Model: Turtles and Grass

Turtles and Grass models a system composed of a population of turtles and a population of grass; within this model turtles (represented by multicolored arrows) move about their environment looking for grass to eat (grass is represented by green squares which turns black when eaten by a turtle). Each unit of grass endows a turtle with a number of energy units, once an energy threshold is reached turtles reproduce. In this model there six parameters that control the behavior of the system, some are held constant and others can be varied to influence system behavior and produce emergent patterns. For each simulation run the duration is 250 time steps.

Constant Parameters
Energy expended by turtles to forage: 1one time step results in a loss of 1 unit of energy in its quest for greener pastures.
Grass Regrowth: If a patch of grass is eaten by a turtle, the grass has a 3% chance of regrowth in each subsequent time step.
Turtle Movement: Each time step a turtle randomly selects a movement angle (from 0 to 360 degrees) and moves that direction one patch.

Adjustable Parameters
Initial Population Number (IPN) of turtles in the system: In this set of simulation runs the number is held constant at 50.
Energy From Grass (EFG): The energy gained by a turtle from eating one patch of grass.
Birth Energy (BE): = 50energy gained from eating the grass, and the birth energy parameter that triggers reproduction and attributes a level of energy to new turtle when it “hatches”.

Q1: How does the “energy-from-grass” parameter influence the emergent patterns observed in the model simulation?

The Energy From Grass (EFG) parameter controls the amount of energy a turtle gains from by eating the grass. Adjusting the EFG (while holding other parameters constant: Initial Population Number = 50; Birth Energy = 50) results in three modes of system behavior.
1. EFG ＜2
When the energy from grass is very low the turtle population cannot gain enough energy to maintain foraging behavior nor reproduce. With this control, turtles go extinct in this environment almost immediately.

Video and Graph for EFG = 1:

 

2. EFG ≥2 & <6
Here the EFG is enough to maintain a stable population of turtles. In this mode the grass population is either higher than the turtle population (low EFG values), or when the EFG nears 6 the turtle population begins to equal the grass population.

3. EFG ≥7
Once the EFG = 7 a threshold is crossed and the system exhibits a pattern that is essentially constant for the remaining range of EFG values, with the pattern becoming pronounced as the EFG value increases. In this pattern, when the simulation begins, the turtle population begins to increase almost immediately while the grass population begins to decrease precipitously. Within 10 time steps the population trend lines cross and the grass population declines to a stable but low population compared to the turtles. The turtle population, increasing rapidly, peaks and within 40 – 50 time steps, begins to drop until the population becomes stable for the remainder of the simulation. As the EFG nears 100, the stable population of turtles can exceed 3000 individuals (peak > 4000), while the grass population will typically be less 60 individuals and intermittently dropping to 1 or even 0 before regrowth.

Video and Graph for EFG = 7:

 

Video and Graph for EFG = 66:

 

Q2: How does the “birth-energy” parameter influence the emergent patterns observed in the model simulation?

The Birth Energy (BE) parameter is both a threshold value for triggering reproduction and the initial value of energy that a turtle “owns” when it “hatches”. In this set of model runs the population curves observed in the condition EFG ≥7 (explained above) are typical and observed throughout the range of BE (with the exception of BE = 0). However, altering the BE parameter changes the shape of the characteristic curves. The changes seen include: shifts in the number of individuals in the system; the timing of peak population; and the stable population value. The other adjustable variables are held constant (IPN = 50; EFG =50).

BE = 0 – 1
BE of 0 results in a quickly stabilizing pattern with ~60 turtles and ~500 patches of grass. Adding just one level of energy changes the dynamics of the system completely, the grass population falls quickly to a steady value of ~50, where it remains for the remainder of the simulation runs. The turtle population also quickly becomes stable with a population ~1600.

 

BE ≥ 2
Once the BE parameter reaches 2 the characteristic curve begins to become evident, the characteristic population peak and drop are observed followed by a stabilization of the turtle population to ~1600 individuals.

 

BE 3 – 65
Within this range the most dramatic peak values are achieved. Lower values allow the turtle population peak to be reached quite quickly (~15 time steps) with the number of individuals exceeding 5000 before stabilizing. This peak and drop characteristic is due to the low reproduction threshold resulting in a very high birthrate before the grass population stabilizes ~ 50.

 

BE ≥ 65
At BE values above 65 the peak seen earlier is essentially gone; the turtle population climbs to its maximum slower and enters a steady state with a population of ~1600 individuals. BE values above 65 produce population curves very similar to the BE = 1. In each of the two scenarios the population graph curves bear a resemblance, but my hunch is that there are quite different reproduction and death behaviors taking place. With BE = 1, I assume that birth and death rates are quite high, while in the high BE ≥ 65 these rates are much lower and the turtles are living longer and reproducing less. Yet, the aggregate behavior of the population looks similar when graphed.

 

Q3: Is the emergent pattern sensitive to the initial population size?

The Initial Population Number (IPN) has no noticeable effect of the emergent properties of the system. Adjusting the IPN throughout the range of possible values produces no differences in the population graph curves. The graphs for IPN = 1 and IPN = 100 (shown below) have population curves that are nearly identical. The population values seen in the counters, too, corroborate the behavioral similarity of the system respective to the IPN.