2 Consumer, 2 Resurce Model

his model has 2 consumers (e.g. different species of herbivores) in exploitation competition for two resources (e.g. plant species). This model is similar to the 2 consumer, 1 resource model on a different page at this site, with the exception of adding a 2nd resource and only having linear (Type I) functional responses. This model has been studied by MacArthur and Tilman among others. Tilman studied both unsubstitutable resources (e.g. Nitrogen vs Phosphorous) which result in isoclines that make right-angled "L"-shapes. The isoclines of this model (book Figure 7.8) of substitutable resources are better known. T his model is nearly identical but it models substitutable resources (e.g. two plant species) that produces linear isoclines. But the overall dynamics are similar whether the resources are substitutable or unsubstitutable - just the isocline shape changes.

Parameters are similar to the 2 consumer, 1 resource model except the resources have a supply rate, S, instead of a growth rate, r, and carrying capacity, K. And aij gives the attack rate of Consumer j on Resource i. The green dot represents the Supply rate of the two resources (its position relative to the R1 and R2 axes match S1 and S2 respecitively. When the model is running the two green arrows pointing to the left and down are the consumption vectors (indicating relative consumption of R1 vs R2 for each consumer and determined by the aij). The third green arrow points to the supply rate. The two dashed red lines extend back from the consumption vectors.

\begin{gather*} \frac{dR_1}{dt} = R_1 \Big( \left( S_1 - R_1 \right) - a_{11} N_1 - a_{12} N_2 \Big) \\ \frac{dR_2}{dt} = R_2 \Big( \left( S_2 - R_2 \right) - a_{21} N_1 - a_{22} N_2 \Big) \\ \frac{dN_1}{dt} = N_1 \left( f_1 a_{11} R_1 + f_1 a_{21} R_2 - m_1 \right) \\ \frac{dN_2}{dt} = N_2 \left( f_2 a_{12} R_1 + f_2 a_{22} R_2 - m_2 \right) \\ \end{gather*}

Like the Lotka-Volterra competition model, the consumers can coexist or one can go extinct. There are several possible outcomes depending on both the relative positions of the ZNGIs AND the nature of the consumption vectors relative to the supply vectors

  1. If one consumer ZNGI is completely inside the other than the one closer to the origin will win (the other consumer goes extint) because the species closer to the origin has a lower R* for both resources (in these diagrams R1* for each consumer species is where its ZNGI interacts the R1 axis).
  2. If the two ZNGI cross, then the outcome depends on the relation of the consumption vectors and the supply rate (green dot). Key is that coexistence occurs only if each species consumes proportionally more of the resource that most limits it.
    1. If the supply rate lies below one ZNGI but above the other ZNGI the species with the ZNGI above the supply rate will go extinct
    2. If the supply rate (green dot) lies between the two consumption vectors (projected backwards by dashed red lines) when the resources are at where the two ZNGI cross, then the two species reach the equilibrium where the ZNGIs cross and coexistence occurs.
    3. If the supply rate lies too close to one axis for the supply rate to stay between the two red dashed lines at the equilibrium, but the supply rate can lie between the red dashed lines with both species coexisting then the species will coexist on one ZNGI but not where the two cross
    4. If the supply rate is so close to one axes of the other that it cannot lie between the two red dashed lines, then one species goes extinct