Archives For predator mathematics

click image to open the model

Predator-prey models are argubly the building blocks of the bio- and ecosystems as biomasses are grown out of their resource masses. Species compete, evolve and disperse simply for the purpose of seeking resources to sustain their struggle for their very existence. Depending on their specific settings of applications, they can take the forms of resource-consumer, plant-herbivore, parasite-host, tumor cells (virus)-immune system, susceptible-infectious interactions, etc. They deal with the general loss-win interactions and hence may have applications outside of ecosystems. When seemingly competitive interactions are carefully examined, they are often in fact some forms of predator-prey interaction in disguise.

Typical predator-prey models consist of 2 populations, where the predator affects the prey (through killing) and viceverse (no prey, no food). These dynamics can be represented with mathematical equations and run through simulation models (mainly System Dynamics).
In this adapted model, the predator does not die if there is not enough prey, they just migrate outside of the system. If there’s a lot of prey, they migrate into the system. This very small adaptation to the dynamics may lead to more structural possible outcomes.

Run the model with me and I’ll show you what I’m talking about!
When you first run the model, you get a transitional behavior, and a permanent, cyclic one, in which both populations have alternating peaks and valleys.

Overshoot and Collapse

Now what happens when you increase the Birth Rate (of prey)? You end with nothing! Why is this? Delays… You see that due to a high birth rate, prey population increases greatly, bringing a huge amount of predators into the system. This increase in predators makes the killing increase over births, reducing greatly the prey population. But by the time the predators realize that their food is depleting and leave the system… it is already too late.

Reduced Delays

Let’s try another thing now, let’s reduce the delay. Start the model over, and set the Migration Delay to the far left (0.01). You’ll immediately see that oscillations cease and equilibrium is found. Now try to increase birth rate, and see what happens. A new equilibrium is found, no collapse!

Play Around

There is another key behavior (the damped oscillator) which you can achieve by slightly reducing the delay. Anyhow, you can try out different scenarios, I’ll explain the controls. We already covered birth rate and migration delay. 
  • Desired Level of Attraction: is the amount of prey the predators see as “comfortable”. More than this, they inmigrate, less than this and they will emigrate.
  • Death Rate: is the natural death rate for prey (unaffected by predators).
  • Hunt Rate: is the rate at which predators “kill” prey.
  • Predator Perception Sensitivity: is an indication of how quickly or slowly predators are wanting to enter or leave the system. It is lightly different to Migration Delay. The first is the intention to migrate, the latter is the actual migration.

Outside Predator-Prey

Predator-prey structures can easily be extrapolated to other areas of application. The concept that delays make for oscillations and can potentially lead to overshoot-and-collapse behaviors has nothing to do with the predator-prey system. We can see the effect of delays, for example, in supply chain behaviors, where small shifts in consumer demand usually lead to huge shifts upstream. 
Whenever there is an action to be made, usually this action is based on some kind of input (which may take the form of gut-feeling, metrics, symptoms, etc… you name it). The thing is that there is a delay between the fact, and the moment in which we receive information for this input. How big or small the delay is depends on numerous factors, but for sure, there is a delay. 
After reading this, I hope that next time you take an action, you stop to think on delays and avoid over-reactions.