{\overline{r}}_{n}^{m}\}$.
- Create bootstrap sample points$${y}_{i}^{m}=f({t}_{i},{\widehat{\theta}}^{0})+{r}_{i}^{m}$$
- Obtain a new estimate ${\widehat{\theta}}^{m+1}$ from the bootstrapping sample $\left\{{y}_{i}^{m}\right\}$ using OLS.
- Set $m=m+1$ and repeat steps 3–5 until $m\ge 1000$ (this can be any large value, but for these experiments we used $M=1000$).

#### 2.4. Model Comparison: Nested Restraint Tests

#### 2.5. Akaike Information Criterion

## 3. Models

#### 3.1. Logistic Model

#### 3.2. Bernoulli Model

#### 3.3. Gompertz Growth Model

#### 3.4. Logistic Model: Numerical Discretization

`ode45`algorithm in Matlab. In order to ensure that the logistic model can be coupled to a discrete time model for a D. magna population in which the population size is updated once per day [5], we investigated the logistic model using a forward Euler scheme that was discretized on an hour basis. In this paper, we refer to this discrete Euler-method logistic model as the DEL model. This numerically discretized logistic model is given by the difference equation

## 4. Results

#### 4.1. Data fitting and Model Comparisons

#### 4.2. Uncertainty Analysis

#### 4.2.1. Uncertainty Analysis: Initial Condition

#### 4.2.2. Uncertainty Analysis: Growth Rate

#### 4.2.3. Uncertainty Analysis: Saturation Parameter

#### 4.2.4. Uncertainty Analysis: Bernoulli Model Parameter β

#### 4.3. Coupling to the Discrete-Time Daphnia magna Population Model

## 5. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Plots of forward solutions for the Logistic curve for the three replicates of the data. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as $cells\times {10}^{7}$/L. Data points are shown as “*”.

**Figure 2.**Plots of forward solutions for the discrete Euler-method logistic (DEL) please confirm. model for the three replicates of the data from left to right. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as $cells\times {10}^{7}$/L. Data points are shown as “*”.

**Figure 3.**Plots of forward solutions for the Gompertz curve for the three replicates of the data. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as $cells\times {10}^{7}$/L. Data points are shown as “*”.

**Figure 4.**Plots of forward solutions for the Bernoulli curve for the three replicates of the data. Replicate one is on top and three is on the bottom.The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as $cells\times {10}^{7}$/L. Data points are shown as “*”.

**Figure 5.**Simulations of the coupled daphnia and green algae dynamics model resulting in steady state behavior (

**Left**, $c=0.01$) and sustained oscillations (

**Right**, $c=0.04$).

Replicate | Gompertz | Logistic | Bernoulli | Discrete Euler-Method Logistic (DEL) |
---|---|---|---|---|

1 | −69.4203 | −71.5919 | −69.2189 | −72.6155 |

2 | −84.2435 | −89.0016 | −89.3905 | −90.4114 |

3 | −71.3972 | −74.2560 | −72.4414 | −75.3515 |

**Table 2.**Model comparison ${\widehat{u}}_{n}$ scores for the continuous Logistic model compared to Bernoulli model for each replicate. We also chose to fix the initial condition ${X}_{0}$ at the seed population value to enunciate model improvement if ${X}_{0}$ was unalterable. Note that values less than 2.706 indicate the restricted model is better.

Bernoulli Restricted to: | Bernoulli Restricted to: | |
---|---|---|

Replicate | Logistic | Logistic with X_{0} fixed |

1 | 0.5935 | 0.7233 |

2 | 2.4718 | 3.6216 |

3 | 1.1733 | 3.4118 |

Asymptotic Results: β | Replicate | Estimate | SE |

1 | 2.1646 | 2.5440 | |

2 | 3.4574 | 2.8118 | |

3 | 2.8188 | 2.8758 | |

Bootstrapping Results: β | Replicate | Estimate | SE |

1 | 38.31 | 113.72 | |

2 | 29.78 | 92.89 | |

3 | 43.27 | 113.81 |

Parameter/Variable | Description | Units |
---|---|---|

$p(t,i)$ | Number of daphnids of age i | # of daphnids |

$N\left(t\right)$ | Total population size at time t $:={\sum}_{i=1}^{{i}_{max}}p(t,i)$ | # of daphnids |

q | Density-dependent fecundity constant | dimensionless |

$\alpha \left(i\right)$ | Density-independent fecundity rates | # neonates·daphnid^{−1}·day^{−1} |

μ | Density-independent survival rate | day^{−1} |

τ | Delay for density-dependent fecundity | days |

c | Density-dependent survival constant | dimensionless |

$M\left(t\right)$ | Total biomass at time t $:={\sum}_{i=1}^{{i}_{max}}p(t,i)\frac{k{Z}_{0}{e}^{ri}}{k+{Z}_{0}({e}^{ri}-1)}$ | mm |

k | Average maximum daphnid size (major axis) | mm |

r | Average daphnid growth rate | mm/hour |

${Z}_{0}$ | Average neonate size (major axis) | mm |

R | Intrinsic growth rate of algae | cells $\xb7{10}^{7}\xb7$L^{−1}·day^{−1} |

K | Algal population carrying capacity | cells $\xb7{10}^{7}\xb7$L^{−1} |

δ | Density dependent predation constant | mm^{−1}·cells$\xb7{10}^{-7}$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Stemkovski, M.; Baraldi, R.; Flores, K.B.; Banks, H.T.
Validation of a Mathematical Model for Green Algae (*Raphidocelis Subcapitata*) Growth and Implications for a Coupled Dynamical System with *Daphnia Magna*. *Appl. Sci.* **2016**, *6*, 155.
https://doi.org/10.3390/app6050155

**AMA Style**

Stemkovski M, Baraldi R, Flores KB, Banks HT.
Validation of a Mathematical Model for Green Algae (*Raphidocelis Subcapitata*) Growth and Implications for a Coupled Dynamical System with *Daphnia Magna*. *Applied Sciences*. 2016; 6(5):155.
https://doi.org/10.3390/app6050155

**Chicago/Turabian Style**

Stemkovski, Michael, Robert Baraldi, Kevin B. Flores, and H.T. Banks.
2016. "Validation of a Mathematical Model for Green Algae (*Raphidocelis Subcapitata*) Growth and Implications for a Coupled Dynamical System with *Daphnia Magna*" *Applied Sciences* 6, no. 5: 155.
https://doi.org/10.3390/app6050155