Lotka-Volterra Competition
Abundance
Time
Click isocline figure to set initial abundance and start dynamics.
Current abundances: N1: 0.1 || N2: 0.1
Use sliders to change model parameter values.
N1 Parameters
N2 Parameters
The Lotka-Volterra competition models(Book Figure 7.2) model competition between two species. There are six parameters:
- r1 - population growth rate of species 1
- r2 - population growth rate of species 2
- K1 - carrying capacity for species 1
- K2 - carrying capacity for species 2
- α12 - competitive effect of species 2 on species 1 population growth rate
- α21 - competitive effect of species 1 on species 2 population growth rate
Four outcomes can occur:
- Both species coexist at the equilibrium (where the isoclines cross) no matter where the dynamics start (if isoclines cross and N2 can invade when rare - i.e. N2 isocline in dashed blue is outside the red N1 isocline on the N1 axis where N2 is rare and vice versa)
- One species wins and the other goes extinct (the equilibrium where the two isoclines cross is unstable). The winner depends on where the dynamics start (Occurs when the isoclines cross and N2 cannot invade when rare - i.e. blue dashed N2 isocline is inside the N1 isocline along the N1 axis where N2 is rare and vice versa)
- Species 1 wins and species 2 goes extinct (if species 1 isocline lies entirely above species 2 iscoline)
- Species 2 wins and species 1 goes extinct (if species 2 isocline lies entirely above species 1 isocline)
\begin{gather*}
\frac{dN_1}{dt} = r_1 N_1 \left( \frac{K_1 - N_1 - \alpha_{12} N_2}{K_1} \right) \\
\frac{dN_2}{dt} = r_2 N_2 \left( \frac{K_2 - N_2 - \alpha_{21} N_1}{K_2} \right)
\end{gather*}