Fundamentals of Predator-Prey Relationships: Lecture Notes

Lotka-Volterra Model

  • Developed independently by:
    • Volterra: Italian mathematician, based model on shark catches in Fiume.
    • Lotka: Austrian demographer, ecologist, chemist, and mathematician (moved to America), developed model from a theoretical standpoint.
  • The models were very similar and developed at the same time, thus the Lotka-Volterra model was born.
  • Famously applied to Lynx and Snowshoe Hare dynamics.
  • Data from Hudson Bay Trading Company fur records (1847-1903).

Prey in the Lotka-Volterra Model

  • Prey grow exponentially in the absence of predators (r*N), without density dependence.
  • Predators control prey density.
  • Factors affecting this control:
    • Prey density.
    • Predator density.
    • Predator hunting efficiency.

Predators in the Lotka-Volterra Model

  • Growth rate of the predator population is dependent on the prey.
  • Factors affecting this:
    • Prey density.
    • Predator density.
    • Predator hunting efficiency.
    • Prey conversion efficiency (not every calorie consumed is absorbed).

Predator-Prey Model: Population Growth Rate

  • Prey population growth rate loss due to predator consumption rate.
  • Predator population growth rate is directly related to prey capture.
  • Includes:
    • Prey.
    • Predator.
    • Conversion rate of prey.
    • Hunting efficiency.
    • Predator death rate.

Predator-Prey Model: Mass Action

  • The relationship between variables is called "mass action."
  • Mass action = homogeneous mixing.
  • A very common function in many types of models.

Predator-Prey Model: Dynamics

  • Higher prey conversion leads to less prey and more predators.
  • Timing of the cycles is also changed.
  • Lotka-Volterra style equations lead to cycles.

Dynamics of Snowshoe Hare and Lynx

  • Data taken from the Hudson Bay Trading Company Records.
  • Show cycles of hare and lynx populations over time.

Host-Parasitoid Dynamics

  • Illustrates oscillations in host and parasitoid populations over generations.

Lotka-Volterra Models: What Can They Tell Us?

  • Effects of:
    • Increasing prey density.
    • Increasing predator density.
    • Long/short-lived predators.
    • Efficient/inefficient predators.
    • High nutritive value (high conversion) of prey to new predators.
  • BUT IS IT REALISTIC?

Is the Lynx-Hare Interaction a Lotka-Volterra Relationship?

  • We can make the model fit the data.
  • Lynx populations peak a little after the hares.
  • BUT snowshoe hares cycle on islands without lynx present.
  • What else could influence prey populations?
    • Density dependence (general).
    • Food availability.
    • Pathogens.

Examining the Real Impact of Predation (and Food) in the Snowshoe Hares

  • Krebs et al (1995) Impact of Food and predation on the snowshoe hare cycle. Science 269 (5227): 1112-1115
  • What they did:
    • Added food to sites.
    • Excluded predators.
    • Excluded predator & added food.
    • Added fertilizer.
    • Left things alone (control).

Snowshoe Hare Investigation: Findings

  • Krebs et al. (1995) Science 269 (5227): 1112-1115
  • Giving food increased hares x 3.
  • Excluding predators increased hares X 2.
  • Excluding predator and increasing food increased hares x 11.
  • Effects are more than additive (multiplicative?).
  • Complex interaction between predation and food.

Ground Squirrel - Food / Predation Study

  • Karels et al. (2000) J of Anim Ecol 59 (2): 235-47
  • Demonstrates the impact of predator exclosure and food supplementation on ground squirrel population density.

How Good Is the Lotka-Volterra Model?

  • Lotka-Volterra does give one explanation for predator-prey cycles.
  • Matches the snowshoe hare – lynx data BUT there are obviously other things going on.
  • Not all predator-prey relationships show cycles.
  • Some show more stable dynamics.
  • The model doesn’t take into account influences on the prey other than predation
    • Prey density dependence
    • Pathogen influences
    • Prey competition with other species
  • Doesn’t (in simplest form) consider predator handling efficiency of the prey.

Adding Prey Density Dependence to the Lotka-Volterra Model

\frac{dN}{dt} = r \cdot N (1 - \frac{N}{K}) - a \cdot N \cdot P

  • Leads to damped oscillations reaching a steady state.
  • Incorporates carrying capacity (K) for the prey.

Predator – Handling Efficiency (Holling Type I)

  • Work carried out by Rigler (1961).
  • Linear increase in feeding with prey density until a maximum.
  • Maximum is a limit e.g. daphnia can only swallow at a certain rate.

Predator – Handling Efficiency (Holling Type II)

  • An asymptotic relationship.
  • Increased consumption as density increase.
  • BUT the rate of consumption slows as prey increases.
  • Predators take a certain amount of time to HANDLE prey.
  • At first, the availability of prey is the over-riding time constraint – got to catch them to eat them.
  • BUT when they’re easy to catch, then the time to process them becomes the more important constraint.

Predator – Handling Efficiency (Holling Type III)

  • Sigmoidal relationship between host/prey density and consumption or attack rate.
  • At low prey density, the host has to increase its search effort.
  • This means that section A of the curve approximates exponential increase.
  • As prey/host density increases, the Holling type II response results.

Holling Feeding Responses

  • Illustrates the three types of functional responses: Type I (linear), Type II (asymptotic), and Type III (sigmoidal).

Holling’s Functional Responses Type II

  • Number attacked vs. Host Density.
  • % Mortality: Inverse density-dependent.
  • Th low, Th high (T_h = handling time).

Holling’s Functional Responses Type III

  • Number attacked vs. Host Density.
  • % Mortality: Density-dependent at low density, inverse density-dependent at high density.

Incorporating Holling Type II

  • Wilkinson M.H.F. (2006) Mathematical Modelling of Predatory Prokaryotes. In: Jurkevitch E. (eds) Predatory Prokaryotes. Microbiology Monographs, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/7171_054
  • h = handling time.
  • Damped oscillations reaching an equilibrium.
  • LIMIT cycles.

Summary

  • Predators and prey often shown linked cycles.
  • These linked cycles can be modeled using Lotka-Volterra predator-prey models.
  • Many natural systems do fit the model (or its extensions).
  • But other influences, such as food supply, can interact with and alter the relationship between the species.
  • The model can describe non-cycling populations if density-dependence is included in the prey.
  • Further modifications to alter the prey handling times allows both cyclic and stable dynamics under different conditions.
  • Prey handling efficiency and form varies across species.