← MATH 448

Project Ideas

A handful of threads that came up during lecture and were worth more than a footnote.

A small system with chaos

Lecture 4 · dynamical systems

Simulate the 3D Lorenz system and compare its route to chaos with the discrete logistic model's period-doubling cascade.

From Lecture 4, “Dimension and chaos.”

Replicator dynamics with three strategies

Lecture 7 · evolutionary game theory

Extend the replicator equation to three strategies and use the Jacobian's eigenvalues to determine whether Rock–Paper–Scissors' interior equilibrium attracts, repels, or neither.

From Lecture 7, “One more direction.”

Conservation: which life stage to target

Lecture 7 · structured populations

Implement the paper's elasticity formula, reproduce their Figure 3, and apply it to another published stage-structured model.

Crouse, Crowder & Caswell (1987), Ecology 68(5), 1412–1423. See the Loggerhead Turtle Simulator.

Invasive species: competitive exclusion or coexistence?

Lecture 9 · nonlinear systems

Replace predation with competition in the Lotka–Volterra equations and classify which of the four possible outcomes — exclusion either way, stable coexistence, or founder control — an invasive species colonizing a native one's habitat would produce.

From Lecture 9, the Lotka–Volterra toolkit applied to competition instead of predation.

When does a population reach a tipping point?

Lecture 6 · harvesting and bifurcation

Let carrying capacity decline slowly to represent a warming climate, and find the critical rate of decline beyond which a population can no longer track its shrinking equilibrium and collapses instead of gradually adjusting.

Scheffer et al. (2009), “Early-warning signals for critical transitions,” Nature 461, 53–59.