Part 4: Interactions Between Species and Community Structure

This part explores how species interactions shape population and community dynamics, using competition and predator-prey models.

By the end of Part 4, you should be able to:

• Predict the outcome of competition from the isocline intercepts (\(K_1\), \(K_1/\alpha\), \(K_2\), \(K_2/\beta\)).
• Interpret competition and predator-prey dynamics from model outputs.
• Explain why Lotka-Volterra predator-prey cycles are neutrally stable in continuous time, and why the discrete-time version spirals outwards.
• Reason about stability and equilibrium in multi-species systems.

Reading: Neal, chapters Interspecific Competition, Predator-Prey Interactions and Species Interactions and Community Structure.

Notation used in Part 4 (following Neal)

• Competition: \(\frac{dN_1}{dt} = r_1 N_1 \frac{K_1 - N_1 - \alpha N_2}{K_1}\) and \(\frac{dN_2}{dt} = r_2 N_2 \frac{K_2 - N_2 - \beta N_1}{K_2}\). \(\alpha\) is the per capita effect of species 2 on species 1, and \(\beta\) the effect of species 1 on species 2. The Excel worksheet calls these \(\alpha_{12}\) and \(\alpha_{21}\).
• Predator-prey: \(\frac{dH}{dt} = rH - aHP\) and \(\frac{dP}{dt} = cHP - dP\), with isoclines \(P = r/a\) (prey) and \(H = d/c\) (predator). The Excel worksheet and the R chapters use \(V\) (prey), \(C\) (predator), \(R\), \(a\), \(f\) and \(q\), where \(V = H\), \(C = P\), \(R = r\), \(af = c\) and \(q = d\).