Lotka–Volterra Predator–Prey Dynamics: Assumptions, Equilibria, and Cycles

All flashcards

What are the standard Lotka–Volterra predator–prey equations?

$\frac{dx}{dt}=\alpha x-\beta xy$ and $\frac{dy}{dt}=\delta xy-\gamma y$.

What do $x$ and $y$ represent in the Lotka–Volterra model?

$x$ is prey population density; $y$ is predator population density.

What does $\beta$ represent in the prey equation?

The strength of prey losses associated with predator–prey encounters.

What does $\delta$ represent in the predator equation?

The effect of prey encounters on predator population growth.

What does it mean that the Lotka–Volterra model is deterministic?

Given its parameters and initial populations, the model determines the trajectory.

Quiz questions

1. For positive parameters, what are the prey and predator densities at the Lotka–Volterra coexistence equilibrium?Answer: $x^*=\gamma/\delta$ and $y^*=\alpha/\beta$
2. How is the extinction equilibrium $(0,0)$ classified in the standard Lotka–Volterra model?Answer: A saddle point, stable in one direction and unstable in another
3. What do positive-population trajectories do around the coexistence equilibrium in the standard Lotka–Volterra model?Answer: They follow closed periodic orbits around the equilibrium.
4. What happens to the conserved quantity $V(x,y)$ along a positive-population Lotka–Volterra trajectory?Answer: It remains constant, with its value set by the initial populations.
5. In the standard model, what is the equilibrium effect of increasing prey growth rate $\alpha$?Answer: Predator equilibrium density rises, while prey equilibrium density is unchanged.