Models of interspecific interactions

Predator-prey dynamics and competition

Introduction

Lotka and Volterra developed models for both predator-prey dynamics and competitive interactions.

As usual, these models were developed as continuous-time models.

We will focus on discrete-time versions (\(t = 1, 2, \ldots\)).

We will ignore potential extensions with stochasticity, age structure, spatial structure, etc…

Predator-Prey dynamics

How should predator-prey dynamics operate?

Lynx-hare cycles

Lotka-Volterra predator-prey model

Model for prey \[ N^{prey}_{t+1} = N^{prey}_t + N^{prey}_t (r^{prey} - k^{pred} N^{pred}_t) \]

Model for predator \[ N^{pred}_{t+1} = N^{pred}_t + N^{pred}_t (b^{pred}N^{prey}_t - d^{pred}) \]

• Model is based on geometric growth
• \(r^{prey}\): growth rate of the prey in the absence of predators
• \(k^{pred}\): kill rate
• \(b^{pred}\): effect of prey on predator birth rate
• \(d^{pred}\): predator mortality rate

Equilibrium

Equilibrium for prey occurs when: \[ N^{pred} = \frac{r^{prey}}{k^{pred}} \]

Equilibrium for predators occurs when \[ N^{prey} = \frac{d^{pred}}{b^{pred}} \]

However, it is rare that both equilibrium conditions will be met at the same time, and so the populations will cycle.

Model predicts population cycles

Isle Royale wolves and moose

Isle Royale Website

Huffaker’s Lab Experiments

Competition

Competition

Lotka-Volterra competition model

Model for species A \[ N^A_{t+1} = N^A_t + r^A N^A_t(K^A - N^A_t - \alpha^B N^B_t) / K^A \]

Model for species B \[ N^B_{t+1} = N^B_t + r^B N^B_t(K^B - N^B_t - \alpha^A N^A_t) / K^B \]

• Model based on logistic growth
• The \(\alpha\) parameters are competition coefficients determining how strongly each species affects the other

Equilibrium

Equilibrium for species A \[ N^A = \frac{K^A - \alpha^B K^B}{1 - \alpha^A \alpha^B} \]

Equilibrium for species B \[ N^B = \frac{K^B - \alpha^A K^A}{1 - \alpha^A \alpha^B} \]

Outcomes

Three posssible outcomes:

1. Stable coexistence
2. Competitive exclusion
3. Unstable equilibrium

Competitive exclusion principle Two species with the same niche cannot coexist on the same limiting resource.

Outcomes

Don’t forget about intraspecific competition

Summary

Predator-prey model is extension of geometric growth

• Predators and prey limit each other’s growth potential

Competition model is extension of logistic growth

• Competitors influence each other’s density-dependent regulation process

These models could be extended to include:

• More species
• Stochasticity
• Age structure

• Harvest
• Spatial structure
• Additional forms of density dependence