04 – The resource-consumer competition model

In this session, we discussed about the resource-consumer model.

Materials

The video record of the seminar can be found HERE.

The Mathematica notebook is HERE.

Model

The two-species resource-consumer competition model is defined as

$$\begin{array}{rl}\frac{d{N}_1}{dt}& =({\mu }_1-m)N_1\\ \frac{d{N}_2}{dt}& =({\mu }_2-m)N_2\\ \frac{dR}{dt}& =a(R_{in}-R)-V_1{N}_1-V_2{N}_2\end{array}$$

where $N_1$, $N_2$ are the population sizes of species 1 and 2, respectively. ${\mu }_1$ and ${\mu }_2$ are the growth rates while $V_1$ and $V_2$ are the consumption rates. $m$ is the background mortality rate. For the dynamics of the resource, $R_{in}$ denotes for the inflow concentration and $a$ is the dilution rate.

The growth rate often follows the functional form of Mechanis-Menten-Monod:

$$\begin{array}{r}\mu :={\mu }_{max}\frac{R}{R+K}\end{array}$$

where ${\mu }_{max}$ is the maximum growth rate when the resource is adequate. $K$ is the half-saturation rate. Therefore, the growth rate when the resoruce is at low density is given by the slope of this function at low $R$

$$\frac{{\mu }_{max}}{K}$$

Given the nonlinearity of the growth rate, the curves may cross. So, imagine a situation where Species 1 has a larger growth rate when the resource is sufficient while Species 2 has a higher growth rate when the resource is scarce,

$${\mu }_{1,max}>{\mu }_{2,max};\frac{{\mu }_{1,max}}{K_1}<\frac{{\mu }_{2,max}}{K_2}$$

which species will win?

The simulation

Try on this interactive plot and see which species would win.

The zero-net-growth-isocline (ZNGI) analysis of a two-species-two-resource model

The ZNGI analysis we talked in the previous lecture on the Lotka-Volterra model can also infer the outcome of the competition on the resource-consumer model. For simplicity, we just assume that the growth rate is linear.

The following figure compares the ZNGIs analysis of the essential and substitutable resource models (Vincent et al. 1996).

ZNGIs of essential and substitutable models

The essential resource model

The two-species-two-essential-resource model is given by

$$\begin{array}{rl}\frac{d{N}_1}{dt}& =(min({\mu }_{11}{R}_1,{\mu }_{12}{R}_2)-m)N_1\\ \frac{d{N}_2}{dt}& =(min({\mu }_{21}{R}_1,{\mu }_{22}{R}_2)-m)N_2\\ \frac{d{R}_1}{dt}& =a(R_{1,in}-R_1)-min(V_{11}{R}_1,V_{12}{R}_2)N_1-min(V_{21}{R}_1,V_{22}{R}_2)N_2\\ \frac{d{R}_2}{dt}& =a(R_{2,in}-R_2)-min(V_{11}{R}_1,V_{12}{R}_2)N_1-min(V_{21}{R}_1,V_{22}{R}_2)N_2\end{array}$$

The substitutable resource model

The two-species-two-substitutable-resource model is given by

$$\begin{array}{rl}\frac{d{N}_1}{dt}& =({\mu }_{11}{R}_1+{\mu }_{12}{R}_2-m)N_1\\ \frac{d{N}_2}{dt}& =({\mu }_{21}{R}_1+{\mu }_{22}{R}_2-m)N_2\\ \frac{d{R}_1}{dt}& =a(R_{1,in}-R_1)-V_{11}{R}_1{N}_1-V_{21}{R}_1{N}_2\\ \frac{d{R}_2}{dt}& =a(R_{2,in}-R_2)-V_{12}{R}_2{N}_1-V_{22}{R}_2{N}_2\end{array}$$

The ZNGI analysis

We take the essential resource competition model as an example. The ZNGIs of the two species are given by

$$\begin{array}{rl}min({\mu }_{11}{R}_1,{\mu }_{12}{R}_2)-m& =0\\ min({\mu }_{21}{R}_1,{\mu }_{22}{R}_2)-m& =0\end{array}$$

which gives the two “L” shaped ZNGIs in the plot.

If the two ZNGIs cross, the crossing point is a potential coexistence destination. Whether it is a stable equilibrium relies on the impact vectors and the resource supply rates. See Lecture 2 for details.

For the substitutable resource model, you can play with the model by switching the tab to “Substitutable”.

References

Vincent, T. L. S., D. Scheel, J. S. Brown, and T. L. Vincent. “Trade-Offs and Coexistence in Consumer-Resource Models: It All Depends on What and Where You Eat.” The American Naturalist 148, no. 6 (1996): 1038–58. http://www.jstor.org/stable/2463561.

Chase, Jonathan M., and Mathew A. Leibold. Ecological Niches: Linking Classical and Contemporary Approaches. University of Chicago Press, 21 Feb 2013, 2003. doi:10.7208/chicago/9780226101811.001.0001. https://doi.org/10.7208/chicago/9780226101811.001.0001.