Lotka-Volterra Predator-Prey Dynamical Model

The following are the Lotka-Volterra equations which can be used to model populations of predators and prey over time:

\[\begin{aligned} \dot{x} &= \alpha x - \beta x y\\ \dot{y} &= \delta x y - \gamma y \end{aligned}\]

where:

• \(x\) is the population size of prey (e.g. rabbits)
• \(y\) is the population size of predators (e.g. foxes)

the components of the model represent:

• \(\alpha x\): the growth rate of prey
• \(\beta x y\): the death rate of prey
• \(\delta x y\): the growth rate of predators
• \(\gamma y\): the death rate of predators

The assumptions of the model are:

• prey find ample food at all times
• the food supply of predators depends entirely on the prey population
• the rate of change of the population is proportional to its size
• the environment does not change

Simulate the behaviour of this system over time. Start with the following values for the initial states, and the constants:

• \((x_{0},y_{0}) = (0.5,0.5)\)
• \(\alpha=0.1\), \(\beta=0.1\), \(\delta=0.1\), \(\gamma=0.1\)
• set up a time vector from 0 to 500 in steps of 1

1. Plot the population size of predators over time. Overlay a plot of the population size of prey over time.
2. Plot the trajectory of the system in state-space, in other words plot \(x(t)\) on the horizontal axis and \(y(t)\) on the vertical axis.
3. Increase the \(\alpha\) constant to \(0.2\) and re-run the simulation. What happens?
4. Set all constants to 0.20 and re-run the simulation. What happens?
5. Try the following: \((\alpha,\beta,\delta,\gamma) = (0.2,0.2,0.2,0.0)\). What happens and why?