Tutorial: Predator-prey models

We will work with two versions of a simple predator-prey model based on Lotka-Volterra^[1][2][3] and foraging arena assumptions^[4]. The models can be developed using the equations in the previous section, but to simplify it, we have made R-code for the two versions of the predator-prey model. You can download this code from this link, and, e.g., use it in R-Studio.

In these models (both included in the R file), we have for clarity here separated the components of the predator prey models, e.g., so that the addition terms for prey and predators are called births, and the subtraction terms are called deaths.

Lotka-Volterra models have stable equilibrium points for the predator and prey, which can be calculated as follows,

• For the prey: Eq.prey = predator death rate / (predator search rate * predator growth efficiency)
• For the predator: Eq.pred = prey birth rate / predator search rate
• Eq.prey and Eq.pred are calculated in the R code, check the Environment > Values window once you have the code running.

We suggest you try the following,

• Run the code
• Does the Lotka-Volterra model (“mass action” in code) and the foraging arena model give the same results?
• Look through the code, find out what it is doing
• Note the differences between the two model formulations, including the settings for the vulnerability (vone, vtwo)
• Try different levels of carrying capacity (CC), including setting CC to 0
  • What does the code do when CC==0?
• What is the effect of setting handling time >0?
  • Check the code for handling time (ht). What impact does handling time have on the number of prey eaten?
• Try changing all of the input parameters, and reflect on the impact and why
  • Do this one by one, so you can go back to the original state
• The foraging arena model can be invoked by changing the vulnerability (vone) setting to a low value (≥1), e.g., 2
  • High settings for vone implies “top-down” control, and will make the model behave like a Lotka-Volterra mass action model. Low settings (≥1) implies “bottom-up” control. Much more about this in the Ecosim parts of this textbook.
• Change input parameters as above to see the difference between the mass action and the foraging arena model formulations.
• Paradox of Enrichment: Rosenzweig described the Paradox of Enrichment where increased prey productivity may cause instability for predators. This is one of the unexplained issues with mass-action models. Simulate this in the predator-prey model by increasing prey productivity (birth rate, br) along with carrying capacity (K). Compare the results in the mass-action and foraging arena formulations of the predator prey model.
• Play!