Classwork 2

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Directions: Add work to this notebook to solve the problems below.

1. Lotka–Volterra predator–prey model

The Lotka–Volterra predator–prey model is the differential equation: \begin{align*} x'(t) = \alpha x(t) - \beta x(t) y(t), \\ y'(t) = -\gamma y(t) + \delta x(t) y(t). \\ \end{align*} Here $x(t)$ represents the population density of prey over time, and $y(t)$ represents the population density of predators over time. The parameters:

• $\alpha$ represents the natural growth rate of the population of prey (absent predation)
• $\beta$ represents the effect that the presence of predators has on the population of prey.
• $\gamma$ represents the predators natural death rate.
• $\delta$ represents the growth rate of the population of predators, which depends on how much prey is available.

Consider the case when $\alpha=2/3$, $\beta=\frac{4}{3}$, $\gamma=1$, and $\delta=1$ with initial conditions $x(0)=2$ and $y(0)=2$.

Create plots of the solution of this differential equation. Plot $x(t)$, $y(t)$, as well as the parameterized curve $t \mapsto \big(x(t), y(t)\big).$