# 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: ${f}_{S}(t,s)=r\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">K-s$, $r>0$, $K>0$.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

- Angulo, O.; López-Marcos, J.C.; López-Marcos, M.A. Analysis of an efficient integrator for a size-structured population model with a dynamical resource. Comput. Math. Appl.
**2014**, 68, 941–961. [Google Scholar] [CrossRef] [Green Version] - Abia, L.M.; Angulo, O.; López-Marcos, J.C.; López-Marcos, M.A. Long-Time Simulation of a Size-Structured Population Model with a Dynamical Resource. Math. Model Nat. Phenom.
**2010**, 5, 1–21. [Google Scholar] [CrossRef] [Green Version] - Angulo, O.; López-Marcos, J.C.; López-Marcos, M.A. Numerical approximation of singular asymptotic states for a size-structured population model with a dynamical resource. Math. Comput. Model.
**2011**, 54, 1693–1698. [Google Scholar] [CrossRef] - de Roos, A.M.; Persson, L. Size-dependent life-history traits promote catastrophic collapses of top predators. Proc. Nat. Acad. Sci. USA
**2002**, 99, 12907–12912. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sánchez-Sanz, J.; Getto, P. Numerical bifurcation analysis of physiologically structured populations: Consumer–resource, cannibalistic and trophic models. Bull Math. Biol.
**2016**, 78, 1546–1584. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cuesta, J.A.; Delius, G.W.; Law, R. Sheldon spectrum and the plankton paradox: Two sides of the same coin—A trait-based plankton size-spectrum model. J. Math. Biol.
**2017**, 76, 67–96. [Google Scholar] [CrossRef] [PubMed] - Pang, J.; Chen, J.; Liu, Z.; Bi, P.; Ruan, S. Local and global stabilities of a viral dynamics model with infection-age and immune response. J. Dyn. Diff. Equat.
**2019**, 31, 793–813. [Google Scholar] [CrossRef] - Lafferty, K.; De Leo, G.; Briggs, C.; Dobson, A.; Gross, T.; Kuris, A. A general consumer-resource population model. Science
**2015**, 349, 854–857. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aylaj, B.; Noussair, A. Global weak solution for a multistage physiologically structured population model with resource interaction. Nonlinear Anal. Real. World Appl.
**2010**, 11, 1670–1684. [Google Scholar] [CrossRef] - Angulo, O.; Crauste, F.; López-Marcos, J.C. Numerical integration of an erythropoiesis model with explicit growth factor dynamics. J. Comput. Appl. Math.
**2018**, 330, 770–782. [Google Scholar] [CrossRef] [Green Version] - Kooijman, S.A.L.M.; Metz, J.A.J. On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals. Ecotoxicol. Environ. Saf.
**1984**, 8, 254–274. [Google Scholar] [CrossRef] - Thieme, H.R. Well-posedness of physiologically structured population models for daphnia magna. J. Math. Biol.
**1988**, 26, 299–317. [Google Scholar] [CrossRef] [Green Version] - Diekmann, O.; Getto, P.; Gyllenberg, M. Stability and bifurcation analysis of volterra functional equations in the light of suns and stars. SIAM J. Math. Anal.
**2007**, 39, 1023–1069. [Google Scholar] [CrossRef] - Diekmann, O.; Gyllenberg, M. Equations with infinite delay: Blending the abstract and the concrete. J. Differential. Equat.
**2012**, 252, 819–851. [Google Scholar] [CrossRef] [Green Version] - Diekmann, O.; Gyllenberg, M.; Metz, J.A.J.; Nakaoka, S.; de Roos, A.M. Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example. J. Math. Biol.
**2010**, 61, 277–318. [Google Scholar] [CrossRef] [PubMed] - de Roos, A.M.; Metz, J.A.J.; Evers, E.; Leipoldt, A. A size dependent predator-prey interaction: Who pursues whom? J. Math. Biol.
**1990**, 28, 609–643. [Google Scholar] [CrossRef] - Breda, D.; Diekmann, O.; Gyllenberg, M.; Scarabel, F.; Vermiglio, R. Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis. SIAM J. Appl. Dyn. Syst.
**2016**, 15, 1–23. [Google Scholar] [CrossRef] - Breda, D.; Getto, P.; Sánchez-Sanz, J.; Vermiglio, R. Computing the eigenvalues of realistic Daphnia models by pseudospectral methods. SIAM J. Sci. Comput.
**2015**, 37, A2607–A2629. [Google Scholar] [CrossRef] [Green Version] - de Roos, A.M. Numerical methods for structured population models: The escalator boxcar train. Numer. Methods Partial Differ. Equat.
**1988**, 4, 173–195. [Google Scholar] [CrossRef] - Banks, R.B. Growth and Diffusion Phenomena; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]

**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