Consumer Resource, 1 Resource

2 Consumer - 1 Resource

Abundance

Time

Click isocline figure to set initial abundance and start dynamics.

Current abundances: R: 0.1 || N₁: 0.1 || N₂: 0.1

Use sliders to change model parameter values.

R parameters

r: 0.2

K: 10

N₁ parameters

a₁: 0.4

f₁: 0.1

h₁: 0

m₁: 0.1

N₂ parameters

a₂: 0.2

f₂: 0.3

h₂: 0

m₂: 0.05

This is a mechanism-specific model of competition (compare to the phenomenological model of Lotka-Volterra competion). It models exploitation competition where two species are competing by both eating one resource. It matches book figures 7.4-7.5 in the book (also Figure 7.3 for empirical examples of population growth rates as a function of resources). R is the population size of the resource (e.g. a plant or phytoplankton). N₁ and N₂ are the populations of two species of consumers (e.g. herbivores). Equally the two consumers could be carnivores and the one resource could be an herbivore. The parameters are:

r - population growth rate of the resource
K - carrying capacity for the resource (which grows logistically when not consumed)
a₁, a₂ - the attack rate (consumption efficiency) of each consumer species on the resource
f₁, f₂ - the conversion efficiency of resource individuals eaten into consumer individuals for each consumer species
h₁, h₂ - the handling time of each consumer species for the resource. When this is zero there is a Type I (linear) functional response. When it is >0, there is a Type II (saturating) functional response.
m₁, m₂ - the density independent mortality rate of each consumer species on the resource

2 Consumer - 1 Resource Model

\begin{gather*} \frac{dN_1}{dt} = f_1 \frac{a_1 R}{1+ h_1 a_1 R} N_1 - m_1 N_1 \\ \frac{dN_2}{dt} = f_2 \frac{a_2 R}{1+ h_2 a_2 R} N_2 - m_2 N_2 \\ \frac{dR}{dt} = r R \left( \frac{K-R}{K} \right) - \frac{a_1 R}{1 + h_1 a_1 R} N_1 - \frac{a_2 R}{1 + h_2 a_2 R} N_2 \end{gather*}

Unlike other figures on this website, the left graph is NOT a phase-plane (each state variable appears on a different axis). It is a plot of per-capita growth rate vs resource level (again compare with Figures 7.3-7.5). One consumer species is in red, one is in blue. Note that in Figures 7.4-7.5 the reproduction and mortality portions of the consumer growth rates were plotted separately giving an equilibrium where they cross. Here the reproduction and mortality (which is negative) are summed together. So they key point is where each consumer species growth rate crosses the zero line (dashed line in the plot). The resource level where a consumer's growth line crosses zero is the R* for that species.

There are several increasingly more complex versions of this model

The resource dynamics are much faster than the consumer dynamics. This was the original model by MacArthur (1970) and several later authors. It gives nice stability properties in that the model always converges. Whichever species has the lower R* will coexist with the resource. The species with the higher R* will go extinct. More generally only as many consumer species can coexist as there are distinct resources (i.e. one consumer in this model). Such time-scale separated dynamics are not modelled on this page.
Consumers and resource have population dynamics on similar time scales. Both species have linear Type I functional responses (i.e. h₁=h₂=0). This model usually converges to the same equilibrium as the first model (whichever consumer has the higher R* goes extinct)
One species has a Type I functional response while the other has a type II saturating response (one h is 0, the other is >0). This is the model that Armstrong and McGehee (1980) examined. As with predator prey models, introducing a nonlinear functional response makes the dynamics tend not to go to a stable equilibrium point but cycle. This can allow two consumer species to coexist on only one resource.
When both species have Type II functional responses (both h>0), the dynamics can quickly get quite complex.