# Computational Study on the Dynamics of a Consumer-Resource Model: The Influence of the Growth Law in the Resource

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## Abstract

**:**

## 1. Introduction

## 2. The Daphnia magna Model and the Growth of the Resource

- (1)
- The exponential growth: ${f}_{S}(t,s)=r\phantom{\rule{0.166667em}{0ex}}s$, $r\in \mathbb{R}$.This is the most basic and simplest growth function and, in essence, it is the growth relationship considered by Malthus (see [20], for instance). The parameter r is the specific growth rate. When $r\ne 0$, there is only an equilibrium for the differential equation associated: the trivial solution. If $r<0$, the trivial equilibrium is an attractor. If $r>0$, this solution is unbounded.
- (2)
- The chemostat growth function: ${f}_{S}(t,s)=r\phantom{\rule{0.166667em}{0ex}}\left(K-s\right)$, $r>0$, $K>0$.This is a modification of the exponential growth function with a limited increment that depends on the carrying capacity K. There is a stable nontrivial equilibrium that corresponds with the carrying capacity. When the ${s}^{0}<K$ population increases, and the growth is slower because the available resources decrease. In the case ${s}^{0}>K$, population decreases to the carrying capacity. It also can represent a population, given by s, that is growing up in an environment in which a constant quantity of feeding is provided (the carrying capacity) and r represents the dilution rate [20].
- (3)
- The Gompertz growth function: ${f}_{S}(t,s)=r\phantom{\rule{0.166667em}{0ex}}s\phantom{\rule{0.166667em}{0ex}}log\left({\displaystyle \frac{K}{s}}\right)$, $r>0$, $K>0$.The Gompertz equation was formulated originally as a law of decreasing survivorship, but it has also been employed to model the growth of plants, tumours, and fisheries [20]. There is a limited carrying capacity. The population has a similar behaviour as when the biological growth is described with the well-known logistic growth law because there are two equilibria: the unstable trivial equilibrium and a stable nontrivial one.

## 3. Numerical Experimentation

#### 3.1. Malthusian Growth

#### 3.2. Chemostat Growth

#### 3.3. Gompertz Growth

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Numerical Method

## References

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**Figure 1.**Malthusian growth. Behaviour of the numerical solution depending on r. ${t}_{down}$ (solid line): time at which the approximated resource shows an exponential decay, its value is lower than ${10}^{-20}$ (extinction). ${t}_{up}$ (dashed line): time at which it shows an exponential growth, its value is higher than ${10}^{20}$ (unbounded solution). Theoretical values in the absence of the consumer (dotted line): ${t}_{up}^{*}={\displaystyle \frac{1}{r}}\phantom{\rule{0.166667em}{0ex}}log\left({\displaystyle \frac{{10}^{20}}{{s}^{0}}}\right)$, $r>0$, and ${t}_{down}^{*}={\displaystyle \frac{1}{r}}\phantom{\rule{0.166667em}{0ex}}log\left({\displaystyle \frac{{10}^{-20}}{{s}^{0}}}\right)$, $r<0$.

**Figure 2.**Chemostat growth. $r=4.5$ and $K=3.4$. Stable equilibria: consumer extinction. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.77,3.4,0)$.

**Figure 3.**Chemostat growth. $r=4.5$ and $K=10$. Stable equilibria. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.80,4.09,199.60)$.

**Figure 4.**Chemostat growth. $r=4.5$, $K=10$, and ${s}^{0}=20$. Stable equilibria. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.80,4.09,199.60)$.

**Figure 5.**Chemostat growth. $r=4.5$ and $K=20.2$. Stable equilibria. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.80,4.09,543.85)$.

**Figure 6.**Gompertz growth. $r=3$ and $K=3.4$. Stable equilibria. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.77,3.40,0)$.

**Figure 7.**Gompertz growth. $r=3$ and $K=10$. Stable equilibria. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: orbit followed towards the equilibrium $(0.80,4.09,82.28)$.

**Figure 8.**Gompertz growth. $r=3$ and $K=14.73$. Stable limit cycle. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: stable limit cycle.

**Figure 9.**Gompertz growth. $r=3$. Stable limit cycle. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: stable limit cycle. Values of K from top to bottom: $K=15.21$, $K=15.33$, $K=15.45$, and $K=15.93$.

**Figure 10.**Gompertz growth. $r=3$ and $K=20.2$. Stable limit cycle. Plot on the

**left**: evolution of the structure of the consumer; plot in the

**centre**: evolution of the maximum size, resource, and total consumer population to the equilibrium; plot on the

**right**: stable limit cycle.

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Abia, L.M.; Angulo, Ó.; López-Marcos, J.C.; López-Marcos, M.Á.
Computational Study on the Dynamics of a Consumer-Resource Model: The Influence of the Growth Law in the Resource. *Mathematics* **2021**, *9*, 2746.
https://doi.org/10.3390/math9212746

**AMA Style**

Abia LM, Angulo Ó, López-Marcos JC, López-Marcos MÁ.
Computational Study on the Dynamics of a Consumer-Resource Model: The Influence of the Growth Law in the Resource. *Mathematics*. 2021; 9(21):2746.
https://doi.org/10.3390/math9212746

**Chicago/Turabian Style**

Abia, Luis M., Óscar Angulo, Juan Carlos López-Marcos, and Miguel Ángel López-Marcos.
2021. "Computational Study on the Dynamics of a Consumer-Resource Model: The Influence of the Growth Law in the Resource" *Mathematics* 9, no. 21: 2746.
https://doi.org/10.3390/math9212746