The Basic Predator-Prey Model
Abundance
Time
Click isocline figure to set initial abundance and start dynamics.
Current abundances: N: 0.1 || P: 0.1
Use sliders to change model parameter values.
N Birth
P Feeding on N
P Death
There are a number of increasingly complex versions of predator prey models. In the book we cover three:
- Volterra model (Book Figure 5.5) - the prey grows exponentially or equivalently is not self limited (i.e. "Exponential growth" box is checked).
This effectively the same as K=∞. Also the predator has no handling time for the prey (h=0). The dynamics are a neutral limit cycle (the
population loops around the equilibrium point not getting closer or further away). Only 4 parameters matter:
- r - the population growth rate of the prey
- a - the attack rate (how fast predators attack prey)
- m - the fraction of predators that die each time step
- f - the conversion effienecy (how many predators are produced by consuming one prey)
- Limited prey (Book Figure 5.7) - the prey grows logistically to a carrying capcity of K (uncheck "Exponential growth" box and set K slider). Keeping handling time h=0. This model shows convergence (the dynamics circle around and converges towards the equilibrium)
- Rosenzweig-MacArthur model (Book Figure 5.8) - keep logistically growing prey (0<K<∞) ("Exponential growth" box unchecked). Handling time is now non-zero (use slider to make h>0). This model can show convergence or divergence (dynamics circle around the equilibrium in increasingly large cycles until it comes close to an axis).
Model #1 - Volterra
\begin{gather*} \frac{dN}{dt} = r N - a N P \\ \frac{dP}{dt} = f a N P - m P \end{gather*}Model #2 - limited prey
\begin{gather*} \frac{dN}{dt} = r N \left( \frac{K - N}{K} \right) - a N P \\ \frac{dP}{dt} = f a N P - m P \end{gather*}Model #3 - Rosenzweig-MacArthur
\begin{gather*} \frac{dN}{dt} = r N \left( \frac{K - N}{K} \right) - \frac{a N P}{1 + a h N} \\ \frac{dP}{dt} = f \frac{a N P}{1 + a h N} - m P \end{gather*}